= kl

For a point source at rest in a stratified moving medium with Vz = 0, the functional f is given by

f[r, Z, t; c(z), e(z), V.l(z)] = p exp( -iWIt) . where p is determined by equations (4.42), (4.39) and (4.36). We recall that these equations are valid in the high-frequency approximation and far away from turning points. Substituting the value of f into equation (5.25), one obtains the expression for the sound field of a point source moving in the horizontal plane Z = Zo with constant velocity U.l in a stratified moving medium:

Here, eo = Q(zo), qo = q(zo), z> = max(z, zo) and z< = min(z, zo); w and q are determined by equations (5.27) and (5.28) respectively, where Vz = Uz = O. The Jacobian J in equation (5.29) is given by

= 2[fh ~ f Z> dz ( a2ul) _ (fz> av.l )2J J WI 2 3 1 + 2 2 2 3 dz • z< c q z< q C q z< C q

(5.30)

where V.l = [(V.l-U.l}2 - (a· (V.l-u.l}}2Ia2]1/2. The vectors a and r in equation (5.29) are related by

(5.31)

Now we transform equation (5.31) to an equation for a sound ray in a stratified moving medium. To do this, the vector U.lt is expressed in the form

U.lt = (t - tj)U.l + tjU.l . Here, t j is the time of emission of the sound wave that arrives at the point of observation at time t. In the above equation, the vector tjU.l = rj determines the position of the point source in the horizontal plane at time tj' and t - tj is the time of sound wave propagation from the point of emission (rj, zo) to the observation point (r, z). It follows from the arguments of section 3.3 that t - tj = ('V - 'VI}/co, where 'V - 'VI is an increment of the eikonal along the path. For a stratified medium, the increment 'V - 'VI is given by the integral on the right-hand side of equation (3.59), where Vz = 0, w is determined by

equation (S.27), and z and ZI are replaced by z< and z> respectively. Substituting U.Lt = r j + (III - III, )U.L/ Co into equation (S.31) yields

(S.32)

This is the equation for a sound ray r = r(z, a) in a stratified moving medium. Equation (S.32) relates the two vectors r and a entering into equation (S.29) for the field of a moving source.