Fourier transform and Fourier series act as important tools for determining the solution to a large class of boundary value problems (BVP) in semi-infinite domains and in semi-infinite strips. Depending on the behavior of the function and/or its derivative at one end of the boundary, and knowing the functional behavior at the far field, appropriate integral transform is applied to reduce the dimension of the partial differential equation in a half-plane or quarter plane for a BVP. In the present chapter, expansion formulae for the velocity potentials are presented in half plane and quarter plane for a class of BVP arising in the broad area of wave structure interaction satisfying higher order boundary condition on the structural boundaries. Further, expansion formulae for the corresponding BVPs in infinite and semi-infinite strips are derived which are generalisations of the classical eigenfunction expansion method. Various characteristics of the eigen-system associated with the expansion formulae are discussed. Application of these types of expansion formulae in several cases are illustrated through physical problems of practical interest.