Applied Underwater Acoustics

3.3. Normal Mode Propagation Models

The wave number integral in Eq. (3.47), for the pressure in the fluid with a symmetric point source according to f = f(x) = −M grad(δ(x−x_s)) where x_s = (0, 0, z_s) is in the water and the time factor exp(−iωt) with ω > 0 is omitted, may be rewritten as

p(r,z)=−12∫−∞∞r4(k,z)H0(1)(kr)kdk,

(3.74)

since r₄(k, z) is even in k. The integrand is analytic in k, except at the branch cuts induced by the vertical wave numbers iγ and iν for the lower solid half-space. These branch cuts are typically chosen as the hyperbolic curves where Re(γ) or Re(ν) vanishes. According to the residue theorem, Eq. (3.74) can be replaced by ...