One-Way Wave Equation Derived from Impedance Theorem

Oskar Bschorr,Hans-Joachim Raida

doi:10.3390/acoustics2010012

Abstract

The wave equations for longitudinal and transverse waves being used in seismic calculations are based on the classical force/moment balance. Mathematically, these equations are 2nd order partial differential equations (PDE) and contain two solutions with a forward and a backward propagating wave, therefore also called “Two-way wave equation”. In order to solve this inherent ambiguity many auxiliary equations were developed being summarized under “One-way wave equation”. In this article the impedance theorem is interpreted as a wave equation with a unique solution. This 1st order PDE is mathematically more convenient than the 2nd order PDE. Furthermore the 1st order wave equation being valid for three-dimensional wave propagation in an inhomogeneous continuum is derived.

Highlights

The n-th order differential equation has n linearly independent solutions
For the calculation of three- dimensional wave propagation in an inhomogeneous continuum a huge number of additional equations were developed being summarized under the title “One-way wave equation”
The impedance theorem is equivalent with a 1st order partial differential equation (PDE) that can be interpreted as a One-way = (x,t) =

Summary

Introduction

The n-th order differential equation has n linearly independent solutions. In the classical 2nd order wave equation with n = 2, the solutions are the two waves running in forward and backward direction. Due to the obvious ambiguity, a factorization [1] of the 2nd order partial differential equation (PDE) into two 1st order PDEs has been known (but not further applied) by seismicians for many decades: ∂2 ∂t2 − c2 ∂2 ∂x2 s= ∂ ∂t c ∂ ∂x + ∂ c ∂x s=0 (1). From this attempt two One-way wave equations result, but the original Equation (1) is only valid for straight wave propagation in a homogeneous continuum. A recent approach uses anti–sound techniques to extinguish backward travelling disturbing waves [2]

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Discussion

Conclusion