Ray–mode analogy: Consider a isovelocity waveguide bounded above and below by pressure-release surfaces.

a. Draw a diagram (see Fig. 2.20) with a “ray” reflecting with phase change, first from the bottom, and then from the surface. Construct a wave front perpendicular to this ray such that it intersects both the ray when it is incident on the bottom and after it is reflected from the surface. What is the condition for angle and frequency that this wavefront be the result of perfect constructive interference?

b. What are the normal modes and eigenvalues of a waveguide with the above boundary conditions? (Note that Sect. 5.4 discusses the rigid bottom case).

c. Compare the two results.

d. Now assume that the bottom is a fluid and consider a ray more grazing than critical. It will be perfectly reflected but will undergo a phase change at the bottom given by the results in problem 1.5. What is the condition for perfect constructive interference. Compare this result with (5.81).

e. Which is a better approximation of a shallow water environment: a waveguide with a rigid or pressure-release b