The Stretched String Problem

We’ll start our discussion of GYRE by considering the analogous (but much simpler) problem of finding normal-mode eigenfrequencies and eigenfunctions for waves on a stretched string clamped at both ends. Let the string have mass per unit length \(\rho\) and tension \(T\); then, the wave equation describing the transverse string displacement \(y(x,t)\) at spatial position \(x\) and time \(t\) is

\[\npderiv{y}{x}{2} = - \frac{1}{c^{2}} \npderiv{y}{t}{2},\]

with \(c \equiv (T/\rho)^{1/2}\). If the string is clamped at \(x=0\) and \(x=L\), then the wave equation together with the boundary conditions

\[y(0,t) = 0 \qquad y(L,t) = 0\]

comprise a two-point boundary value problem (BVP).