Analysis of Wave Constants in the Laplace Equation Solution for Deep Water with Wave Amplitude or Wave Energy as Input

Syawaluddin Hutahaean

wave constants, wave number, wave constant G, wave period

In the velocity potential solution of the Laplace equation obtained using the separation of variables method, three wave constants arise: wavelength, wave period, and the wave constant G. The wave constant G represents the rate of wave energy transmission. Unlike these constants, wave amplitude is not part of the solution constants but serves as an input parameter. Therefore, the wave constants should be expressed as functions of the wave amplitude. This study derives analytical expressions for wavelength, wave constant G, and wave period in deep water with wave amplitude as the governing variable. The relationship between wavelength and wave amplitude is obtained using the Kinematic Free Surface Boundary Condition. The relationship between wave constant G and wave amplitude is derived from a modified Euler momentum conservation equation together with a wave amplitude function, which relates the three wave constants to the wave amplitude. The wave period is then determined using the equations for wave constant G and the wave amplitude function. After establishing the wave constants as functions of wave amplitude, the study further formulates wave amplitude as a function of wave energy. The resulting amplitude is then used to calculate the three wave constants. This approach can also be applied to analyze waves generated by ship motion, where the input energy corresponds to the ship’s kinetic energy. The method is further extended to long waves, particularly tsunamis and sneaker waves.

Received: 23 Feb 2026, Received in revised form: 25 Mar 2026, Accepted: 02 Apr 2026, Available online: 06 Apr 2026

10.22161/ijaers.134.4