# Guided (channel) waves and normal-mode propagation

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 13 485 - 496 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 13.3a

In Figure 13.3a the first arrival (${\displaystyle f_{o}=40\ {\rm {Hz}}}$) has traveled at the velocity 2.7 km/s; find the water depth.

### Background

A wave guide is a layer in which a wave can propagate with little loss of energy. In a water layer nearly total reflection occurs at both boundaries, at the surface because of the very large impedance contrast and at the bottom reflection beyond the critical angle. The phase is inverted at the surface because the reflectivity is nearly ${\displaystyle -1}$, but not at the seafloor (beyond the critical angle) until the angle becomes very large.

Figure 13.3b(i) shows waves bouncing back and forth in a wave guide. For certain incident angles ${\displaystyle \theta }$ and frequencies ${\displaystyle f}$, they interfere constructively. In Figure 13.3b(ii), ${\displaystyle AC}$ is a wavefront traveling upward at the angle ${\displaystyle \theta }$. The previous cycle of a parallel wavefront that passed through the position earlier followed paths such as EFGH and BDA and thus coincides with ${\displaystyle AC}$. Clearly ${\displaystyle (EF+FG+GH)=BD+DA}$. ${\displaystyle DA=h/\cos \theta }$, ${\displaystyle BD=DA\cos 2\theta }$, so ${\displaystyle (BD+DA)=2h/\cos \theta }$. Taking into account the phase reversal at the surface the condition for constructive interference is

{\displaystyle {\begin{aligned}(BD+DA)=2h\cos \theta =(2n+1)\lambda /2,\end{aligned}}}

 {\displaystyle {\begin{aligned}{\mbox{so}}\qquad \qquad h=(2n+1)\lambda /4\cos \theta .\end{aligned}}} (13.3a)

Writing ${\displaystyle V}$ for the phase velocity, the frequencies that are reinforced are

 {\displaystyle {\begin{aligned}f_{n}=V/\lambda =(2n+1)V/4h\cos \theta ,\quad n=0,1,2,\ldots .\end{aligned}}} (13.3b)

In addition to the upgoing waves parallel to ${\displaystyle AC}$ and ${\displaystyle A^{\prime }C^{\prime }}$ in Figure 13.3b(iii), a second downgoing set ${\displaystyle PQ}$ and ${\displaystyle P^{\prime }Q^{\prime }}$ will combine with the set ${\displaystyle AC}$ to build up the energy traveling in the direction ${\displaystyle RR^{\prime }}$ along the wave guide. Energy travels from ${\displaystyle R}$ to ${\displaystyle R^{\prime }}$ in the time that wavefront ${\displaystyle AC}$ moves to ${\displaystyle A^{\prime }C^{\prime }}$, so the phase velocity of the energy traveling along ${\displaystyle RR^{\prime }}$, ${\displaystyle V_{g}}$, is

 {\displaystyle {\begin{aligned}V_{g}=V/\sin \theta .\end{aligned}}} (13.3c)
Figure 13.3a.  Wave-layer channel wave, source at 4 km (after Clay and Medwin, 1977).
Figure 13.3b.  (i) Raypaths in a wave guide; (ii) showing reinforcement of reflection wavefronts; (iii) relation between phase and group velocities.

Since both ${\displaystyle V_{g}}$ and ${\displaystyle f}$ are functions of ${\displaystyle \theta }$, ${\displaystyle V_{g}}$ is dispersive with a group velocity ${\displaystyle U}$ given by equation (2.7a):

 {\displaystyle {\begin{aligned}U=V_{g}+f{\frac {{\rm {d}}V_{g}}{{\rm {d}}f}}.\end{aligned}}} (13.3d)

The derivative is always negative for a water channel wave (Figure 13.3c), so ${\displaystyle U.

If the wave-guide effect had been due to reflection beyond the critical angle at both boundaries, as with a low-velocity coal seam, the phase would not have been reversed at either boundary and equations (13.3a,b) would reduce to

 {\displaystyle {\begin{aligned}h=n\lambda /2\cos \theta ,\quad f_{n}=nV/2h\cos \theta .\end{aligned}}} (13.3e)

### Solution

Equation (13.3c) gives

{\displaystyle {\begin{aligned}\sin \theta =V/V_{g}=1.5/2.7,\quad \theta =33.7^{\circ },\end{aligned}}}

and we use equations (13.3b,e) with ${\displaystyle n=0}$ to get ${\displaystyle h}$:

{\displaystyle {\begin{aligned}h=V/4f_{0}\cos \theta =1500/4\times 40\cos 33.7^{\circ }=11.3\ {\rm {m}}.\end{aligned}}}

Figure 13.3c.  Phase and group velocities ${\displaystyle (V,U)}$ versus normalized frequency where ${\displaystyle \alpha _{2}/\alpha _{1}=2/3}$, ${\displaystyle \sigma _{1}=0.5}$, ${\displaystyle \sigma _{2}=0.25}$, and ${\displaystyle \rho _{2}/\rho _{1}=2.5}$ (from Ewing, Jardetsky, and Press, 1957).

## Problem 13.3b

What frequency is reinforced when ${\displaystyle \theta =40^{\circ }}$?

### Solution

We use equation (13.3b) with ${\displaystyle n=0}$, ${\displaystyle h=11.3\ {\rm {m}}}$, ${\displaystyle \theta =40^{\circ }}$:

{\displaystyle {\begin{aligned}f=V/4h\cos \theta =1500/4\times 11.3\cos 40^{\circ }=43.3\ {\rm {Hz}}.\end{aligned}}}

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