In this study, we consider the ideal fluid model of an inviscid fluid, assuming that the fluid motion is adiabatic; the flow is irrotational, that is, the individual fluid particles do not rotate; vorticity ω̃=0; and the flow is incompressible, in which the density of fluid particles does not vary significantly with fluid motion and can be considered constant throughout the fluid volume and throughout the motion. We start with equations representing continuity, conservation of momentum, conservation of entropy, and streamline equations, respectively. It is then reduced to a standard system of equations describing motion in two dimensions, defined by the Laplace equation with appropriate kinematic and dynamic boundary conditions, in terms of velocity potential and surface elevation. Finally, the one-dimensional nonlinear Korteweg–De Vries (KdV) equation is derived. Then, we further investigate the interaction of multiple periodic waves using the KdV equation and explain the interaction wave energy transfer procedure between the primary and higher order harmonics, and the Phillips [“On the dynamics of unsteady gravity waves of finite amplitude. I. The elementary interactions,” J. Fluid Mech. 9, 193–217 (1960)] wave resonance criterion is employed for capturing the periodic wave interaction whose energy conversion is analyzed via Fourier spectra. It is also found that for solitons, multiple collisions of different solitons eventually regain their original shape and that higher-energy solitons have faster velocities than lower-energy solitons, which, to the best of our knowledge, is overlooked.

In this study, the propagation of a fundamental plane mode in a bifurcated waveguide structure with soft–hard boundaries is analyzed by using the Helmholtz equation. The explicit solution is given to this bifurcated spaced waveguide problem by means of matching the potential across the boundary of continuity. Amplitudes of the reflected field in all those regions have been evaluated, and the energy balance has been derived. We have observed the reflection of the acoustic wave against the wavenumber and shown its variation with the duct width. Convergence of the problem has been shown graphically. In our analysis, we notice that the reflected amplitude decreases as the duct spacing increases; as a result, the acoustic energy will increase as the duct spacing increases. It is expected that our analysis could be helpful to give better understanding of wave reflection in an exhaust duct system. We then reduce the linear acoustic wave equation to the Kadomtsev–Petviashvili (KP) equation. Multiple-periodic wave interaction solutions of the KP nonlinear wave equation are investigated, and the energy transfer mechanism between the primary and higher harmonics is explained, which, to the best of our knowledge, is overlooked.

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