This paper describes the derivation and implementation of combined wave diffraction-refraction in a spectral directional wave model for coastal applications. The wave refraction is included in the total derivative of wave-action while diffraction is formulated as wave energy diffusion in the wave-action balance equation. The combined diffraction-refraction model equation conserves the total wave-action under no net energy gain or loss. The numerical solution scheme for the model equation is simple and computation is stable. The method is demonstrated in a two-dimensional (2D) steady-state spectral model for a long wedge, an idealized shoal and a coastal harbor physical model.
Wave diffraction theory for the interaction with structures has been studied by many researchers in the past following the early work by Sommerfeld (1896). Combined wave diffraction-refraction is more challenging mathematically and analytical solutions are scarce (Penny and Price, 1944). The original work for combined diffraction-refraction by Berkhoff (1972, 1982) and recent investigations are mainly based on either Boussinesq or Mild-slope equation (Nwogu and Demirbilek, 2001) using phase-resolving wave models that require advanced numerical techniques to calculate solutions. Because the phase-resolving models are computationally intensive, their applications are often limited to local small areas. For larger coastal areas, the spectral wave models that neglect wave phase calculations are popular as they are computationally more efficient than the phase-resolving models. However, the implementation of wave diffraction in today's coastal spectral wave models is more or less based on empirical treatment or simple formulation extending the features of phase-resolving models (Chawla et al. 1998; Mase, 2001). The present paper describes a theoretical derivation of wave diffraction combined with wave refraction for spectral wave models. The approach of diffraction formulation is consistent with the well-known Helmholtz equation, and its implementation in the wave action balance equation conserves wave energy for combined diffraction-refraction.