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Concept# One-way wave equation

Summary

A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resulting from superposition of two waves in opposite directions. In the one-dimensional case, the one-way wave equation allows wave propagation to be calculated without the mathematical complication of solving a 2nd order differential equation. Due to the fact that in the last decades no 3D one-way wave equation could be found numerous approximation methods based on the 1D one-way wave equation are used for 3D seismic and other geophysical calculations, see also the section .
The scalar second-order (two-way) wave equation describing a standing wavefield can be written as:
where is the coordinate, is time, is the displacement, and is the wave velocity.
Due to the ambiguity in the direction of the wave velocity, , the equation does not contain information about the wave direction and therefore has solutions propagating in both the forward () and backward () directions. The general solution of the equation is the summation of the solutions in these two directions:
where and are the displacement amplitudes of the waves running in and direction.
When a one-way wave problem is formulated, the wave propagation direction has to be (manually) selected by keeping one of the two terms in the general solution.
Factoring the operator on the left side of the equation yields a pair of one-way wave equations, one with solutions that propagate forwards and the other with solutions that propagate backwards.
The forward- and backward-travelling waves are described respectively,
The one-way wave equations can also be physically derived directly from specific acoustic impedance.
In a longitudinal plane wave, the specific impedance determines the local proportionality of pressure and particle velocity :
with = density.
The conversion of the impedance equation leads to:
A longitudinal plane wave of angular frequency has the displacement .

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