6

Normal Modes

6.1 Introduction

In many practical situations, the refractive properties of the medium depend on one coordinate. In such cases the wave equation can be separated in a suitable coordinate system, and the solution described by a factor that represents a propagating wave in two dimensions and time and a factor that is a solution to a one-dimensional ordinary differential equation containing the index of refraction. This equation is similar in form to Schrödinger’s time-independent eigenvalue equation for the wave function of a particle in a potential well. The set of solutions to this equation are referred to as normal modes and can be used to build up a complete solution to the acoustic wave equation for a collection of point sources embedded in the medium. Due to the nature of the potential well and possible boundary conditions, the spectrum of these modes may be continuous or discrete. There exist a handful of example refractive index functions for which exact solutions to this equation may be found in terms of special functions. More general cases require approximation or numerical methods. This chapter introduces normal mode theory applied to the second-order wave equation for the pressure field with a depth-dependent refractive index. Some cases leading to exact solutions are presented to illustrate the general features of the modes. To solve more complex problems, this chapter presents perturbation theory and applies it to several problems with boundary conditions. Afterward the relaxation technique is presented and discussed as a numerical method for solving the eigenvalue problem with boundary conditions and nontrivial refractive index. The final section contains a brief description of the theory behind coupled modes and their use in extending mode theory to more general environments.

6.2 Mode Theory

Consider an environment described by c = cz, υ0 = 0, and ρ0 = constant. The pressure field obeys the wave equation (6.1):

Applying separation of variables in x , φx = XxYyZz, leads to a set of ordinary differential equations (ODE) for each factor:

The separation constants kx2 and ky2 have been introduced, having the physical interpretation of wavevector components in the x and y directions, respectively. The transverse wavenumber, kT = kx ky 0T , kT2 kx2 + ky2, appears in (6.5). The solution for the transverse propagation is given by a plane wave:

When c = constant, the term in brackets in (6.5) is identified as the wavenumber kz2, and the separation constants obey the dispersion relation

A complete solution is a plane wave traveling in the k direction. In the general case a position-dependent wavenumber is defined:

With this definition (6.5) is written in compact form:

Solutions to (6.9) may have bound (decaying) or unbound (oscillatory) behavior depending on the sign of kz2. Since this is no longer constant, there may be locations where the behavior changes from one type to the other. The zeros of kz2 will be determined by the specific nature of the sound speed function c(z). Consider a single transition point at some arbitrary value, z = z, for which kz2z = 0. From (6.9) it follows that this is a point of inflection of the function Z. For values of z such that kz2 > 0, the solution will have oscillatory behavior, and for kz2 < 0 it will have exponential behavior.

6.3 Profile Models

Physical models of refraction in underwater acoustics frequently involve mathematical models of speed profiles that tend to infinity at z ∞. Simple models with this behavior include the following:

A specific example from underwater acoustics is the SOFAR channel described by the Garrett–Munk profile:

The variable η = 2z− z0z0 is defined in (6.13). This model has free parameters, namely, c0, ε, and z0. Typical values found in the literature are c0 = 1500 m/s, ε = 0.00737, and z0 = 1300 m. For the profile presented in (6.13), the positive z-axis is directed downward, a common definition used in underwater acoustics.

For very large values of |z|, all of these profiles approach infinity and the modes vanish. At these extreme limits, sound is trapped in a waveguide-like structure due solely to the refractive nature of the sound speed. At the other extreme one often encounters the purely mathematical situation where cz = 0 at some value, z. In this case kz2 ∞ and the field must vanish as z z. This is a mathematical artifact of certain models and does not represent anything physical. An example with this behavior is the linear sound speed profile:

for z < − c0ε.

It is possible to find examples of c(z) for which (6.9) becomes one of the well-known equations defining a special function. Consider a physical model where the sound speed profile can be written as a constant, c0, plus a function of position that is small relative to c0, at least in some interval z z1,z2:

Under this assumption (6.8) is approximated by a first-order Taylor expansion in czc0:

Using (6.16) and (6.17) in (6.9) and a little rearranging gives the following form for the vertical mode equation:

The “potential energy” function, U(z), and energy, Ez, appearing in (6.18) are defined as

The special form of (6.18) is reminiscent of Schrödinger’s time-independent equation. It should be pointed out that the expansion in (6.16) is not necessary to massage (6.9) into this form. The identification of U and E with potential and energy level is for illustrative purposes, the quantities do not necessarily have units of energy.

A reasonable question is whether or not the modes predicted using (6.16) will describe solutions to the original form, (6.8). It is clear that in general the two sets will not match since at some point the condition cz c0 may be violated. This is not a concern if ideal boundary conditions are applied in the interval [z1, z2] where the approximation holds. To better under-

Comparing (6.21) with (6.18) with (6.16) serving as the potential, it is clear that the differences (and similarities) can be understood by comparing the following two effective potentials:

Figure 6.2 Effect potentials for the linear sound speed profile at various strengths