4.3 Separation of Variables

Separation of variables is a method designed to reduce the full homogeneous PDE into a set of ordinary differential equations. The goal is to reduce the multidimensional differential equation into several one-dimensional differential equations. The original equation is expressed as

This method requires the equation to be linear but the coefficients may be function of position and time. It is assumed that the field can be factored, such that each factor depends on only one coordinate:

The partial derivatives of the original function are proportional to ordinary derivatives:

Expressions similar to (4.42) exist for the other derivatives and the reader can derive expressions for the mixed partials. After applying differential operators to (4.40), a factor of ψ is divided through. For example,

If the resulting equation can be grouped into additive factors, each dependent on a single variable, then each factor can be set equal to a constant:

This section provides a review of the three most common separable coordinates: Cartesian, cylindrical, and spherical [5].

4.4.1 Cartesian Coordinates

When the speed of sound is constant, the function U(xk) in (4.54) is a constant, and the factor ω2c2 = k2 can be grouped with any one of the three derivative terms. In anticipation of defining the wavevector, the separation constants for X and Y are defined as − kx2 and − ky2, respectively. Combining this with the equation for Z leads to the following set of separated equations:

In (4.58) the separation constants for X and Y have been used to replace terms with x and y dependence in the original equation. Identifying the term in parenthesis in (4.58) with kz2, the constraint equation, (4.49), becomes

Equations (4.56)–(4.58) are recognized as the equation for a simple harmonic oscillator. The two independent solutions for each can be expressed in terms of sine and cosine or complex exponentials. The latter is presented here as

Each solution represents a sinusoidal wave traveling along one of the Cartesian axes. Combing the results into a complete solution leads to the familiar expression for a plane wave solution:

The previous expression also reinforces a standard convention that waves are traveling in the direction of k . This is a good place to mention conventions. The choice of sign on the time exponential is completely arbitrary. Here, the choice of minus sign is made. With this comes

For a constant sound speed, the equation for Z can be solved exactly:

For free waves, α < k so this solution is a plane wave traveling in the z direction. The solution to (4.72) is

These are traveling waves in the ± φ directions. For these waves to be continuous, periodic boundary conditions are applied, Φφ = Φφ + 2π:

To satisfy (4.76), the separation constant is restricted to be an integer, β = n = 0 1,…. The remaining equation for R is

The second derivative has been expanded and the equation multiplied through by ρ2R. Defining a new variable, ξ = αρ, this equation becomes Bessel’s equation:

This equation has two independent solutions for each n, referred to as Bessel’s function and Neumann’s function. Each is defined by an infinite series:

(4.80)

In (4.80) ψ(x) is the psi function [7]. Linear combinations of these functions define incoming and outgoing traveling waves propagating along the ρ direction:

The function Hn1 represents outward traveling waves in the physics convention. To illustrate this, consider the limiting behavior of these functions for large values of ξ:

A complete solution can be built up from these solutions. The solution for n = 0, and α = 0 gives 2-dim Green’s function, αH01,2ξ, where α is a constant. Following the same convention as for plane waves, the outgoing solution is Hn1, while the incoming solution is Hn2. The former can be thought of as circular waves emanating from the origin and traveling out to infinity, while the latter travels toward the origin from infinity.

4.4.3 Spherical Coordinates

Spherical coordinates r, θ, φ are given by the following transformation:

The Laplacian operator in spherical coordinates has the following form:

A solution of the form ψ = RrΘθΦφ is assumed and used to separate the wave equation: