Turán–Kubilius inequality

US Army bombers flying over near-periodic swell in shallow water, close to the Panama coast (1933). The sharp crests and very flat troughs are characteristic for cnoidal waves.

In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.

The cnoidal wave solutions were derived by Korteweg and de Vries, in their 1895 paper in which they also propose their dispersive long-wave equation, now known as the Korteweg–de Vries equation. In the limit of infinite wavelength, the cnoidal wave becomes a solitary wave.

The Benjamin–Bona–Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg–de Vries equation, and is another uni-directional wave equation with cnoidal wave solutions. Further, since the Korteweg–de Vries equation is an approximation to the Boussinesq equations for the case of one-way wave propagation, cnoidal waves are approximate solutions to the Boussinesq equations.

Cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics.

A cnoidal wave, characterised by sharper crests and flatter troughs than in a sine wave. For the shown case, the elliptic parameter is m = 0.9.

Crossing swells, consisting of near-cnoidal wave trains. Photo taken from Phares des Baleines (Whale Lighthouse) at the western point of Île de Ré (Isle of Rhé), France, in the Atlantic Ocean.

Background

Korteweg–de Vries, and Benjamin–Bona–Mahony equations

Validity of several theories for periodic water waves, according to Le Méhauté (1976). The light-blue area gives the range of validity of cnoidal wave theory; light-yellow for Airy wave theory; and the dashed blue lines demarcate between the required order in Stokes' wave theory. The light-gray shading gives the range extension by numerical approximations using fifth-order stream-function theory, for high waves (H > ¼ H_breaking).

The Korteweg–de Vries equation (KdV equation) can be used to describe the uni-directional propagation of weakly nonlinear and long waves—where long wave means: having long wavelengths as compared with the mean water depth—of surface gravity waves on a fluid layer. The KdV equation is a dispersive wave equation, including both frequency dispersion and amplitude dispersion effects. In its classical use, the KdV equation is applicable for wavelengths λ in excess of about five times the average water depth h, so for λ > 5 h; and for the period τ greater than $7\sqrt{h/g}$ with g the strength of the gravitational acceleration. To envisage the position of the KdV equation within the scope of classical wave approximations, it distinguishes itself in the following ways:

• Korteweg–de Vries equation — describes the forward propagation of weakly nonlinear and dispersive waves, for long waves with λ > 7 h.
• Shallow water equations — are also nonlinear and do have amplitude dispersion, but no frequency dispersion; they are valid for very long waves, λ > 20 h.
• Boussinesq equations — have the same range of validity as the KdV equation (in their classical form), but allow for wave propagation in arbitrary directions, so not only forward-propagating waves. The drawback is that the Boussinesq equations are often more difficult to solve than the KdV equation; and in many applications wave reflections are small and may be neglected.
• Airy wave theory — has full frequency dispersion, so valid for arbitrary depth and wavelength, but is a linear theory without amplitude dispersion, limited to low-amplitude waves.
• Stokes' wave theory — a perturbation-series approach to the description of weakly nonlinear and dispersive waves, especially successful in deeper water for relative short wavelengths, as compared to the water depth. However, for long waves the Boussinesq approach—as also applied in the KdV equation—is often preferred. This is because in shallow water the Stokes' perturbation series needs many terms before convergence towards the solution, due to the peaked crests and long flat troughs of the nonlinear waves. While the KdV or Boussinesq models give good approximations for these long nonlinear waves.

The KdV equation can be derived from the Boussinesq equations, but additional assumptions are needed to be able to split off the forward wave propagation. For practical applications, the Benjamin–Bona–Mahony equation (BBM equation) is preferable over the KdV equation, a forward-propagating model similar to KdV but with much better frequency-dispersion behaviour at shorter wavelengths. Further improvements in short-wave performance can be obtained by starting to derive a one-way wave equation from a modern improved Boussinesq model, valid for even shorter wavelengths.

Cnoidal waves

Cnoidal wave profiles for three values of the elliptic parameter m.

blue	: m = 0,
red	: m = 0.9 and
black	: m = 0.99999.

The cnoidal wave solutions of the KdV equation were presented by Korteweg and de Vries in their 1895 paper, which article is based on the PhD thesis by de Vries in 1894. Solitary wave solutions for nonlinear and dispersive long waves had been found earlier by Boussinesq in 1872, and Rayleigh in 1876. The search for these solutions was triggered by the observations of this solitary wave (or "wave of translation") by Russell, both in nature and in laboratory experiments. Cnoidal wave solutions of the KdV equation are stable with respect to small perturbations.

The surface elevation η(x,t), as a function of horizontal position x and time t, for a cnoidal wave is given by:

$\eta (x,t)={\eta }_2+H{\mathrm{cn}}^2\left(\overline{)\begin{array}{cc}2K(m)\frac{x-ct}{\lambda }& m\end{array}}\right),$

where H is the wave height, λ is the wavelength, c is the phase speed and η₂ is the trough elevation. Further cn is one of the Jacobi elliptic functions and K(m) is the complete elliptic integral of the first kind; both are dependent on the elliptic parameter m. The latter, m, determines the shape of the cnoidal wave. For m equal to zero the cnoidal wave becomes a cosine function, while for values close to one the cnoidal wave gets peaked crests and (very) flat troughs. For values of m less than 0.95, the cnoidal function can be approximated with trigonometric functions.

An important dimensionless parameter for nonlinear long waves (λ PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

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$U=\frac{H{\lambda }^2}{h^3}=\frac{H}{h}{\left(\frac{\lambda }{h}\right)}^2.$

For small values of U, say U < 5, a linear theory can be used, and at higher values nonlinear theories have to be used, like cnoidal wave theory. The demarcation zone between—third or fifth order—Stokes' and cnoidal wave theories is in the range 10–25 of the Ursell parameter. As can be seen from the formula for the Ursell parameter, for a given relative wave height H/h the Ursell parameter—and thus also the nonlinearity—grows quickly with increasing relative wavelength λ/h.

Based on the analysis of the full nonlinear problem of surface gravity waves within potential flow theory, the above cnoidal waves can be considered the lowest-order term in a perturbation series. Higher-order cnoidal wave theories remain valid for shorter and more nonlinear waves. A fifth-order cnoidal wave theory was developed by Fenton in 1979. A detailed description and comparison of fifth-order Stokes' and fifth-order cnoidal wave theories is given in the review article by Fenton.

Cnoidal wave descriptions, through a renormalisation, are also well suited to waves on deep water, even infinite water depth; as found by Clamond. A description of the interactions of cnoidal waves in shallow water, as found in real seas, has been provided by Osborne in 1994.

