Wavelet Acoustics

Summary

In this chapter we construct wavelet representations for acoustics along similar lines as was done for electromagnetics in Chapter 9. Acoustic waves are solutions of the wave equation in space-time. For this reason, the same conformal group C that transforms electromagnetic waves into one another also transforms acoustic waves into one another. Hence the construction of acoustic wavelets and the analysis of their scattering can be done along similar lines as was done for electromagnetic wavelets. Two important differences are: (a) Acoustic waves are scalar-valued rather than vector-valued. This makes them simpler to handle, since we do not need to deal with polarization and matrix-valued wavelets, (b) Unlike the electromagnetic wavelets, acoustic wavelets are necessarily associated with nonunitary representations of C. This means, for example, that Lorentz transformations change the norm energy of a solution, and relatively moving reference frames are not equivalent. There is a unique reference frame in which the energy of the wavelets is lowest, and that defines a unique rest frame. The nonunitarity of the representations can thus be given a physical interpretation: unlike electromagnetic waves, acoustic waves must be carried by a medium, and the medium determines a rest frame. We construct a one-parameter family of nonunitary wavelet representations, each with its own resolution of unity. Again, the tool implementing the construction is the analytic-signal transform.