In
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, the
Biot
The British Indian Ocean Territory (BIOT) is an Overseas Territory of the United Kingdom situated in the Indian Ocean, halfway between Tanzania and Indonesia. The territory comprises the seven atolls of the Chagos Archipelago with over 1,000 ...
–Tolstoy–Medwin (BTM) diffraction model describes
edge diffraction. Unlike the
uniform theory of diffraction In numerical analysis, the uniform theory of diffraction (UTD) is a high-frequency method for solving electromagnetic scattering problems from electrically small discontinuities or discontinuities in more than one dimension at the same point. UTD i ...
(UTD), BTM does not make the
high frequency
High frequency (HF) is the ITU designation for the range of radio frequency electromagnetic waves (radio waves) between 3 and 30 megahertz (MHz). It is also known as the decameter band or decameter wave as its wavelengths range from one to ten ...
assumption (in which edge lengths and distances from source and receiver are much larger than the wavelength). BTM sees use in acoustic simulations.
Impulse response
The
impulse response
In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...
according to BTM is given as follows:
[Calamia 2007, p. 183.]
The general expression for
sound pressure
Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average or equilibrium) atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone ...
is given by the
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
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
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
.
:
where
:
See also
*
Uniform theory of diffraction In numerical analysis, the uniform theory of diffraction (UTD) is a high-frequency method for solving electromagnetic scattering problems from electrically small discontinuities or discontinuities in more than one dimension at the same point. UTD i ...
Notes
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.
Signal processing
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