Periodic wave solutions

Korteweg–de Vries equation

The Korteweg–de Vries equation (KdV equation), as used for water waves and in dimensional form, is:

${\partial }_t\eta +\sqrt{gh}{\partial }_x\eta +\frac{3}{2}\sqrt{\frac{g}{h}}\eta {\partial }_x\eta +\frac{1}{6}{h}^2\sqrt{gh}\underset{x}{\overset{3}{\partial }}\eta =0,$

where

η	: surface elevation, a function of x and t, with the positive direction upwards (opposing gravity),
x	: horizontal coordinate,
t	: time,
g	: the value of Earth's gravity,
h	: the mean water depth, and
∂x and ∂t	: partial derivative operators with respect to x and t.

Non-dimensionalisation

All quantities can be made dimensionless using the gravitational acceleration g and water depth h:

$\tilde{\eta }=\frac{\eta }{h},$ Template:Pad $\tilde{x}=\frac{x}{h}$ Template:Pad and Template:Pad $\tilde{t}=\sqrt{\frac{g}{h}}t.$

The resulting non-dimensional form of the KdV equation is

${\partial }_{\tilde{t}}\tilde{\eta }+{\partial }_{\tilde{x}}\tilde{\eta }+\frac{3}{2}\tilde{\eta }{\partial }_{\tilde{x}}\tilde{\eta }+\frac{1}{6}\underset{\tilde{x}}{\overset{3}{\partial }}\tilde{\eta }=0,$

In the remainder, the tildes will be dropped for ease of notation.

Relation to a standard form

The form

${\partial }_{\hat{t}}\varphi +6\varphi {\partial }_{\hat{x}}\varphi +\underset{\hat{x}}{\overset{3}{\partial }}\varphi =0$

is obtained through the transformation

$\hat{t}=\frac{1}{6}t,$ Template:Pad $\hat{x}=x-t$ Template:Pad and Template:Pad $\varphi =\frac{3}{2}\eta ,$

but this form will not be used any further in this derivation.

Fixed-form propagating waves

Periodic wave solutions, travelling with phase speed c, are sought. These permanent waves have to be of the following:

$\eta =\eta (\xi )$ Template:Pad with $\xi$ the wave phase: Template:Pad $\xi =x-ct.$

Consequently, the partial derivatives with respect to space and time become:

${\partial }_x\eta ={\eta }^{\prime }$ Template:Pad and Template:Pad ${\partial }_t\eta =-c{\eta }^{\prime },$

where η’ denotes the ordinary derivative of η(ξ) with respect to the argument ξ.

Using these in the KdV equation, the following third-order ordinary differential equation is obtained:

$\frac{1}{6}{\eta }^{‴}+(1-c){\eta }^{\prime }+\frac{3}{2}\eta {\eta }^{\prime }=0.$

Integration to a first-order ordinary differential equation

This can be integrated once, to obtain:

$\frac{1}{6}{\eta }^{″}+(1-c)\eta +\frac{3}{4}{\eta }^2=\frac{1}{4}r,$

with r an integration constant. After multiplying with 4 η’, and integrating once more

$\frac{1}{3}{\left({\eta }^{\prime }\right)}^2+2(1-c){\eta }^2+{\eta }^3=r\eta +s,$

Cubic polynomial f(η) as encountered in periodic wave solutions of the Korteweg–de Vries equation and Benjamin–Bona–Mahony equation.

with s another integration constant. This is written in the form

The cubic polynomial f(η) becomes negative for large positive values of η, and positive for large negative values of η. Since the surface elevation η is real valued, also the integration constants r and s are real. The polynomial f can be expressed in terms of its roots η₁, η₂ and η₃:

Because f(η) is real valued, the three roots η₁, η₂ and η₃ are either all three real, or otherwise one is real and the remaining two are a pair of complex conjugates. In the latter case, with only one real-valued root, there is only one elevation η at which f(η) is zero. And consequently also only one elevation at which the surface slope η’ is zero. However, we are looking for wave like solutions, with two elevations—the wave crest and trough (physics)—where the surface slope is zero. The conclusion is that all three roots of f(η) have to be real valued.

Without loss of generality, its is assumed that the three real roots are ordered as:

${\eta }_1\ge {\eta }_2\ge {\eta }_3.$

Solution of the first-order ordinary-differential equation

Now, from equation (Template:EquationNote) it can be seen that only real values for the slope exist if f(η) is positive. This corresponds with η₂ ≤ η≤ η₁, which therefore is the range between which the surface elevation oscillates, see also the graph of f(η). This condition is satisfied with the following representation of the elevation η(ξ):

in agreement with the periodic character of the sought wave solutions and with ψ(ξ) the phase of the trigonometric functions sin and cos. From this form, the following descriptions of various terms in equations (Template:EquationNote) and (Template:EquationNote) can be obtained:

$\begin{array}{rl}\eta -{\eta }_1& =-({\eta }_1-{\eta }_2){\sin }^2\psi (\xi ),\\ \eta -{\eta }_2& =+({\eta }_1-{\eta }_2){\cos }^2\psi (\xi ),\\ \eta -{\eta }_3& =({\eta }_1-{\eta }_3)-({\eta }_1-{\eta }_2){\sin }^2\psi (\xi ),& & \text{and}\\ {\eta }^{\prime }& =-2({\eta }_1-{\eta }_2)\sin \psi (\xi )\cos \psi (\xi ){\psi }^{\prime }(\xi )& & \text{with}{\psi }^{\prime }(\xi )=\frac{\text{d}\psi (\xi )}{\text{d}\xi }.\end{array}$

Using these in equations (Template:EquationNote) and (Template:EquationNote), the following ordinary differential equation relating ψ and ξ is obtained, after some manipulations:

$\frac{4}{3}{\left(\frac{\text{d}\psi }{\text{d}\xi }\right)}^2=({\eta }_1-{\eta }_3)-({\eta }_1-{\eta }_2){\sin }^2\psi (\xi ),$

with the right hand side still positive, since η₁ − η₃ ≥ η₁ − η₂. Without loss of generality, we can assume that ψ(ξ) is a monotone function, since f(η) has no zeros in the interval η₂ < η < η₁. So the above ordinary differential equation can also be solved in terms of ξ(ψ) being a function of ψ:

$\frac{1}{\mathrm{\Delta }}\frac{\text{d}\xi }{\text{d}\psi }=\pm \frac{1}{\sqrt{1-m{\sin }^2\psi }},$

with:

${\mathrm{\Delta }}^2=\frac{4}{3}\frac{1}{{\eta }_1-{\eta }_3}$ Template:Pad and Template:Pad $m=\frac{{\eta }_1-{\eta }_2}{{\eta }_1-{\eta }_3},$

where m is the so-called elliptic parameter, satisfying 0 ≤ m ≤ 1 (because η₃ ≤ η₂ ≤ η₁). If ξ = 0 is chosen at the wave crest η(0) = η₁ integration gives

with F(ψ|m) the incomplete elliptic integral of the first kind. The Jacobi elliptic functions cn and sn are inverses of F(ψ|m) given by

$\cos \psi =\mathrm{cn}\left(\overline{)\begin{array}{cc}\frac{\xi }{\mathrm{\Delta }}& m\end{array}}\right)$ Template:Pad and Template:Pad $\sin \psi =\mathrm{sn}\left(\overline{)\begin{array}{cc}\frac{\xi }{\mathrm{\Delta }}& m\end{array}}\right).$

With the use of equation (Template:EquationNote), the resulting cnoidal-wave solution of the KdV equation is found

$\eta (\xi )={\eta }_2+({\eta }_1-{\eta }_2){\mathrm{cn}}^2\left(\overline{)\begin{array}{cc}\frac{\xi }{\mathrm{\Delta }}& m\end{array}}\right).$

What remains, is to determine the parameters: η₁, η₂, Δ and m.

