Inverse Problem for the One‐dimensional Wave Equation

Summary The problems of determination of velocity‐depth functions from travel‐time curves or from dispersion curves show that the solution of an inverse problem may not be unique. We study here, as a preliminary analogy of such problems, the derivation of the unknown density function for an inhomogeneous string capable of small transverse vibrations, with one end fixed and one free. A unit impulse is applied at the free end, and the subsequent motion of the free end is observed. We prove that the density as function of position on the string is uniquely determined by these observations, under certain conditions. If a more general disturbance is applied, and similar observations are made at an arbitrary point of the string, is the determination of density still unique? We show that it is, provided all modes of free oscillation of the string are excited when the string is symmetrical with free ends. Further, we examine the stability of our solution. Could very large variations of density correspond to small variations in the observed motion? If so, a solution from actual data, liable to error, would be useless. We show that the solution is stable for a wide class of strings provided the observation point does not coincide with a node of one of the first N modes, and that these modes are excited ‘distinctly enough'. The choice of N and the meaning of ‘distinctly enough ‘are fully explained.