Abstract
Abstract We present homogenization technique for the uniformly discretized wave equation, based on the derivation of an effective equation for the low-wavenumber component of the solution. The method produces a down-sampled, effective medium, thus making the solution of the effective equation less computationally expensive. Advantages of the method include its conceptual simplicity and ease of implementation, the applicability to any uniformly discretized wave equation in 1-D, 2-D or 3-D, and the absence of any constraints on the medium properties. We illustrate our method with a numerical example of wave propagation through a 1-D multiscale medium and demonstrate the accurate reproduction of the original wavefield for sufficiently low frequencies.
Highlights
Small-scale structure affects waves in a similar way as an effective structure with smooth variations
SUMMARY We present homogenization technique for the uniformly discretized wave equation, based on the derivation of an effective equation for the low-wavenumber component of the solution
We illustrate our method with a numerical example of wave propagation through a 1-D multiscale medium and demonstrate the accurate reproduction of the original wavefield for sufficiently low frequencies
Summary
Small-scale structure affects waves in a similar way as an effective structure with smooth variations. This fortunate fact allows us to understand and model their propagation without knowing or describing subwavelength details. Fichtner & Igel 2008; Capdeville et al 2010b). Capdeville et al 2010a, 2015) or the Born approximation (Jordan 2015). They operate on the level of the partial differential equation (PDE), that is, on the differential form of the wave equation, given in the frequency domain and in 1-D by
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