Biot–Tolstoy–Medwin diffraction model explained

In applied mathematics, the Biot–Tolstoy–Medwin (BTM) diffraction model describes edge diffraction. 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

p(t)=

infty
\int
0

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

(rS,\thetaS,zS)

where the

z

-axis is considered to lie on the edge and

\theta

is measured from one of the faces of the wedge.

(rR,\thetaR,zR)

\thetaW

and from this the wedge index

\nu=\pi/\thetaW

c

as an integral over edge positions

z

h(\tau)=-

\nu
4\pi
\sum
\phii=\pi\pm\thetaS\pm\thetaR
z2
\int
z1

\delta\left(\tau-

m+l
c

\right)

\betai
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

\delta

is the Dirac delta function.

\betai=

\sin(\nu\phii)
\cosh(\nuη)-\cos(\nu\phii)

where

η=\cosh-1

ml+(z-zS)(z-zR)
rSrR

See also

References

Notes and References

  1. Calamia 2007, p. 182.
  2. Calamia 2007, p. 183.