Computational Acoustics by Bergman David R

Author:Bergman, David R.
Language: eng
Format: epub
ISBN: 9781119277279
Publisher: John Wiley & Sons, Inc.
Published: 2018-02-27T00:00:00+00:00

7.2 High Frequency Expansion of the Wave Equation

Classic treatments of ray theory are usually restricted to the Helmholtz equation with a position‐dependent refractive index. Since so much effort was invested in developing a wave equation for more generic circumstances, the derivation will go as far as possible without neglecting any terms. The only simplifying assumption made here is that the wave equation takes the following approximate form:

(7.1)

The acoustic field is assumed to have the following form:

(7.2)

The amplitude, A, and phase, ϕ, are functions of position and time as well as frequency. This ansatz is inserted into (7.1), and the real and imaginary parts separated with an overall phase are factored out:

(7.3)

The phase depends linearly on frequency, and the amplitude is expanded in an infinite series in inverse powers of frequency:

(7.4)

(7.5)

Inserting (7.5) and (7.4) into (7.3) leads to an infinite series in the parameter :

(7.6)

The coefficients of each term must independently vanish, leading to a hierarchy of equations for the fields {An}. The lowest‐order equation, that is, the coefficient of ω2, in the set is called the eikonal equation:

(7.7)

The reader will recognize this as the characteristic equation. The next term in the series arises from the coefficient of ω1 and is referred to as the transport equation:

(7.8)

For all other powers of , we get a series of equations relating the amplitudes of order n and n + 1:

(7.9)

Equation (7.8) states that the covariant divergence of some vector field in space‐time is zero:

(7.10)

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