Suppression of nonlinear forced waves by input shaping

Abstract

When a subcritical stratified flow is disturbed by a bottom obstacle, oscillatory lee waves generally arise due to a resonance mechanism. Such waves play an important part in meteorological events, and sometimes may be disastrous. Here, using the forced Korteweg–de Vries (fKdV) equation as a simple model, we demonstrate the possibility of suppressing the oscillatory wave tail by slightly modifying the preexistent forcing. For linear monochromatic waves, this is done by shaping the forcing (or input) of the wave system into a superposition of two identical ‘partial forcings’ that are separated in space by an odd multiple of the half-wavelength of the excited wave. The two excited wave components then annihilate each other through destructive interference. In the weakly nonlinear–weakly dispersive parameter regime of the fKdV model, the resultant wave amplitude formally is exponentially small with respect to the dispersion parameter, and is calculated here using the techniques of exponential asymptotics. It then transpires that under the influence of nonlinearity the wave suppression scheme not only remains effective, but also appears to be quite tolerant of the practically inevitable input shaping errors. Exact numerical results also support these findings.