Multiscale Investigation of Solutions of the Wave Equation

We consider here an initial value problem for the homogeneous wave equation with constant coefficients in three spatial dimensions, that is, 27.1 $$\begin{cases}u_{tt} - c^2 (u_{xx} + u_{yy} + u_{zz}) = 0,\\ u|_{t=0} = w({\bf r}), \quad \left.\frac{\partial u}{\partial t}\right|_{t=0} = v({\bf r}).\end{cases}$$ The number of dimensions is not essential, and the method proposed can be generalized with minor changes to the case of an arbitrary number of spatial dimensions. We suppose that the initial data for the problem (27.1) has a complicated multiscale structure, i.e., the initial data possesses rapid changes of local frequency, a high degree of localization, singularities, discontinuities, and sharp edges. An example of such data is presented in Figure 27.1. We also note that this image is represented in discrete, not analytic, form. The most convenient mathematical apparatus for describing initial data of this kind is a continuous wavelet transform [AnMu04]. Not only does the wavelet transform contain complete information about the local structure of the data, i.e., it has an inverse, but it is also known to be the most adequate transform for qualitative analysis of the data. When the initial data has a multiscale structure, the wave field is also multiscale at any time. This means that different spatial scales of a wave field at a fixed time may have localization in different spatial areas. Then it is useful to know the time evolution of the wavelet transform taken with respect to the spatial coordinates. We offer an analytic formula for the time dependency of the wavelet transform, which does not require the calculation of the wave field itself.