Preliminary notes for acoustic waves:

c = speed of sound

f = frequency

k = wavenumber

lambda = wavelength

r = radius

wavelength is equal to the speed of sound divided by frequency

lambda = c / f

wavenumber is equal to two pi divided by wavelength

k = 2 pi / lambda

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Consider an acoustic pressure field acting on a structural surface, resulting in structural vibration.

The acoustic field will have some spatial variation, where the phase varies with wavenumber and location. This is a modeling challenge.

A typical analysis method is to divide the surface into an array of patches. Then the pressure amplitude is varied by patch according to the analyst’s spatial correlation assumption.

The patch density must allow for at least four patches per the acoustical wavelength of interest, as a rule-of-thumb per my colleagues Bruce LaVerde and Paul Blelloch.

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Now consider a diffuse acoustic field (DAF) where the sound waves arrive at a point from all directions.

The spatial cross-correlation for this case is typically assumed as the sinc function:

sin(kr)/kr

As an example, the sinc functions is used in NASA/TM—2008-215167, Initial Assessment of the Ares I–X Launch Vehicle Upper Stage to Vibroacoustic Flight Environments

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The spatial cross-correlation for a turbulent boundary layer (TBL) is more complicated. There are also separate coefficients for “Along Flow” and “Across Flow.

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Space Engineering: Mechanical shock design and verification handbook, ECSS-E-HB-32-25A, 2015.

This handbook gives a recommendation for shock finite element analysis, including models with propagating bending waves.

*Section 9.2.4*

*It is generally considered that, for a mesh definition, a wavelength should be approximated by a minimum of 4 to 6 elements, in order to avoid numerical filtering or reflection of incident waves but the use of a mesh with at least 8 elements per wavelength is strongly recommended.*

*Section 9.4.3*

*Use a sufficient number of elements to describe the expected response of an FE subsystem. A good rule of thumb is to use between 6 and 10 linear elements per propagating wavelength within an FE subsystem. In addition, you should try to ensure that you can capture the local evanescent fields surrounding any discontinuities or junctions. Additional mesh density criteria also apply when resolving curved components using linear elements. *

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An evanescent field is an oscillating field that does not propagate as a wave but whose energy is spatially concentrated in the vicinity of the source

– Tom Irvine