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From undergraduate physics (a few years ago for me) I roughly remember how to derive reflection and transmission coefficients for a wave travelling from one medium to another with a different impedance; e.g. a wave travelling along a thin string connected to a thicker string.

I can also vaguely remember solving the wave equation in 2D (maybe 3D) for waves in a box.

In both the examples above the boundary is clearly defined and the media on either side of the boundary can be thought of as extending away to infinity. I am interested in cases where there are perturbations of the impedance over different length scales. Let's say the thick and the thin string interchange every λ/8. How does the wave "see" the changes"?

Practically, I am thinking about seismic exploration for hydrocarbons where the dominant frequency of the sound pulse may be around 20 Hz, a typical rock velocity may be 2000 m/s therefore the "wavelet" is about 100 m long. However the rocks can be layered on a scale much less than 100 m (e.g. look at Grand Canyon pictures) and therefore the impedance medium varies on all length scales, both shorter and longer than the wavelet length. In the hydrocarbon example a hole may be drilled and the rock properties can be observed on a vertical scale of less than a metre. However typically we apply a moving average of the rock properties over a scale roughly equal to the wavelet length/duration, and roughly guess an attenuation factor before modelling this as an "effective medium". Is this theoretically sound?

Can anyone provide a discussion of what the key concepts to read up on are, or point me in the direction of a paper, online description etc?

Thanks,

BOYLANATOR