Relationships between the cnoidal-wave parameters

First, since η₁ is the crest elevation and η₂ is the trough elevation, it is convenient to introduce the wave height, defined as H = η₁ − η₂. Consequently, we find for m and for Δ:

$m=\frac{H}{{\eta }_1-{\eta }_3}$ Template:Pad and Template:Pad $\frac{{\mathrm{\Delta }}^2}{m}=\frac{4}{3H}$ Template:Pad so Template:Pad $\mathrm{\Delta }=\sqrt{\frac{4}{3}\frac{m}{H}}.$

The cnoidal wave solution can be written as:

$\eta (\xi )={\eta }_2+H{\mathrm{cn}}^2\left(\overline{)\begin{array}{cc}\frac{\xi }{\mathrm{\Delta }}& m\end{array}}\right).$

Second, the trough is located at ψ = ½ π, so the distance between ξ = 0 and ξ = ½ λ is, with λ the wavelength, from equation (Template:EquationNote):

$\frac{1}{2}\lambda =\mathrm{\Delta }F\left(\overline{)\begin{array}{cc}\frac{1}{2}\pi & m\end{array}}\right)=\mathrm{\Delta }K(m),$ Template:Pad giving Template:Pad $\lambda =2\mathrm{\Delta }K(m)=\sqrt{\frac{16}{3}\frac{m}{H}}K(m),$

where K(m) is the complete elliptic integral of the first kind. Third, since the wave oscillates around the mean water depth, the average value of η(ξ) has to be zero. So

$\begin{array}{rl}0& =\underset{0}{\overset{\lambda }{\int }}\eta (\xi )\text{d}\xi =2\underset{0}{\overset{\frac{1}{2}\lambda }{\int }}[{\eta }_2+({\eta }_1-{\eta }_2){\mathrm{cn}}^2\left(\overline{)\begin{array}{cc}\frac{\xi }{\mathrm{\Delta }}& m\end{array}}\right)]\text{d}\xi \\ & =2\underset{0}{\overset{\frac{1}{2}\pi }{\int }}[{\eta }_2+({\eta }_1-{\eta }_2){\cos }^2\psi ]\frac{\text{d}\xi }{\text{d}\psi }\text{d}\psi =2\mathrm{\Delta }\underset{0}{\overset{\frac{1}{2}\pi }{\int }}\frac{{\eta }_1-({\eta }_1-{\eta }_2){\sin }^2\psi }{\sqrt{1-m{\sin }^2\psi }}\text{d}\psi \\ & =2\mathrm{\Delta }\underset{0}{\overset{\frac{1}{2}\pi }{\int }}\frac{{\eta }_1-m({\eta }_1-{\eta }_3){\sin }^2\psi }{\sqrt{1-m{\sin }^2\psi }}\text{d}\psi =2\mathrm{\Delta }\underset{0}{\overset{\frac{1}{2}\pi }{\int }}[\frac{{\eta }_3}{\sqrt{1-m{\sin }^2\psi }}+({\eta }_1-{\eta }_3)\sqrt{1-m{\sin }^2\psi }]\text{d}\psi \\ & =2\mathrm{\Delta }[{\eta }_{3}K(m)+({\eta }_1-{\eta }_3)E(m)]=2\mathrm{\Delta }[{\eta }_{3}K(m)+\frac{H}{m}E(m)],\end{array}$

where E(m) is the complete elliptic integral of the second kind. The following expressions for η₁, η₂ and η₃ as a function of the elliptic parameter m and wave height H result:

${\eta }_3=-\frac{H}{m}\frac{E(m)}{K(m)},$ Template:Pad ${\eta }_1=\frac{H}{m}(1-\frac{E(m)}{K(m)})$ Template:Pad and Template:Pad ${\eta }_2=\frac{H}{m}(1-m-\frac{E(m)}{K(m)}).$

Fourth, from equations (Template:EquationNote) and (Template:EquationNote) a relationship can be established between the phase speed c and the roots η₁, η₂ and η₃:

$c=1+\frac{1}{2}({\eta }_1+{\eta }_2+{\eta }_3)=1+\frac{H}{m}(1-\frac{1}{2}m-\frac{3}{2}\frac{E(m)}{K(m)}).$

The relative phase-speed changes are depicted in the figure below. As can be seen, for m > 0.96 (so for 1 − m < 0.04) the phase speed increases with increasing wave height H. This corresponds with the longer and more nonlinear waves. The nonlinear change in the phase speed, for fixed m, is proportional to the wave height H. Note that the phase speed c is related to the wavelength λ and period τ as:

$c=\frac{\lambda }{\tau }.$

Résumé of the solution

All quantities here will be given in their dimensional forms, as valid for surface gravity waves before non-dimensionalisation. Template:Hidden end

Relative phase speed increase of cnoidal wave solutions for the Korteweg–de Vries equation as a function of 1−m, with m the elliptic parameter.
The horizontal axis is on a logarithmic scale, from 10⁻⁶ to 10⁰=1.
The figure is for non-dimensional quantities, i.e. the phase speed c is made dimensionless with the shallow-water phase speed $\sqrt{gh}$, and the wave height H is made dimensionless with the mean water depth h.

The cnoidal-wave solution of the KdV equation is:

$\eta (x,t)={\eta }_2+H{\mathrm{cn}}^2\left(\overline{)\begin{array}{cc}\frac{x-ct}{\mathrm{\Delta }}& m\end{array}}\right),$

with H the wave height—the difference between crest and trough elevation, η₂ the trough elevation, m the elliptic parameter, c the phase speed and cn one of the Jacobi elliptic functions. The trough level η₂ and width parameter Δ can be expressed in terms of H, h and m:

${\eta }_2=\frac{H}{m}(1-m-\frac{E(m)}{K(m)}),$ Template:Pad and Template:Pad $\mathrm{\Delta }=\frac{\lambda }{2K(m)}=h\sqrt{\frac{4}{3}\frac{mh}{H}},$

with K(m) the complete elliptic integral of the first kind and E(m) the complete elliptic integral of the second kind. Note that K(m) and E(m) are denoted here as a function of the elliptic parameter m and not as a function of the elliptic modulus k, with m = k².

The wavelength λ, phase speed c and wave period τ are related to H, h and m by:

$\lambda =h\sqrt{\frac{16}{3}\frac{mh}{H}}K(m),$ Template:Pad $c=\sqrt{gh}[1+\frac{H}{mh}(1-\frac{1}{2}m-\frac{3}{2}\frac{E(m)}{K(m)})]$ Template:Pad and Template:Pad $\tau =\frac{\lambda }{c},$

with g the Earth's gravity.

