Applied Underwater Acoustics

3.2. Wave Number Integration or Spectral Methods

The solution of the basic equations of motion (3.3) is greatly simplified when the medium parameters are independent of the horizontal spatial coordinates x and y. Such a laterally homogeneous, or range-invariant, medium is now assumed. The symmetric point source with f(x, t) = −M grad(δ(x − x_s)) exp(−iωt) and ω > 0 is stipulated, with x_s = (0, 0, z_s) in the water. Introducing a cylindrical coordinate system (r, ϕ, z) with r = (x² + y²)^1/2 (for radius) and ϕ (for azimuth) in the horizontal plane, and corresponding basis vectors e_r, e_ϕ, e_z, the wave field is independent of ϕ. Omitting the time-harmonic factor exp(−iωt), as usual, the displacement vector u and the traction vector at horizontal planes ...