One Cannot Hear the Shape of a Drum

We begin by considering the one-dimensional analogue of Kac's question: a vibrating string of length L. Idealize this string by the interval [0,L] and represent the possible configuration of the vibrating string as a function f(x,t) defined for x (the position variable) in [0,L] and any non-negative number t (the time variable). Since the endpoints are fixed, f must satisfy the boundary conditions f(0,t) = f(L,t) = 0 for all values of t. It can be shown that such a vibrating string must also satisfy the wave equation ∂2f/∂t2 = ∂2f/∂x2 where ∂ is the change in the given variable.