Most often, the known wave parameters are the wave height H, mean water depth h, gravitational acceleration g, and either the wavelength λ or else the period τ. Then the above relations for λ, c and τ are used to find the elliptic parameter m. This requires numerical solution by some iterative method.

Benjamin–Bona–Mahony equation

The Benjamin–Bona–Mahony equation (BBM equation), or regularised long wave (RLW) equation, is in dimensional form given by:

${\partial }_t\eta +\sqrt{gh}{\partial }_x\eta +\frac{3}{2}\sqrt{\frac{g}{h}}\eta {\partial }_x\eta -\frac{1}{6}{h}^2{\partial }_t\underset{x}{\overset{2}{\partial }}\eta =0.$

All quantities have the same meaning as for the KdV equation. The BBM equation is often preferred over the KdV equation because it has a better short-wave behaviour.

Derivation

The derivation is analogous to the one for the KdV equation. The dimensionless BBM equation is, non-dimensionalised using mean water depth h and gravitational acceleration g:

${\partial }_t\eta +{\partial }_x\eta +\frac{3}{2}\eta {\partial }_x\eta -\frac{1}{6}{\partial }_t\underset{x}{\overset{2}{\partial }}\eta =0.$

This can be brought into the standard form

${\partial }_{\hat{t}}\phi +{\partial }_{\hat{x}}\phi +\phi {\partial }_{\hat{x}}\phi -{\partial }_{\hat{t}}\underset{\hat{x}}{\overset{2}{\partial }}\phi =0$

through the transformation:

$\hat{t}=\sqrt{6}t,$ Template:Pad $\hat{x}=\sqrt{6}x$ Template:Pad and Template:Pad $\phi =\frac{3}{2}\eta ,$

but this standard form will not be used here.

Analogue to the drivation of the cnoidal wave solution for the KdV equation, periodic wave solutions η(ξ), with ξ = x−ct are considered Then the BBM equation becomes a third-order ordinary differential equation, which can be integrated twice, to obtain:

$\frac{1}{3}c{\left({\eta }^{\prime }\right)}^2=f(\eta )$ Template:Pad with Template:Pad $f(\eta )=-{\eta }^3+2(c-1){\eta }^2+r\eta +s.$

Which only differs from the equation for the KdV equation through the factor c in front of (η′)² in the left hand side. Through a coordinate transformation β = ξ / $\sqrt{c}$ the factor c may be removed, resulting in the same first-order ordinary differential equation for both the KdV and BBM equation. However, here the form given in the preceding equation is used. This results in a different formulation for Δ as found for the KdV equation:

$\mathrm{\Delta }=\sqrt{\frac{4}{3}\frac{mc}{H}}.$

The relation of the wavelength λ, as a function of H and m, is affected by this change in $\mathrm{\Delta }:$

$\lambda =\sqrt{\frac{16}{3}\frac{mc}{H}}K(m).$

For the rest, the derivation is analogous to the one for the KdV equation, and will not be repeated here.

Résumé

The results are presented in dimensional form, for water waves on a fluid layer of depth h. Template:Hidden end

The cnoidal wave solution of the BBM equation, together with the associated relationships for the parameters is:

$\begin{array}{rl}\eta (x,t)& ={\eta }_2+H{\mathrm{cn}}^2\left(\overline{)\begin{array}{cc}\frac{x-ct}{\mathrm{\Delta }}& m\end{array}}\right),\\ {\eta }_2& =\frac{H}{m}(1-m-\frac{E(m)}{K(m)}),\\ \mathrm{\Delta }& =h\sqrt{\frac{4}{3}\frac{mh}{H}\frac{c}{\sqrt{gh}}}& & =\frac{\lambda }{2K(m)},\\ \lambda & =h\sqrt{\frac{16}{3}\frac{mh}{H}\frac{c}{\sqrt{gh}}}K(m),\\ c& =\sqrt{gh}[1+\frac{H}{mh}(1-\frac{1}{2}m-\frac{3}{2}\frac{E(m)}{K(m)})]& & \text{and}\\ \tau & =\frac{\lambda }{c}.\end{array}$

The only difference with the cnoidal wave solution of the KdV equation is in the equation for the wavelength λ. For practical applications, usually the water depth h, wave height H, gravitational acceleration g, and either the wavelength λ, or—most often—the period (physics) τ are provided. Then the elliptic parameter m has to be determined from the above relations for λ, c and τ through some iterative method.

Example

Parameter relations for cnoidal wave solutions of the Korteweg–de Vries equation. Shown is −log₁₀ (1−m), with m the elliptic parameter of the complete elliptic integrals, as a function of dimensionless period τ √(g/h) and relative wave height H / h. The values along the contour lines are −log₁₀ (1−m), so a value 1 corresponds with m = 1 − 10⁻¹ = 0.9 and a value 40 with m = 1 − 10⁻⁴⁰.

In this example, a cnoidal wave according to the Korteweg–de Vries (KdV) equation is considered. The following parameters of the wave are given:

• mean water depth h = Template:Convert,
• wave height H = Template:Convert,
• wave period τ = 7 s, and
• gravitational acceleration g = 9.81 m/s² (32 ft/s²).

Instead of the period τ, in other cases the wavelength λ may occur as a quantity known beforehand.

First, the dimensionless period is computed:

$\tau \sqrt{\frac{g}{h}}=9.80,$

which is larger than seven, so long enough for cnoidal theory to be valid. The main unknown is the elliptic parameter m. This has to be determined in such a way that the wave period τ, as computed from cnoidal wave theory for the KdV equation:

$\lambda =h\sqrt{\frac{16}{3}\frac{mh}{H}}K(m),$ Template:Pad $c=\sqrt{gh}[1+\frac{H}{mh}(1-\frac{1}{2}m-\frac{3}{2}\frac{E(m)}{K(m)})]$ Template:Pad and Template:Pad $\tau =\frac{\lambda }{c},$

is consistent with the given value of τ; here λ is the wavelength and c is the phase speed of the wave. Further, K(m) and E(m) are complete elliptic integrals of the first and second kind, respectively. Searching for the elliptic parameter m can be done by trial and error, or by use of a numerical root-finding algorithm. In this case, starting from an initial guess m_init = 0.99, by trial and error the answer

$m=0.9832$

is found. Within the process, the wavelength λ and phase speed c have been computed:

• wavelength λ = Template:Convert, and
• phase speed c = Template:Convert.

The phase speed c can be compared with its value $\sqrt{gh}$ according to the shallow water equations:

$\frac{c}{\sqrt{gh}}=1.0376,$

showing a 3.8% increase due to the effect of nonlinear amplitude dispersion, which wins in this case from the reduction of phase speed by frequency dispersion.

