The response of closed square and cubical cavities to linear standing waves of second sound in helium ii has been studied. In a square cavity, when the number of half-wavelengths $n$ along one side is even, a linear wave in the $x$ direction couples loosely with the $y$ direction, producing standing waves in both directions at two resonant frequencies, in a narrow doublet. The theory of the coupled waves is analogous to that for coupled LC electric circuits. When $n$ is odd, a linear wave does not couple with the perpendicular directions. However, owing to quadratic terms in the thermohydrodynamic equations, a small-amplitude wave at twice the frequency of the standing wave is generated, and this wave is present along both the $x$ and $y$ axes. A theory of these phenomena is presented, along with frequency, amplitude, phase, and $Q$ measurements. The results are similar in cubical cavities, but the number of available coupled-wave modes is reduced by cancellation of waves, due to the symmetry of the cavity.