Riemann's Method for Integrating the Most General 2nd Order Linear Hyperbolic Equation

In its most general form a linear second order hyperbolic equation is

$$0=\frac{\partial^2\psi}{\partial u\partial v} +D\frac{\partial \psi}{\partial u} +E\frac{\partial \psi}{\partial v} +F\psi\equiv L(\psi)~.$$ (617)

In compliance with standard practice one designates the characteristic coordinates by $u$ and $v$ . The problem to be solved is this:

Given

(a) the differential Eq.(6.17) and

(b) the initial value data (=Cauchy conditions) $\psi (s)$ and its normal derivative $\frac{d\psi(s)}{dn}$ on the given curve in Figure 6.2,

Find: the function $\psi(u,v)$ which satisfies (a) and (b).

Figure 6.2: Characteristic coordinates of a hyperbolic differetial equation in two dimensions.

Riemann's method of solving this problem is a three-step process whose essence parallels the Green's function method described on page :