Now the wavelength is known, the Ursell number can be computed as well:

$U=\frac{H{\lambda }^2}{h^3}=62,$

which is not small, so linear wave theory is not applicable, but cnoidal wave theory is. Finally, the ratio of wavelength to depth is λ / h = 10.2 > 7, again indicating this wave is long enough to being considered as a cnoidal wave.

Solitary-wave limit

For very long nonlinear waves, with the parameter m close to one, m → 1, the Jacobi elliptic function cn can be approximated by

$\mathrm{cn}\left(z\right|m)\approx \mathrm{sech}(z)-\frac{1}{4}(1-m)[\sinh (z)\cosh (z)-z]\tanh (z)\mathrm{sech}(z),$ Template:Pad with Template:Pad $\mathrm{sech}(z)=\frac{1}{\cosh (z)}.$

Here sinh, cosh, tanh and sech are hyperbolic functions. In the limit m = 1:

$\mathrm{cn}\left(z\right|m)\to \mathrm{sech}(z),$

with sech(z) = 1 / cosh(z).

Further, for the same limit of m → 1, the complete elliptic integral of the first kind K(m) goes to infinity, while the complete elliptic integral of the second kind E(m) goes to one. This implies that the limiting values of the phase speed c and minimum elevelation η₂ become:

$c=\sqrt{gh}(1+\frac{1}{2}\frac{H}{h})$ Template:Pad and Template:Pad ${\eta }_2=0.$

Consequently, in terms of the width parameter Δ, the solitary wave solution to both the KdV and BBM equation is:

$\eta (x,t)=H{\mathrm{sech}}^2\left(\frac{x-ct}{\mathrm{\Delta }}\right).$

The width parameter, as found for the cnoidal waves and now in the limit m → 1, is different for the KdV and the BBM equation:

$\mathrm{\Delta }=h\sqrt{\frac{4h}{3H}}$ Template:Pad	: KdV equation, and
$\mathrm{\Delta }=h\sqrt{\frac{4h}{3H}\frac{c}{\sqrt{gh}}}$ Template:Pad	: BBM equation.

But the phase speed of the solitary wave in both equations is the same, for a certain combination of height H and depth h.

Limit of infinitesimal wave height

For infinitesimal wave height the results of cnoidal wave theory are expected to converge towards those of Airy wave theory for the limit of long waves λ PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

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Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

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The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER h. First the surface elevation, and thereafter the phase speed, of the cnoidal waves for infinitesimal wave height will be examined.

Surface elevation

The Jacobi elliptic function cn can be expanded into a Fourier series

$\mathrm{cn}(z|m)=\frac{\pi }{\sqrt{m}K(m)}\sum _{n=0}^{\mathrm{\infty }}\mathrm{sech}((2n+1)\frac{\pi K^{\prime }(m)}{2K(m)})\cos ((2n+1)\frac{\pi z}{2K(m)}).$

K’(m) is known as the imaginary quarter period, while K(m) is also called the real quarter period of the Jacobi elliptic function. They are related through: K’(m) = K(1−m)

Since the interest here is in small wave height, corresponding with small parameter m PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

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The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER 1, it is convenient to consider the Maclaurin series for the relevant parameters, to start with the complete elliptic integrals K and E:

$\begin{array}{rl}K(m)& =\frac{\pi }{2}[1+{\left(\frac{1}{2}\right)}^{2}m+{\left(\frac{1\cdot 3}{2\cdot 4}\right)}^2{m}^2+{\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)}^2{m}^3+\cdots ],\\ E(m)& =\frac{\pi }{2}[1-{\left(\frac{1}{2}\right)}^2\frac{m}{1}-{\left(\frac{1\cdot 3}{2\cdot 4}\right)}^2\frac{m^2}{3}-{\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)}^2\frac{m^3}{5}-\cdots ].\end{array}$

Then the hyperbolic-cosine terms, appearing in the Fourier series, can be expanded for small m PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

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Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER 1 as follows:

$\mathrm{sech}((2n+1)\frac{\pi K^{\prime }(m)}{2K(m)})=2\frac{q^{n+\frac{1}{2}}}{1+q^{2n+1}}$ Template:Pad with the nome q given by Template:Pad $q=\exp (-\pi \frac{K^{\prime }(m)}{K(m)}).$

The nome q has the following behaviour for small m:

$q=\frac{m}{16}+8{\left(\frac{m}{16}\right)}^2+84{\left(\frac{m}{16}\right)}^3+992{\left(\frac{m}{16}\right)}^4+\cdots .$

Consequently, the amplitudes of the first terms in the Fourier series are:

$n=0$	: Template:Pad	$\frac{\pi }{\sqrt{m}K(m)}\mathrm{sech}\left(\frac{\pi K^{\prime }(m)}{2K(m)}\right)=1-\frac{1}{16}m-\frac{9}{16}{m}^2+\cdots ,$
$n=1$	: Template:Pad	$\frac{\pi }{\sqrt{m}K(m)}\mathrm{sech}\left(\frac{3\pi K^{\prime }(m)}{2K(m)}\right)=\frac{1}{16}m+\frac{1}{32}{m}^2+\cdots ,$
$n=2$	: Template:Pad	$\frac{\pi }{\sqrt{m}K(m)}\mathrm{sech}\left(\frac{5\pi K^{\prime }(m)}{2K(m)}\right)=\frac{1}{256}{m}^2+\cdots .$

So, for m PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

As a public-listed company and a pioneer in the improvement of landed properties in Singapore, we have now been constructing some of Singapore's nicely-recognized, established residential estates for over 50 years. Our many developments in Singapore are an affidavit of our steady want to create not just high quality houses however communities for you and your family members Hotel Properties Limited (HPL) was listed on the Inventory Trade of Singapore in 1982. Beginning with simply the Hilton Lodge in Singapore, HPL has expanded rapidly through the years. As we speak HPL has pursuits in 19 resorts with almost 4000 rooms in 8 countries. An Choice or Settlement/Contract for the Buy of a Home or Flat will be formedin many ways e.g. A gaggle of persons, whether in partnership or otherwise

Certainly one of Asia's premier property firms, Keppel Land is recognised for its sterling portfolio of award-profitable residential developments and funding-grade business properties as well as excessive requirements of company governance and transparency. Keppel Land is without doubt one of the largest listed property corporations by total assets on the Singapore Change. The Group's total assets amounted to about $13.eight billion as at 31 March 2014. Additionally it is a component of a number of stock indices including the FTSE ST Real Property Index, FTSE ST China Prime Index, FTSE All-World Index, FTSE Asia Pacific ex-Japan Index, FTSE EPRA/NAREIT World Real Estate Index and EPRA/NAREIT Index. WOODSVALE PERSONAL CONDOMINIUM CONDOMINIUM WOODSVALE CLOSE, SINGAPORE (DISTRICT thirteen) Industrial

