FY 1976

Tsunami scattering by an island surrounded by water of variable depth

Shaw, R.P.

In Proceedings of the International Union of Geodesy and Geophysics Tsunami Symposium, Bulletin 15, Royal Society of New Zealand, Wellington, New Zealand, 20 January–1 February 1974, 133–140 (1976)

Several numerical schemes are available for the scattering of time harmonic and transient water waves by islands. Variable water depth, irregular island shapes, even nonlinear effects may be included in these schemes without changing their basic character as finite difference algebraic equations approximating some governing partial differential equations. However, for an infinite ocean, or at least one large enough to make the use of a finite difference mesh everywhere impractical, these schemes must all be terminated at some fictitious outer boundary. The 'outer boundary condition', which requires a complete system of algebraic equations for solution, is usually approximated on some physical grounds. In essence, a local impedance or relationship between dependent variables and their derivatives is guessed. An alternative method is proposed here in which the outer 'boundary' condition is written exactly for those problems where the water depth beyond this fictitious outer 'boundary' may be taken as constant and where the wave heights may be taken as small enough to justify linearization there. These conditions in fact define the region in which this outer 'boundary' may be drawn. Solutions interior to this 'boundary' proceed by the usual methods. At this 'boundary', the Kirchhoff time-retarded integral or Helmholtz integral equation is used to replace the entire infinite domain of fluid exterior to this 'boundary' by a single equation, involving an integral over the 'boundary' of the dependent variables and their derivatives, i.e., a 'global' impedance matching. This integral equation, which represents the exact relationship between the dependent variables and their derivatives at this 'boundary', may itself be approximated by linear algebraic equations. These equations are coupled to the algebraic equations in the interior domain by use of finite difference approximations or normal derivatives. The net result is a deterministic sort of algebraic equations for values of the dependent variables at the mesh points. Examples are given for the time harmonic case for a constant depth and a parabolic depth variation.