The application of a number of the classical methods of mathematical physics to
modeling non-stationary fluid-structure interaction (FSI) is considered. The
particular emphasis is on the examining the interaction between thin-walled
cylindrical structures and weak shock waves, a problem that has been for
decades regarded as `classical' for the type of FSI in question. Although the
solutions themselves are not overly complex, obtaining computable expressions
is anything but trivial. A number of significant challenges that one encounters
are discussed, and some efficient approaches to dealing with them are proposed,
sometimes with interesting theoretical results being a spin-off. The numerical
aspects of the simulations based on the solutions discussed are often not
trivial as well, and are examined in some detail. The results of the
simulations are compared to many available experimental data, and an excellent
agreement is observed. Then, the simulations are used to obtain a number of
results of practical significance, some directly relevant to the safety
improvement of engineering systems. It is therefore demonstrated that the
classical analytical solutions are a powerful and efficient tool of
mathematical modeling of FSI when the loads are limited to weak shock waves, a
result that (hopefully) helps to somewhat reinstate the once-undisputed
reputation of the classical analytical techniques.