Asymptotic analysis for time harmonic wave problems with small wave number

We study the asymptotic behavior of the solution to some time harmonic wave problems when the wave number is taken as a small asymptotic parameter. Our basic strategy is to introduce suitable Lagrangian multipliers into the governing equations, and transforming them into saddle point problems. These saddle point problems are uniformly invertible with respect to the wave number k∈[0,k0]k∈[0,k0], with k0k0 being an arbitrary but fixed positive number. The asymptotic expansion is then derived by the standard regular perturbation technique.