Modal decomposition is often applied in elastodynamics and acoustics for the solution of problems related to propagation of mechanical disturbances in waveguides. One of the key elements of this method is the solution of an eigenvalue problem for obtaining the roots of the dispersion equation, which signify the wavenumbers of the waves that may exist in the system. For non-dissipative media, the wavenumber spectrum consists of a finite number of real roots supplemented by infinitely many imaginary and complex ones. The former refer to the propagating modes in the medium, whereas the latter constitute the so-called evanescent spectrum. This study investigates the significance of the evanescent spectrum in structure-waveguide interaction problems. Two cases are analysed, namely, a beam in contact with a fluid layer and a cylindrical shell interacting with an acousto-elastic waveguide. The first case allows the introduction of a modal decomposition method and the establishment of appropriate criteria for the truncation of the modal expansions in a simple mathematical framework. The second case describes structure-borne wave radiation in an offshore environment during the installation of a pile with an impact hammer—a problem that has raised serious concerns in recent years due to the associated underwater noise pollution.