Introduction to Partial Differential Equations
Exercise Sheet 14
Ross Ogilvie 2nd December,
2024
______________________________________________________________________________________________________________________________
In this exercise we will apply the method of descent to solve the wave equation on for a particular set of initial conditions. The idea is to help you understand the key ideas and notation of the method. It is a combination of results from Sections 5.1–5.
Consider the wave equation on with initial conditions
and likewise let and be the spherical means of and respectively. Explain why and (or give the definition of bar). Show that
You may use the following geometric fact: for , the surface area of the part of the sphere with is . (4 points)
Note that there are no -derivatives in this PDE, so we can think of it as a family of PDEs in the variables parametrised by . (2 points)
(4 points)
Observe that does not depend on . So by part (b) we have a solution to the 2-dimensional wave equation:
This solution has jump discontinuities, but this is unsurprising since the initial conditions also had them.
Let us work with for simplicity. If we apply the Fourier transform to the wave equation, we arrive at
This leads us to define the spectral energy density