Applied Underwater Acoustics

3.4. Parabolic Equation Methods

The Helmholtz equation (3.98) accommodates out-going as well as in-coming (reflected, backscattered) waves, since a second derivative ∂²p/∂r² with respect to range is involved. In a medium where the range dependence is weak and smooth, an analysis of mode-coupling equations shows that the energy of the backscattered waves is very small [75] (Section 7.1.3). The clay island example in Section 3.3.5.2 supports this statement. An approximation where the backscattered waves are neglected is thus of interest, with a corresponding parabolic or, with a more correct term, one-way wave equation that only includes first derivatives ∂p/∂r with respect to range. Its solution could be conveniently marched outward in range, starting ...