The cost of pleasure: about some catastrophes in time-harmonic wave scattering

The paper is concerned with a review of several situations taking place in the time-harmonic wave scattering (~e^-iωt) when solutions to certain wave scattering problems do not exist. Each of these situations is closely tied to the violation of solution uniqueness, due to a presence of an eigenvalue, in terms of either natural frequency or propagation constant, simple or multiple, coinciding with the corresponding parameter of the excitation field. Then the reason of solution non-existence call be easily understood. From the physical point of view, the mentioned catastrophes are always a result of joint action of several major assumptions, such as harmonic time-dependence, absence of losses or nonlinearities, and infinite extension of the scatterer. Thus, a loss of solution existence, at certain discrete values of parameters, is the cost of the pleasure of working with a simplified problem