Consider sound waves propagating in an atmosphere with a horizontal wind. Assume that the sound speed C, as measured in the air’s local rest frame, is constant. Let the wind velocity u = u_xe_x increase linearly with height z above the ground: u_x = Sz, where S is the constant shearing rate. Consider only rays in the x-z plane.

(a) Give an expression for the dispersion relation ω = Ω(x, t ; k).

(b) Show that k_x is constant along a ray path, and then demonstrate that sound waves will not propagate when

(c) Consider sound rays generated on the ground that make an angle θ to the horizontal initially. Derive the equations describing the rays, and use them to sketch the rays, distinguishing values of θ both less than and greater than π/2. 

@ kx - ux(z)