LINCOLN RESIDENCES, THE NON-PUBLIC CONDOMINIUM APARTMENT SURREY STREET, SINGAPORE (DISTRICT eleven) LUCIDA NON-PUBLIC CONDOMINIUM APARTMENT SUFFOLK ROAD, SINGAPORE (DISTRICT 11) LUMOS, THE PRIVATE CONDOMINIUM APARTMENT LEONIE HILL, SINGAPORE (DISTRICT 09) LUXURIE, THE PRIVATE CONDOMINIUM CONDOMINIUM COMPASSVALE BOW, SINGAPORE (DISTRICT 19) M66 NON-PUBLIC CONDOMINIUM RESIDENCE MOONSTONE LANE, SINGAPORE (DISTRICT 12) MARINA BAY SUITES PRIVATE CONDOMINIUM CONDOMINIUM CENTRAL BOULEVARD, SINGAPORE (DISTRICT 01) MEIER SUITES PERSONAL CONDOMINIUM house in singapore MARGATE STREET, SINGAPORE (DISTRICT 15) MKZ, THE NON-PUBLIC CONDOMINIUM CONDO MACKENZIE STREET, SINGAPORE (DISTRICT 09) MONTCLAIR @ WHITLEY CLUSTER STRATA HOUSE WHITLEY HIGHWAY, SINGAPORE (DISTRICT 11) Condominiums by District

The Singapore Property Awards recognise excellence in actual estate development initiatives or individual properties in terms of design, aesthetics, functionality, contribution to the constructed atmosphere and neighborhood at massive. It represents an outstanding achievement which developers, professionals and property house owners aspire to achieve. It bestows upon the winner the correct to use the coveted award emblem recognised extensively throughout the FIABCI network.

ADRIA NON-PUBLIC CONDOMINIUM CONDOMINIUM DERBYSHIRE HIGHWAY, THOMSON ROAD, SINGAPORE (DISTRICT eleven) AMBER RESIDENCES (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE AMBER STREET, SINGAPORE (DISTRICT 15) ARC AT TAMPINES GOVERNMENT CONDOMINIUM APARTMENT TAMPINES AVENUE 8, SINGAPORE (DISTRICT 18) ARDMORE RESIDENCE NON-PUBLIC CONDOMINIUM CONDO ARDMORE PARK, SINGAPORE (DISTRICT 10) ARISTO @ AMBER , THE PRIVATE CONDOMINIUM CONDOMINIUM AMBER STREET, SINGAPORE (DISTRICT 15) ASPEN LINQ NON-PUBLIC CONDOMINIUM RESIDENCE INSTITUTION HILL, SINGAPORE (DISTRICT 09) BARTLEY RESIDENCES NON-PUBLIC CONDOMINIUM HOUSE BARTLEY HIGHWAY, SINGAPORE (DISTRICT 19) BEACON HEIGHTS PERSONAL CONDOMINIUM CONDO MAR THOMA STREET, SINGAPORE (DISTRCT 12) title searches and authorized requisitions on the property; and

Hongkong Land is a number one property funding, administration and growth group with a serious portfolio in Hong Kong and different property pursuits in Asia. As considered one of Singapore's largest property gamers, Singapore Land (SingLand) is synonymous with premier property developments in both prime and suburban areas. YHS made its first foray into property improvement with Tivoli Gardens, a 59-unit landed estate in District 19. This was adopted by the launch of The Sterling, a freehold condominium in Bukit Timah, and landed projects resembling Tai Keng Villas, Parry Inexperienced, Chuan Villas and Princeton Vale. Its newest growth is JARDIN, an exclusive property nestled alongside Bukit Timah/Dunearn Highway Learn More Can Singapore safely deflate its property market?

GPS Funding Sales operates like an entrepreneur group drawing on wealth of experiences with in depth insight locally and having a global perspective. We are in a position to provide investment methods that tailor-made to the clients profile. Our purchasers starting from Wealth Fund managers, Multinational Companies, Small Medium Enterprise companies, Property Developers and Ultra Networth Individual. En-bloc Sales Department Sustainability, property growth, sustainability initiatives, tripple backside line, measuring sustainability, reporting, metrics and benchmarks When you've loved what you've got read thus far why not sign up for our FREE property alert and online journal PropertyWire Confidential. District 23, ninety nine years Leasehold condominium BUILD TO ALTER 1 the Jacobi elliptic function has the first Fourier series terms:

$\begin{array}{rlrl}\mathrm{cn}(z|m)& =(1-\frac{1}{16}m-\frac{9}{16}{m}^2+\cdots & & )\cos \alpha z\\ & +(\frac{1}{16}m+\frac{1}{32}{m}^2+\cdots & & )\cos 2\alpha z\\ & +(\frac{1}{256}{m}^2+\cdots & & )\cos 3\alpha z+\cdots ,\end{array}$ Template:Pad with Template:Pad $\alpha \equiv \frac{\pi }{2K(m)}.$

And its square is

$\begin{array}{rlrl}{\mathrm{cn}}^2(z|m)& =(\frac{1}{2}-\frac{1}{16}m-\frac{1}{32}{m}^2+\cdots & & )\\ & +(\frac{1}{2}-\frac{3}{512}{m}^2+\cdots & & )\cos 2\alpha z\\ & +(\frac{1}{16}m+\frac{1}{32}{m}^2+\cdots & & )\cos 4\alpha z\\ & +(\frac{3}{512}{m}^2+\cdots & & )\cos 6\alpha z+\cdots .\end{array}$

The free surface η(x,t) of the cnoidal wave will be expressed in its Fourier series, for small values of the elliptic parameter m. First, note that the argument of the cn function is ξ/Δ, and that the wavelength λ = 2 Δ K(m), so:

$\theta \equiv \frac{\pi \xi }{K(m)}\frac{\xi }{\mathrm{\Delta }}=2\pi \frac{\xi }{\lambda }.$ Template:Pad

Further, the mean free-surface elevation is zero. Therefore, the surface elevation of small amplitude waves is

$\eta (x,t)=H(\frac{1}{2}-\frac{3}{512}{m}^2+\cdots )\cos \theta +H(\frac{1}{16}m+\frac{1}{32}{m}^2+\cdots )\cos 2\theta +H(\frac{3}{512}{m}^2+\cdots )\cos 3\theta +\cdots .$

Also the wavelength λ can be expanded into a Maclaurin series of the elliptic parameter m, differently for the KdV and the BBM equation, but this is not necessary for the present purpose.

Note: The limiting behaviour for zero m—at infinitesimal wave height—can also be seen from:

Template:Pad $\mathrm{cn}(z|m)\approx \cos (z)+\frac{1}{4}m(z-\sin (z)\cos (z))\sin (z)+\cdots ,$

but the higher-order term proportional to m in this approximation contains a secular term, due to the mismatch between the period of cn(z|m), which is 4 K(m), and the period 2π for the cosine cos(z). The above Fourier series for small m does not have this drawback, and is consistent with forms as found using the Lindstedt–Poincaré method in perturbation theory.

