**Figure:** Schematic of the isovelocity problem.

The principal numerical problem is to solve for the normal modes
corresponding to Eq. (). The sound speed
profile, assumes a fairly arbitrary form so simple analytical
techniques are generally not useful. On the other hand, it is
instructive to consider some simple profiles in order to understand
the qualitative features of modal problems. The simplest such case is
the * isovelocity* profile with unit density as shown in Fig. 2.2.
The general solution is

where

The surface boundary condition implies that **B=0** while the bottom
boundary condition leads to:

where **D** is the depth of the bottom.
Thus, either **A=0** (the trivial solution) or we must have

that is, **k** must assume particular values,

The corresponding eigenfunctions are given by,

where we have chosen the constant **A** so that the modes have unit
norm as specified in Eq. ().

Equation (), which relates the frequency to
the wavenumber , is known as the * dispersion relation*.
Plots of versus are in turn called the dispersion
curves. The quantities and
are respectively the * phase
velocity* and the * group velocity* of the **m**th mode. The group
velocity is associated with the radial speed of propagation for a pulse.

The eigenvalues divide into two classes corresponding to *
propagating* and * evanescent* modes depending on whether the
argument of the square root in Eq. () is positive or
negative. In either case, the square root admits two values and
. The positions of these eigenvalues are indicated
schematically in Fig. 2.3 by circles. (Their precise positions depend
on the frequency, depth and sound speed.)

For the propagating modes we select the branch which gives an outgoing wave. Since we have suppressed a time dependence of the form we should take the positive value for . These eigenvalues are indicated by the filled circles lying on the positive real axis in Fig. 2.3.

For the evanescent modes we have to choose between roots of the form
**i a** and **-i a** where **a** is a positive real number. These modes have
the property of either growing or decaying in range. In order to have
a bounded solution we take the branch for which lies in the
upper half-plane, i.e. with **a** positive. These
eigenvalues are indicated by the filled circles lying on the positive
imaginary axis in Fig. 2.3.

The real eigenvalues have an upper bound . As we reduce
the frequency, the eigenvalues on the real axis slide to the left and
up the imaginary axis. At a sufficiently low frequency the first
mode will make the transition leaving no propagating modes. The
frequency at which this occurs is called the * cut-off* frequency
for the waveguide.

**Figure:** Location of eigenvalues for the isovelocity problem.

As a concrete example, consider the isovelocity problem with sound
speed , depth , and source frequency
. Selected modes are plotted in Fig. 2.4. Note that
the **m**th mode has **m** zeros.

Substituting the formula for the isovelocity modes given in Eq. () into Eq. () we obtain a representation of the pressure field:

Similarly, from Eq. () we obtain a representation for the
transmission loss as where **I** is an intensity
defined by

In Fig. 2.5 we display the transmission loss for this problem keeping 1, 2 and 3 modes respectively in the modal sum. The source depth is and the receiver depth is in these calculations. Note that as we increase the number of modes the detail in the TL curves also increases. This can be understood by writing the intensity as

where,

and

With just one mode in the series, the complex pressure involves an oscillatory term of the form , however, its envelope (the intensity) is smooth as indicated in Fig. 2.5. With two modes in the series the intensity is seen to include a term giving the two-mode interference pattern in Fig. 2.5(b). Note that the interference pattern occurs over a scale significantly larger than the wavelength. Finally, with 3 modes the interference structure shows a further increase in complexity as shown in Fig. 2.5(c).

**Figure:** Transmission loss for the isovelocity problem using (a) 1
mode, (b) 2 modes and (c) 3 modes.

Many of the properties we see for the isovelocity profile will carry through to more general profiles. On the other hand, while it may still be useful to speak of propagating and evanescent modes, the distinction is blurred when attenuation is included for then all of the modes are displaced into the first quadrant and so all the modes have both a propagating and an evanescent component. Similarly, the cut-off frequency is poorly defined in such cases. These points will be made clearer as we consider more complicated cases.

Tue Oct 28 13:27:38 PST 1997