Inverse Problem for the One-dimensional Wave Equation

Abstract

The problems of determination of velocity-depth functions from traveltime 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.The inverse problem of seismology is to determine the internal structure of the Earth from observations of the surface motion resulting from earthquakes and explosions. Some examples of inverse problems are the determination of velocitydepth functions from travel-time curves or from dispersion curves; and these examples show that the solution of the inverse problem may not be unique. The basic question arises: is it possible to determine the internal constitution of the Earth from observations on its surface?A priori, it is not excluded that differing models of the Earth may be indistinguishable when examined from the surface. The complete mathematical solution of this question is rather complicated. We will choose a simplified case: the inverse problem for the one-dimensional wave equation-' equation of the string ', Many aspects of this simplified problem will be important for the general problem. This simplified problem is also of independent interest, it is a problem of interpretation of normal incidence. 337

On the problem of determining the structure of a layered medium and the shape of an impulse source

“…Thus, the latter are practically impossible to find without severely restricting the class of feasible functions which essentially is what was established in the article cited above. The hypotheses of the uniqueness theorem of [7] cannot be checked by using the data of the problem.…”

“…To determine it we could only use observations of the oscillating boundary of the domain. Gerver considered [7] a similarly formulated problem for the one-dimensional wave equation corresponding to the case that the solution to the problem is independent of x; for instance, if we eliminate the delta function from the boundary condition (1.2). However, in this case the information under measurement is a function of one variable, whereas there are two unknown functions.…”

An Accurate and Efficient Algorithm for Parameter Estimation of 2D Acoustic Wave Equation

“…Consequently, the estimated ) , ( y x c is also accurate as we can see from Fig. 2 that, both maximal and root mean square norms of the error are reduced to the level of 3 …”

“…This technique has been widely used to determine the unknown property of a medium in which the wave is propagated by measuring data on its boundary or a specified location in the domain. A great deal of work have been done and many direct and indirect methods have been reported in the past decades [1][2][3][4]10]. In the wave equation, the unknown coefficient which characterizes the property of the medium is important to the physical process but usually cannot be measured directly, or very expensive to be measured, thus some mathematical method is needed to estimate it.…”

Well-posed Inverse Eigenvalue Problems

“…For inslance returning to the Sturm-Liouville problem, Levitan (1965) has shown that one spectrum together with the slope at the origin of the normalized eigelifunctions are equivalent to two spectra. Krein (1952b) and Gerver (1970) have also shown that the knowledge of the displacement of the free end point of a string set in motion by an impulse at t = 0 is equivalent to two spectra of the vibrating string.…”