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'''Structural acoustics''' is the study of the mechanical [[waves]] in [[structures]] and how they interact with and radiate into adjacent media. The field of structural acoustics is often referred to as vibroacoustics in Europe and Asia.{{citation needed|date=September 2013}} People that work in the field of structural acoustics are known as structural acousticians.{{citation needed|date=September 2013}} The field of structural acoustics can be closely related to a number of other fields of [[acoustics]] including [[noise]], [[Transducer|transduction]], [[underwater acoustics]], and [[physical acoustics]].

==Vibrations in Structures<ref>{{cite |url=http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=ATCODK000002000004000021000001&idtype=cvips&prog=search |title=STRUCTURAL ACOUSTICS TUTORIAL I, VIBRATION IN STRUCTURES |accessdate=2010-08-09 |author=Stephen A. Hambric, Applied Research Lab at The Pennsylvania State University}}</ref>==
===Compressional and Shear Waves (isotropic, homogeneous material)===
Compressional waves (often referred to as [[longitudinal waves]]) expand and contract in the same direction (or opposite) as the wave motion. The wave equation dictates the motion of the wave in the x direction.

:<math> { \partial^2 w \over \partial x ^2 } = {1 \over c_L^2} { \partial^2 w \over \partial t ^2 } </math>

where <math>w</math> is the deformation <math>c_L</math> is the wave speed. This has the same form as the [[acoustic wave equation]] in one-dimension. <math>c_L</math> is determined by properties ([[bulk modulus]] <math>B</math> and [[density]] <math>\rho</math>) of the structure according to

:<math> { c_L } = { \sqrt { B \over \rho } } </math>

When two dimensions of the structure are small with respect to [[wavelength]] (commonly called a beam), the wave speed is dictated by [[Youngs modulus]] <math>E</math> instead of the <math>B</math> and are consequently slower than in infinite media.

Shear waves occur due to the shear stiffness and follows a similar equation, but with the shear deformation occurring in the transverse direction, perpendicular to the wave motion.

:<math> { \partial^2 w \over \partial x ^2 } = {1 \over c_s^2} { \partial^2 w \over \partial t ^2 } </math>

The shear wave speed is governed by the [[shear modulus]] <math>G</math> which is less than <math>E</math> and <math>B</math>, making shear waves slower than longitudinal waves.

===Bending Waves in beams and plates===

Most sound radiation is caused by bending (or flexural) waves, which deform the structure transversely as they propagate. Bending waves are more complicated than compressional or shear waves and depend on material properties as well as geometric properties. They are also [[dispersive]] since different frequencies travel as different speeds. For a thin beam then bending wave speed is defined as

:<math> { c_B } = { \left( { E I \omega^2 \over \rho A } \right)^{1 \over 4} } </math>

and the wave equation is fourth order in space. For a thin plate the bending wave speed is

:<math> { c_B } = { \left( { D \omega^2 \over \rho h } \right)^{1 \over 4} } </math>

where <math>{D}={E h^3 \over 12 ( 1 - \nu^2 ) }</math>.

===Modeling Vibrations===
[[Finite element analysis]] can be used to predict the vibration of complex structures. A finite element computer program will assemble the mass, stiffness, and damping matrices based on the element geometries and material properties, and solve for the vibration response based on the loads applied.

:<math> { [ -\omega^2 \bold{M} + j \omega \bold{B} + (1 + j \eta ) \bold{K} ] } { \bold{d} = \bold{F} } </math>

==Sound-Structure interaction<ref>{{cite |url=http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=ATCODK000003000002000009000001&idtype=cvips |title=STRUCTURAL ACOUSTICS TUTORIAL II, SOUND—STRUCTURE INTERACTION |accessdate=2010-08-09 |author=Stephen A. Hambric and John B. Fahnline, Applied Research Lab at The Pennsylvania State University}}</ref>==
===Fluid-structure Interaction===

When a vibrating structure is in contact with a fluid, the normal particle velocities at the interface must be conserved (i.e. be equivalent). This causes some of the energy from the structure to escape into the fluid, some of which radiates away as sound, some of which stays near the structure and does not radiate away.

===Piston Radiation===

A piston oscillating uniformly in a rigid baffle is the classic example to consider acoustic radiation. For a circular piston that has time harmonic motion, the pressure far away from the piston is found to be

:<math> { p (r, \theta ) } = { j \omega \rho_0 a^2 v_n { J_1 (k a \sin \theta) \over k a \sin \theta } { e^{ j k r } \over r } } </math>

where <math> v_n </math> is the piston velocity (assumed constant over the surface) and <math> J_1 </math> is the first order [[Bessel function]]. This is derived by integrating the far-field pressure contributions of tiny [[point sources]] over the area of the piston. The radiated sound power is related directly to the radiation resistance of the fluid and is calculated as

:<math> { P_{rad} } = { R_0 \left<|v|\right>^2 } </math>

where <math>\left<|v|\right></math> is the spatially and time averaged velocity.

===Structural Wave Radiation===

Since most structures do not vibrate uniformly (as in the case of the baffled piston), but vibrate according as combinations of flexural, compressional and shear waves. The radiation characteristics actually depend strongly on whether the bending wave speed is slower than the [[sound speed]] in the fluid or faster. For subsonic bending waves, the radiation is weak, while supersonic bending waves radiate efficiently.

== See also ==
{{col-begin|width=80%}}
{{col-break}}
*[[Acoustics]]
*[[Acoustic wave equation]]
*[[Lamb wave]]
*[[Linear elasticity]]
*[[Noise control]]
*[[Sound]]
*[[Surface acoustic wave]]
*[[Waves]]
*[[Wave equation]]
{{col-break}}
{{col-end}}

==References==
{{Reflist|2}}
{{cite book |title=Sound Structure Interaction |author=Fahy F., Gardonio P.|url=http://books.google.com/?id=7qCMUfwoQcAC&pg=PA61 |pages=60–61 |isbn=3-540-67458-6 |year=2007 |edition=2nd |publisher=Academic Press}}

==External links==
* [http://www.arl.psu.edu/capabilities/fsm.html?tab=3#TabbedPanels1 arl.psu.edu/structural_acoustics]—Website of the Penn State University's Structural Acoustics Group
* [http://asa.aip.org/ asa.aip.org]—Website of the [[Acoustical Society of America]]

[[Category:Acoustics]]

Calculus of moving surfaces - Revision history

en>Rjwilmsi: Format plain DOIs using AWB (8087)