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Geometrical acoustics

The below discussion is from Landau and Lifshitz.^[2] If the amplitude and the direction of propagation varies slowly over the distances of wavelength, then an arbitrary sound wave can be approximated locally as a plane wave. In this case, the velocity potential can be written as

${\displaystyle \phi =\mathrm {e} ^{\mathrm {i} \psi }}$

For plane wave ${\displaystyle \psi ={\boldsymbol {k}}\cdot {\boldsymbol {r}}-\omega t+\alpha }$, where ${\displaystyle {\boldsymbol {k}}}$ is a constant wavenumber vector, ${\displaystyle \omega }$ is a constant frequency, ${\displaystyle {\boldsymbol {r}}}$ is the radius vector, ${\displaystyle t}$ is the time and ${\displaystyle \alpha }$ is some arbitrary complex constant. The function ${\displaystyle \psi }$ is called the eikonal. We expect the eikonal to vary slowly with coordinates and time consistent with the approximation, then in that case, a Taylor series expansion provides

${\displaystyle \psi =\psi _{o}+{\boldsymbol {r}}\cdot {\frac {\partial \psi }{\partial {\boldsymbol {r}}}}+t{\frac {\partial \psi }{\partial t}}.}$

Equating the two terms for ${\displaystyle \psi }$, one finds

${\displaystyle {\boldsymbol {k}}={\frac {\partial \psi }{\partial {\boldsymbol {r}}}},\quad \omega =-{\frac {\partial \psi }{\partial t}}}$

For sound waves, the relation ${\displaystyle \omega ^{2}=c^{2}k^{2}}$ holds, where ${\displaystyle c}$ is the speed of sound and ${\displaystyle k}$ is the magnitude of the wavenumber vector. Therefore, the eikonal satisfies a first order nonlinear partial differential equation,

${\displaystyle \left({\frac {\partial \psi }{\partial x}}\right)^{2}+\left({\frac {\partial \psi }{\partial y}}\right)^{2}+\left({\frac {\partial \psi }{\partial z}}\right)^{2}-{\frac {1}{c^{2}}}\left({\frac {\partial \psi }{\partial t}}\right)^{2}=0.}$

where ${\displaystyle c}$ can be a function of coordinates if the fluid is not homogeneous. The above equation is same as Hamilton–Jacobi equation where the eikonal can be considered as the action. Since Hamilton–Jacobi equation is equivalent to Hamilton's equations, by analogy, one finds that

${\displaystyle {\frac {\mathrm {d} {\boldsymbol {k}}}{\mathrm {d} t}}=-{\frac {\partial \omega }{\partial {\boldsymbol {r}}}},\quad {\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}={\frac {\partial \omega }{\partial {\boldsymbol {k}}}}}$

Practical applications of the methods of geometrical acoustics can be found in very different areas of acoustics. For example, in architectural acoustics the rectilinear trajectories of sound rays make it possible to determine reverberation time in a very simple way. The operation of fathometers and hydrolocators is based on measurements of the time required for sound rays to travel to a reflecting object and back. The ray concept is used in designing sound focusing systems. Also, the approximate theory of sound propagation in inhomogeneous media (such as the ocean and the atmosphere) has been developed largely on the basis of the laws of geometrical acoustics.^[3][4]

The methods of geometrical acoustics have a limited range of applicability because the ray concept itself is only valid for those cases where the amplitude and direction of a wave undergo little changes over distances of the order of wavelength of a sound wave. More specifically, it is necessary that the dimensions of the rooms or obstacles in the sound path should be much greater than the wavelength. If the characteristic dimensions for a given problem become comparable to the wavelength, then wave diffraction begins to play an important part, and this is not covered by geometric acoustics.^[1]

The concept of geometrical acoustics is widely used in software applications. Some software applications that use geometrical acoustics for their calculations are ODEON, Enhanced Acoustic Simulator for Engineers, Olive Tree Lab Terrain, and COMSOL Multiphysics.

• ODEON Room Acoustics Software
• EASE – Industry Standard for Acoustical Simulation of Rooms
• Olive Tree Lab Terrain