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
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]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
