5.1.1. General Solution to the Wave Equation

The polar coordinates (r, ?) and the coordinate z constitute the coordinate system. The corresponding unit vectors are respectively denoted , and .

Figure 5.1: Cylindrical coordinate system

The usual operators take the following forms:

(5.1)

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

Away from any source, the acoustic pressure satisfies the following equation of propagation:

(5.7)

The solutions to this equation are assumed to be separable and in the form

(5.8)

By using the same approach as in section 4.2.2, equations (4.24) to (4.31) and, applying the same logic, the substitution of solution (5.8) into equation (5.7) leads consecutively to

(5.9)

(5.10)

and finally to

(5.11)

and

(5.12)

where

(5.13)

is the associated equation of dispersion .

The in-plane "wavenumber component" defined by the polar coordinates and denoted here as k _w is independent of the variable r and is given by

(5.14)

where the three "components" k _r, k _? and k _z of the wavenumber are always functions of the quantum number m (unlike the wavenumber itself).

By...