The continuous spectrum, near its infimum, of the Laplacian for certain classes of `doubly-warped' complete Riemannian metrics on the interiors of compact manifolds with boundary is examined using geometric optics. General conditions on the geometry of an algebra of vector fields leading to a calculus of pseudodifferential operators are described and, once verified in this particular case, the associated notion of wavefront set can be used to show the existence of an absolute scattering matrix. This operator is conjectured to be a Fourier integral operator; the confirmation of this conjecture is discussed in the special case of a `double-zero' metric where the fibrations are trivial.