When the compressibility of water is considered, resonant triads can occur with the family of acoustic-gravity waves. The steady-state triad resonance in the acoustic-gravity waves is investigated by solving the water-wave equations as a nonlinear boundary-value problem. Here, two types of triad resonance are considered: one is the two gravity one acoustic triad, which comprises two progressive gravity waves with the same wavelength travelling in opposite directions and a hydroacoustic wave of almost double the frequency; the other one is the one gravity two acoustic triad, which comprising a gravity wave and two hydroacoustic waves of almost double the length travelling in opposite directions. Based on the HAM, it is found that the two types of steady-state resonant acoustic–gravity waves widely exist and the two primary wave components and the resonant wave component might occupy most of the wave energy in both steady-state resonant acoustic–gravity wave systems.
Initially, Longuet-Higgins (1950) proposed that when considering the compressibility of seawater and two gravity wave components with the same wavelength travelling in opposite directions, the nonlinear interaction between them would generate hydroacoustic waves. This type of acoustic-gravity wave not only affects the water at the free surface but also has an impact on the entire water body. Moreover, the gravity wave system will resonate with the generated hydroacoustic waves under certain water depth conditions. However, at that time, Longuet-Higgins (1950) only provided the second-order solution of the acoustic-gravity wave velocity potential and the condition for resonance to occur, without specifically considering the operating conditions when resonance occurs. In recent years, Kadri and Stiassnie (2013) have studied the nonlinear interaction between two opposing gravity waves and the generated hydroacoustic wave when resonance occurs. They found that when the resonance condition is precisely met, the amplitude of the hydroacoustic wave, as a resonant wave, increases over time at the beginning, and after reaching a certain degree, there is a periodic energy conversion with the amplitude of the two primary gravity waves. This conclusion was also confirmed by Kadri and Akylas (2016), who not only found that the energy of a resonant acoustic-gravity wave system varies periodically between different wave components, but also found that the time scale of this periodic change is longer than that of resonant gravity waves. Kadri (2015) provided a more general resonance condition for acoustic-gravity waves based on the research of Longuet-Higgins (1950), pointing out that the resonance condition for acoustic-gravity waves mentioned by Longuet-Higgins (1950) is only one special case. Kadri (2016) showed that a hydroacoustic wave interacting with a surface-gravity wave may generate a second hydroacoustic wave. Interestingly, the two hydroacoustic waves propagate in the same direction with similar wavelengths, that are almost double that of the gravity wave. The above-mentioned studies are all about resonant acoustic gravity wave systems in unsteady states.
