Equal Phase Lines and Phase Invariant in the Sound Field of Deep Water

The spatial–frequency responses of the amplitudes and phases of the sound pressure in deep water have been studied. Analytical relationships have been obtained that make it possible to calculate and compare the amplitude–phase structures of water, leaky, and trapped modes, as well as the sound pressure field formed by the sum of the modes. The calculations were performed using the modified WKB (Wentzel–Kramers–Brillouin) approximation. It has been shown that in deep water, as in shallow water, there are stable equal phase lines along which, under certain conditions, coherent summation of complex Fourier components is possible. To describe the equal phase lines, a differential equation has been obtained that uses the phase invariant, already studied in shallow water, as a basic parameter. This has made it possible to study the properties of the phase invariant corresponding to water, leaky, and trapped modes in all zones of the sound field for deep water as well. It is established that at different distances in the field constructed from the sum of all modes, invariant properties are manifested, primarily, the modes that dominate at these distances. It is shown that leaky modes formed in the near illumination zone and in the shadow zone, formed by steep rays reflected from the bottom, have invariant properties only at large distances from the source. Water and trapped modes have invariant properties in full and at all distances. Recommendations are given on the use of equal phase lines and the phase invariant in processing experimental data and modeling.