Spectral properties of certain stiff problems in elasticity and acoustics
Giuseppe Geymonat, Miguel Lobo-Hidalgo, É. Sanchez-Palencia, Gary F. Roach · Mathematical Methods in the Applied Sciences · 1982
Abstract We consider two vibration problems containing a small parameter → 0: a) Vibration of an elastic, slightly compressible body, and b) acoustic vibration of a slightly viscous compressible barotropic fluid in a vessel. The asymptotics of eigenvalues for problem a) is studied by using a uniformly convergent expansion of the stiff type. After a re‐scaling of the spectral parameter, the problem b) reduces to an analogous problem, and we prove that, as ε → 0, infinitely many eigenvalues converge to 0 (which is an eigenvalue of infinite multiplicity of the corresponding inviscid acoustic problem).