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In [[applied mathematics]], the '''Biot–Tolstoy–Medwin (BTM) diffraction model''' describes [[diffraction]] from a rigid edge. Unlike the [[uniform theory of diffraction]] (UTD), BTM does not make the high frequency assumption (in which edge lengths and distances from source and receiver are much larger than the wavelength). BTM sees use in acoustic simulations.<ref>Calamia 2007, p. 182.</ref> | |||
== Impulse response == | |||
The [[impulse response]] according to BTM is given as follows:<ref>Calamia 2007, p. 183.</ref> | |||
The general expression for [[sound pressure]] is given by the [[convolution]] integral | |||
: <math> | |||
p(t) = \int_0^\infty h(\tau) q (t - \tau) \, d \tau | |||
</math> | |||
where <math>q(t)</math> represents the source signal, and <math>h(t)</math> represents the impulse response at the receiver position. The BTM gives the latter in terms of | |||
* the source position in cylindrical coordinates <math>( r_S, \theta_S, z_S )</math> where the <math>z</math>-axis is considered to lie on the edge and <math>\theta</math> is measured from one of the faces of the wedge. | |||
* the receiver position <math>( r_R, \theta_R, z_R )</math> | |||
* the (outer) wedge angle <math>\theta_W</math> and from this the wedge index <math>\nu = \pi / \theta_W</math> | |||
* the speed of sound <math>c</math> | |||
as an integral over edge positions <math>z</math> | |||
: <math> | |||
h(\tau) = -\frac{\nu}{4\pi} \sum_{\phi_i = \pi \pm \phi_S \pm \phi_R} \int_{z_1}^{z_2} \delta\left(\tau - \frac{m+l}{c}\right) \frac{\beta_i}{ml} \, dz | |||
</math> | |||
where the summation is over the four possible choices of the two signs, <math>m</math> and <math>l</math> are the distances from the point <math>z</math> to the source and receiver respectively, and <math>\delta</math> is the [[Dirac delta function]]. | |||
: <math> | |||
\beta_i = \frac{\sin (\nu \phi_i)}{\cosh(\nu \eta) - \cos(\nu \phi_i)} | |||
</math> | |||
where | |||
: <math> | |||
\eta = \cosh^{-1} \frac{ml + (z - z_S)(z - z_R)}{r_S r_R} | |||
</math> | |||
== See also == | |||
* [[Uniform theory of diffraction]] | |||
== Notes == | |||
{{reflist}} | |||
== References == | |||
* 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}} | |||
Revision as of 17:46, 7 March 2013
In applied mathematics, the Biot–Tolstoy–Medwin (BTM) diffraction model describes diffraction from a rigid edge. Unlike the uniform theory of diffraction (UTD), BTM does not make the high frequency assumption (in which edge lengths and distances from source and receiver are much larger than the wavelength). BTM sees use in acoustic simulations.[1]
Impulse response
The impulse response according to BTM is given as follows:[2]
The general expression for sound pressure is given by the convolution integral
where represents the source signal, and represents the impulse response at the receiver position. The BTM gives the latter in terms of
- the source position in cylindrical coordinates where the -axis is considered to lie on the edge and is measured from one of the faces of the wedge.
- the receiver position
- the (outer) wedge angle and from this the wedge index
- the speed of sound
as an integral over edge positions
where the summation is over the four possible choices of the two signs, and are the distances from the point to the source and receiver respectively, and is the Dirac delta function.
where
See also
Notes
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References
- 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.