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4.7  Rayleigh integral computations

• Summary of theory
• Diffuse field and transmission loss

4.7.1  Summary of theory

The Rayleigh integral provides an approximation of the pressure at point M as

(4.14)

with R(x,M)=|x−M | the distance between emission and target, k=iω/c₀ the wave number, ρ₀ the fluid density, c₀ the fluid celerity.

This integral is implemented in fsc and also in fe2xf frfzr in an optimized form.

4.7.2  Diffuse field and transmission loss

Sound transmission loss corresponds to the ratio between incident and radiated power in a panel.

The incident power is associated with acoustic loading assumed to be due to a diffuse field. This loading is classically estimated using the power spectral density at integration points of the excited surface

(4.15)

where w(g_A)J(g_A) is the surface associated with the the integration rule at Gauss point A, k=ω/c the acoustic wavenumber, γ_AB=sin(k A−B)/kA−B the spatial correlation of pressures between two points, S_ref(ω) the blocking pressure. [] also addresses grazing incidences. The function sin(x)/x is also called sinc in the code.

When using a reduced model, where a kinematic description of motion is of the form {q}=[T]{q_R}, it is more efficient to use the projection of the PSD defined in surface nodes onto the generalized coordinates q_R. Thus

(4.16)

xxx order of operations xxx

An alternative to the definition of a diffuse field through a spectral density matrix, is the use a sum of plane waves, each defined by an origin M, a direction {d_M} and an amplitude a_M(ω). The pressure applied on any point of structure subjected to this wave is given by

(4.17)

From a series of sources, and a loaded surface described by its Gauss points, the computation of the pressure load is done as

(4.18)

In order to compute the radiation response, the next step is to compute the transfer from pressure forces to responses. xxx

Radiation from a surface into a cavity