For infinitesimal wave height, in the limit m → 0, the free-surface elevation becomes:

$\eta (x,t)=\frac{1}{2}H\cos \theta ,$ Template:Pad with Template:Pad $\theta =2\pi \frac{\xi }{\lambda }=2\pi \frac{x-ct}{\lambda }.$

So the wave amplitude is ½H, half the wave height. This is of the same form as studied in Airy wave theory, but note that cnoidal wave theory is only valid for long waves with their wavelength much longer than the average water depth.

Phase speed

The phase speed of a cnoidal wave, both for the KdV and BBM equation, is given by:

$c=\sqrt{gh}[1+\frac{H}{mh}(1-\frac{1}{2}m-\frac{3}{2}\frac{E(m)}{K(m)})].$

In this formulation the phase speed is a function of wave height H and parameter m. However, for the determination of wave propagation for waves of infinitesimal height, it is necessary to determine the behaviour of the phase speed at constant wavelength λ in the limit that the parameter m approaches zero. This can be done by using the equation for the wavelength, which is different for the KdV and BBM equation:

KdV : Template:Pad	$\lambda =h\sqrt{\frac{16}{3}\frac{mh}{H}}K(m),$
BBM : Template:Pad	$\lambda =h\sqrt{\frac{16}{3}\frac{mh}{H}\frac{c}{\sqrt{gh}}}K(m),$

Introducing the relative wavenumber κh:

$\kappa h=\frac{2\pi }{\lambda }h,$

and using the above equations for the phase speed and wavelength, the factor H / m in the phase speed can be replaced by κh and m. The resulting phase speeds are:

KdV : Template:Pad	$c=\sqrt{gh}[1+(\kappa h{)}^2\frac{4}{3{\pi }^2}{K}^2(m)(1-\frac{1}{2}m-\frac{3}{2}\frac{E(m)}{K(m)})],$
BBM :	$c=\sqrt{gh}\frac{1}{1-(\kappa h{)}^2\frac{4}{3{\pi }^2}{K}^2(m)(1-\frac{1}{2}m-\frac{3}{2}\frac{E(m)}{K(m)})}.$

The limiting behaviour for small m can be analysed through the use of the Maclaurin series for K(m) and E(m), resulting in the following expression for the common factor in both formulas for c:

$\gamma =\frac{4}{3{\pi }^2}{K}^2(m)(1-\frac{1}{2}m-\frac{3}{2}\frac{E(m)}{K(m)})=-\frac{1}{6}+\frac{1}{64}{m}^2+\frac{1}{64}{m}^3+\cdots ,$

so in the limit m → 0, the factor γ → −Template:Frac. The limiting value of the phase speed for m PROPERTY builders did not have the simplest year, what with the cooling measures imposed in January and the loan curbs in June, but some still managed to do effectively while others made their first foray abroad.

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Template:Hidden end The phase speeds for infinitesimal wave height, according to the cnoidal wave theories for the KdV equation and BBM equation, are

KdV	: Template:Pad	$c=[1-\frac{1}{6}{\left(\kappa h\right)}^2]\sqrt{gh},$
BBM	:	$c=\frac{1}{1+\frac{1}{6}{\left(\kappa h\right)}^2}\sqrt{gh},$

with κ = 2π / λ the wavenumber and κh the relative wavenumber. These phase speeds are in full agreement with the result obtained by directly searching for sine-wave solutions of the linearised KdV and BBM equations. As is evident from these equations, the linearised BBM equation has a positive phase speed for all κh. On the other hand, the phase speed of the linearised KdV equation changes sign for short waves with κh > $\sqrt{6}$. This is in conflict with the derivation of the KdV equation as a one-way wave equation.

Direct derivation from the full inviscid-flow equations

Undular bore and whelps near the mouth of Araguari River in north-eastern Brazil. View is oblique toward mouth from airplane at approximately Template:Convert altitude.

Cnoidal waves can be derived directly from the inviscid, irrotational and incompressible flow equations, and expressed in terms of three invariants of the flow, as shown by Template:Harvtxt in their research on undular bores. In a frame of reference moving with the phase speed, in which reference frame the flow becomes a steady flow, the cnoidal wave solutions can directly be related to the mass flux, momentum flux and energy head of the flow. Following Template:Harvtxt—using a stream function description of this incompressible flow—the horizontal and vertical components of the flow velocity are +∂_zΨ and −∂_ξΨ, in the ξ and z direction respectively (ξ = x−ct). The vertical coordinate z is positive in the upward direction, opposite to the direction of the gravitational acceleration, and the zero level of z is at the impermeable lower boundary of the fluid domain. While the free surface is at z = ζ(ξ); note that ζ is the local water depth, related to the surface elevation η(ξ) as ζ = h + η with h the mean water depth.

In this steady flow, the discharge Q through each vertical cross section is a constant independent of ξ, and because of the horizontal bed also the horizontal momentum flux S, divided by the density ρ, through each vertical cross section is conserved. Further, for this inviscid and irrotational flow, Bernoulli's principle can be applied and has the same Bernoulli constant R everywhere in the flow domain. They are defined as:

$\begin{array}{rl}Q& =\underset{0}{\overset{\zeta (\xi )}{\int }}{\partial }_z\mathrm{\Psi }\text{d}z,\\ R& =\frac{p}{\rho }+\frac{1}{2}[{\left({\partial }_{\xi }\mathrm{\Psi }\right)}^2+{\left({\partial }_z\mathrm{\Psi }\right)}^2]+gz\text{and}\\ S& =\underset{0}{\overset{\zeta (\xi )}{\int }}[\frac{p}{\rho }+{\left({\partial }_z\mathrm{\Psi }\right)}^2]\text{d}z.\end{array}$

For fairly long waves, assuming the water depth ζ is small compared to the wavelength λ, the following relation is obtained between the water depth ζ(ξ) and the three invariants Q, R and S:

This nonlinear and first-order ordinary differential equation has cnoidal wave solutions.

For very long waves of infinitesimal amplitude on a fluid of depth h and with a uniform flow velocity v, the flow constants are according to the shallow water equations:

$Q_0=vh,$ Template:Pad $R_0=\frac{1}{2}{v}^2+gh$ Template:Pad and Template:Pad $S_0=v^{2}h+\frac{1}{2}g{h}^2.$

Equation (Template:EquationNote) can be brought into non-dimensional form by use of the discharge Q and gravitational acceleration g, and defining the critical depth h_c:

$h_c=\sqrt[3]{\frac{Q^2}{g}},$

related to the critical flow demarcation between subcritical flow and supercritical flow (see also Froude number). Consequently, the non-dimensional form of the equation is

$\frac{1}{3}{\left({\tilde{\zeta }}^{\prime }\right)}^2\approx -{\zeta }^3+2\tilde{R}{\tilde{\zeta }}^2-2\tilde{S}\tilde{\zeta }+1,$

with

$\tilde{\zeta }=\frac{\zeta }{h_c},$ Template:Pad $\tilde{\xi }=\frac{\xi }{h_c},$ Template:Pad $\tilde{R}=\frac{R}{g{h}_c},$ Template:Pad and Template:Pad $\tilde{S}=\frac{S}{g{h}_c^2}.$

