Engquist boundaries for the scalar wave equation

The simplest ``textbook'' boundary condition is that a function should vanish on the boundary. A wave incident onto such a boundary reflects with a change in polarity (so that the incident wave plus the reflected wave will vanish on the boundary). The next-to-simplest boundary condition is the zero-slope condition. It is also a perfect reflector, but the reflection coefficient is +1 instead of -1. Two points at the edge of the differencing mesh are required to represent the zero-slope boundary. The most general boundary condition usually considered is a linear combination of function value and slope. This is also a two-point boundary condition. It so happens that our extrapolation equations (chapter ) contain only a single depth derivative, so that on the z-axis they are a two-point condition. Observing this, Björn Engquist recognized a new application for our extrapolation equations. Many researchers in other disciplines are interested in forward modeling, that is, evolving forward in time with an equation like the scalar wave equation, say, $P_{{x} x}\ +\ P_{{z} z} = P_{{t} t} / v^2$.These people suffer severely the consequences of limited memory. Engquist's idea was that they should use our extrapolation equations for their boundary conditions. (This idea led to his winning the SIAM prize). Suppose they desire an infinite absorbing volume surrounding a box in the (x,z)-plane. Then they need a boundary condition that goes all the way around the box. They could use the downgoing wave equation on the bottom of the box and the upcoming wave equation on the top edge. The sides could be handled analogously with an interchange of x and z. This idea was thoroughly tested and confirmed by Robert Clayton. For an example of one of his comparisons, see Figure 1.

 

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Figure 1 Expanding circular wavefront in a box with absorbing sides (top) and zero-slope sides (bottom). (Clayton)