Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations

[PDEs in] higher dimensions

The pictures below show as examples the solution to the wave equation in 1D, 2D and 3D starting from a stationary square pulse.

In each case a 1D slice through the solution is shown, and the solution is multiplied by r^(d-1). For the wave equation, and for a fair number of other equations, even and odd dimensions behave differently. In 1D and 3D, the value at the origin quickly becomes exactly 0; in 2D it is given by 1-t/Sqrt[t^2-1], which tends to zero only like -1/(2t^2) (which means that a sound pulse cannot propagate in a normal way in 2D).

From Stephen Wolfram: A New Kind of Science [citation]