Derivation

First eliminate the pressure p from the momentum flux S by use of the Bernoulli equation:

$S=R\zeta -\frac{1}{2}g{\zeta }^2+\underset{0}{\overset{\zeta }{\int }}\frac{1}{2}[{\left({\partial }_z\mathrm{\Psi }\right)}^2-{\left({\partial }_{\xi }\mathrm{\Psi }\right)}^2]\text{d}z.$

The streamfunction Ψ is expanded as a Maclaurin series around the bed at z = 0, and using that the impermeable bed is a streamline and the irrotationality of the flow: Ψ = 0 and ∂_z²Ψ = 0 at z = 0:

$\mathrm{\Psi }=z{u}_b(\xi )-\frac{z^3}{3!}{u_b}^{″}(\xi )+\frac{z^5}{5!}{u}_b^{\text{iv}}(\xi )+\cdots ,$

with u_b the horizontal velocity at the bed z = 0. Because the waves are long, h ≫ λ, only terms up to z³ and ζ³ are retained in the approximations to Q and S. The momentum flux S then becomes:

$S=R\zeta -\frac{1}{2}g{\zeta }^2+\frac{1}{2}\zeta u_b^2-\frac{1}{6}{\zeta }^3{u}_b{u_b}^{″}-\frac{1}{6}{\zeta }^3{\left({u_b}^{\prime }\right)}^2+\cdots .$

The discharge Q becomes, since it is the value of the streamfunction Ψ at the free surface z = ζ:

$Q=\zeta u_b(\xi )-\frac{1}{6}{\zeta }^3{u_b}^{″}\xi +\cdots .$

As can be seen, the discharge Q is an O(ζ) quantity. From this, the bed velocity is seen to be

$u_b=\frac{Q}{\zeta }+\frac{1}{6}{\zeta }^2{u_b}^{″}+\cdots .$

Note that Q / ζ is an order one quantity. This relation will be used to replace the bed velocity u_b by Q and ζ in the momentum flux S. The following terms can be derived from it:

$\begin{array}{rl}{u}_b^2& =\frac{Q^2}{{\zeta }^2}+\frac{1}{3}\zeta Q{u_b}^{″}+\cdots ,\\ {u_b}^{\prime }& =-\frac{Q}{\zeta }{\zeta }^{\prime }+\frac{1}{3}\zeta {\zeta }^{\prime }{u_b}^{″}+\frac{1}{6}{\zeta }^2{u_b}^{‴}+\cdots \text{and}\\ {\left({u_b}^{\prime }\right)}^2& =\frac{Q^2}{{\zeta }^4}{\left({\zeta }^{\prime }\right)}^2-\frac{2}{3}\frac{Q}{\zeta }{\zeta }^{\prime }{u_b}^{″}+\cdots .\end{array}$

Consequently, the momentum flux S becomes, again retaining only terms up to proportional to ζ³:

$S\approx R\zeta -\frac{1}{2}g{\zeta }^2+\frac{1}{2}\frac{Q^2}{\zeta }-\frac{1}{6}\frac{Q^2}{\zeta }{\left({\zeta }^{\prime }\right)}^2.$

Which can directly be recast in the form of equation (Template:EquationNote).

Potential energy

The potential energy density

$E_{\text{pot}}=\frac{1}{\lambda }\underset{0}{\overset{\lambda }{\int }}\frac{1}{2}\rho g{\eta }^2(x,t)\text{d}x$

with ρ the fluid density, is one of the infinite number of invariants of the KdV equation. This can be seen by multiplying the KdV equation with the surface elevation η(x,t); after repeated use of the chain rule the result is:

${\partial }_t\left(\frac{1}{2}{\eta }^2\right)+{\partial }_x\{\frac{1}{2}\sqrt{gh}{\eta }^2+\frac{1}{2}\sqrt{\frac{g}{h}}{\eta }^3+\frac{1}{12}{h}^2\sqrt{gh}[\underset{x}{\overset{2}{\partial }}\left({\eta }^2\right)-3{\left({\partial }_x\eta \right)}^2]\}=0,$

which is in conservation form, and is an invariant after integration over the interval of periodicity—the wavelength for a cnoidal wave. The potential energy is not an invariant of the BBM equation, but ½ρg [η² + Template:Frac h² (∂_x η)²] is.

First the variance of the surface elevation in a cnoidal wave is computed. Note that η₂ = −(1/λ) ₀∫^λ H cn²(ξ/Δ|m) dx, cn(ξ/Δ|m)  = cos ψ(ξ) and λ = 2 Δ K(m), so

$\begin{array}{rl}\frac{1}{\lambda }\underset{0}{\overset{\lambda }{\int }}{\eta }^2\text{d}x& =\frac{1}{\lambda }\underset{0}{\overset{\lambda }{\int }}{\{{\eta }_2+H{\mathrm{cn}}^2\left(\overline{)\begin{array}{cc}\frac{\xi }{\mathrm{\Delta }}& m\end{array}}\right)\}}^2\text{d}\xi =\frac{H^2}{\lambda }\underset{0}{\overset{\lambda }{\int }}{\mathrm{cn}}^4\left(\overline{)\begin{array}{cc}\frac{\xi }{\mathrm{\Delta }}& m\end{array}}\right)\text{d}\xi -{\eta }_2^2\\ & =\frac{\mathrm{\Delta }{H}^2}{\lambda }\underset{0}{\overset{\pi }{\int }}{\cos }^4\psi \frac{\text{d}\xi }{\text{d}\psi }\text{d}\psi -{\eta }_2^2=\frac{H^2}{2K(m)}\underset{0}{\overset{\pi }{\int }}\frac{{\cos }^4\psi }{\sqrt{1-m{\sin }^2\psi }}\text{d}\psi -{\eta }_2^2\\ & =\frac{1}{3}\frac{H^2}{m^2}[(2-5m+3m^2)+(4m-2)\frac{E(m)}{K(m)}]-\frac{H^2}{m^2}{(1-m-\frac{E(m)}{K(m)})}^2\end{array}$

The potential energy, both for the KdV and the BBM equation, is subsequently found to be

$E_{\text{pot}}=\frac{1}{2}\rho g{H}^2[-\frac{1}{3m}+\frac{2}{3m}(1+\frac{1}{m})(1-\frac{E(m)}{K(m)})-\frac{1}{m^2}{(1-\frac{E(m)}{K(m)})}^2].$

The infinitesimal wave-height limit (m → 0) of the potential energy is E_pot = Template:Frac ρ g H², which is in agreement with Airy wave theory. The wave height is twice the amplitude, H = 2a, in the infinitesimal wave limit.

See also

• Soliton

Notes and references

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Template:AS ref
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 See Part 2, Chapter 6.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Further reading

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Template:Cite arXiv
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, see pp. 702–714 for cnoidal waves
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010