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).