The behavior of imperfectly trapped high-frequency acoustic waves is examined for a planar model of the Sun in which the solar interior is represented by an adiabatically stratified polytrope and the overlying atmosphere is represented by a semi-infinite isothermal region. In the absence of dissipation within the medium, unattenuated horizontally propagating modes that form standing waves vertically within the interior are found at frequencies below the acoustic cutoff frequency of the upper, isothermal region. The dispersion relation for these modes is simply expressed in terms of confluent hypergeometric functions of the second kind. In the limit of the temperature of the overlying region becoming zero (i.e., absent the atmosphere), these become the well-known modes of the planar polytrope terminated by a free surface. Above the acoustic cutoff frequency of the atmosphere, the atmosphere/interior interface becomes leaky. Consequently, unattenuated propagation horizontally is precluded. The pseudomodes that populate this range of frequencies in the absence of dissipation are excluded from the eigenvalue spectrum but nevertheless remain capable of affecting the distribution of power in the omega-kappa(h) diagram. This influence rapidly diminishes, however, with increasing frequency above the cutoff, leaving the mechanism advanced by Kumar et al., namely interference between initially upward and once-reflected, initially downward-radiated waves, as the principle determiner of ridge location. Unlike the situation for the modally defined peaks below the cutoff frequency, ridge locations under these conditions are sensitive to source depth, a property used by Kumar & Lu to infer a shallow depth for the source of acoustic wave energy in the Sun. The amplitudes of the modal peaks are shown here generally to be modulated by the same factors that determine ridge locations at frequencies well above the cutoff frequency. The absence of a clear indication of such modulation in solar data consequently constitutes further evidence that the acoustic source region in the Sun is located at a shallow depth. In order to reproduce the smooth variation of peak amplitude through the acoustic cutoff exhibited by solar ridges, the model studied here requires appreciable dissipation to be present in the region above the wave source. This also causes the transition from modal resonance to direct/reflected wave interference as the dominant factor determining ridge locations to occupy a considerable range of frequencies on both sides of the cutoff frequency. The transition characteristics are also sensitive to details of the sound-speed variations near the top of the convection zone such as the presence of the thin superadiabatic layer found there in the Sun, a result anticipated by Balmforth & Gough.
