Biot–Tolstoy–Medwin diffraction model

The impulse response according to BTM is given as follows:Calamia 2007, p. 183.

The general expression for sound pressure is given by the convolution integral

: $$

p(t) = \int_0^\infty h(\tau) q (t - \tau) \, d \tau

where $q(t)$ represents the source signal, and $h(t)$ represents the impulse response at the receiver position. The BTM gives the latter in terms of

• the source position in cylindrical coordinates $(\; r\_S,\; \backslash theta\_S,\; z\_S\; )$ where the $z$-axis is considered to lie on the edge and $\backslash theta$ is measured from one of the faces of the wedge.
• the receiver position $(\; r\_R,\; \backslash theta\_R,\; z\_R\; )$
• the (outer) wedge angle $\backslash theta\_W$ and from this the wedge index $\backslash nu\; =\; \backslash pi\; /\; \backslash theta\_W$
• the speed of sound $c$

as an integral over edge positions $z$

: $$

h(\tau) = -\frac{\nu}{4\pi} \sum_{\phi_i = \pi \pm \theta_S \pm \theta_R} \int_{z_1}^{z_2} \delta\left(\tau - \frac{m+l}{c}\right) \frac{\beta_i}{ml} \, dz

where the summation is over the four possible choices of the two signs, $m$ and $l$ are the distances from the point $z$ to the source and receiver respectively, and $\backslash delta$ is the Dirac delta function.

: $$

\beta_i = \frac{\sin (\nu \phi_i)}{\cosh(\nu \eta) - \cos(\nu \phi_i)}

where

: $$

\eta = \cosh^{-1} \frac{ml + (z - z_S)(z - z_R)}{r_S r_R}

• Uniform theory of diffraction

{{reflist}}

• Calamia, Paul T. and Svensson, U. Peter, "Fast time-domain edge-diffraction calculations for interactive acoustic simulations," EURASIP Journal on Advances in Signal Processing, Volume 2007, Article ID 63560.

{{DEFAULTSORT:Biot-Tolstoy-Medwin diffraction model}}

Category:Signal processing

{{signal-processing-stub}}