Time-Dependent Wave Motion in a Running Stream Due to Initial Disturbances in Magnetohydrodynamics

Abstract The generation of two-dimensional time-dependent magnetohydrodynamic surface waves in a flowing stream is investigated in this paper. The waves here arise from some initial disturbances at the free surface of an electrically conducting field-free fluid, which is of finite depth. It is also assumed that a constant magnetic field exists outside the air–water interface. With the help of linearized theory, the present problem is modeled as an initial boundary value problem, and the Laplace–Fourier transform techniques are primarily used to obtain the form of free surface elevation through infinite integrals. These integrals are then interpreted asymptotically for large time t and distance x from the origin, using the method of stationary phase. Dispersion relation, phase, and group velocities are also studied. Numerical results of the asymptotic form of free surface elevation are obtained in a number of figures for different values of current speed, Alfven velocity, and various types of initial disturbances. Graphical representations demonstrate that the presence of a magnetic field has a significant effect on wave motion.