## Info

$ = F(r,d,z) G(r)

Range-dependent (3D)

Range-dependent (3D)

> Ray theory

- Extended to range dependence(2D/3D)

• Multipath expansion

> Parobolic equation

F ^ Amplitude function P ^ Phase function

- Extended to range dependence(2D/3D)

• Multipath expansion

F ^ Normal mode equation ^ Green's function G ^ Bessel equation

^ Hankel function

> Parobolic equation

F ^ Parabolic equation G ^ Bessel equation

^ Hankel function

Figure 4.1 Summary of relationships among theoretical approaches for propagation modeling (adapted from Jensen and Krol, 1975).

### 4.3 Ray-theory models 4.3.1 Basic theory

Ray-theoretical models calculate TL on the basis of ray tracing (National Defense Research Committee, 1946). Ray theory starts with the Helmholtz equation. The solution for $ is assumed to be the product of a pressure amplitude function A = A(x,y, z) and a phase function P = P(x,y, z) : $ = A elP. The phase function (P) is commonly referred to as the eikonal, a Greek word meaning "image." Substituting this solution into the Helmholtz equation (4.3a) and separating real and imaginary terms yields:

Equation (4.4) contains the real terms and defines the geometry of the rays. Equation (4.5), also known as the transport equation, contains the imaginary terms and determines the wave amplitudes. The separation of functions is performed under the assumption that the amplitude varies more slowly with position than does the phase (geometrical acoustics approximation). The geometrical acoustics approximation is a condition in which the fractional change in the sound-speed gradient over a wavelength is small compared to the gradient c/k, where c is the speed of sound and k is the acoustic wavelength. Specifically 1

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