Using the procedure to derive generalized invariants, derive the corresponding invariant quantities in which range can be taken as the independent variable. In particular, show that  Next, show that the ideal waveguide trajectory of constant-intensity interference

maxima along a horizontal array (with elements at r) for a changing sound speed c would be described by

a. Derive a parameter representation for the ray path before the first surface bounce,

Where  is the local grazing angle for the ray.

b. Show that the ray path describes a circular arc and that the center of the circle falls at a depth z D zc, where zc is independent of the launch angle _0.

c. Derive the expression for the range r1._0/, where the ray launched at angle _0 bounces off the sea surface.

d. Derive the parameter representation for the ray in the second ray cycle, i.e., after the first surface bounce.

e. Derive the expression for  in the second ray cycle. Discuss the physical

significance of the points where