Validity criteria for the small slope approximation for backscattering of sound from rough surfaces with a power-law roughness spectrum

The small slope approximation is a commonly used model for scattering of waves from rough surfaces. Its limitations are currently unknown for a power-law roughness spectral density, used to model natural terrain and seafloor roughness. This work compares the small slope approximation to the boundary element method, a numerical solution of the Helmholtz boundary integral equations governing acoustic scattering. These comparisons were used to find the validity limits as a function of dimensionless root mean square height, kh, and root mean square slope, s, where k is the wavenumber. The two lowest terms in the small slope series were used, and both the Dirichlet and fluid-fluid boundary conditions were examined. Both kh and s are found to be important parameters for the validity of the small slope approximation. Different spectral exponents have similar maximum valid kh. The maximum s is almost constant as a function of the power law outer scale, but varies as a function of the spectral exponent. Comparisons with other scattering models are discussed, and aspects of scattering from very large roughness are explored.