The effect of earth curvature on long-range propagation in the ocean can be easily accounted for in acoustic models via a modification of the local sound-speed profile.

a. With r being the horizontal range from a source and R the earth radius, show that the sea surface on a sphere is displaced by 

b. By introducing the transformation

and substituting into (6.9), derive a parabolic wave equation in 

c. Discuss the form of this equation and show that the earth curvature effect can be

accounted for by a small linear increase in sound speed with depth.

d. Estimate the percentage change in convergence-zone ranges due to earth curvature.