The superharmonic instability and wave breaking in Whitham equations

Abstract

The Whitham equation is a model for the evolution of surface waves on shallow water that combines the unidirectional linear dispersion relation of the Euler equations with a weakly nonlinear approximation based on the Korteweg–De Vries equation. We show that large-amplitude, periodic, traveling-wave solutions to the Whitham equation and its higher-order generalization, the cubic Whitham equation, are unstable with respect to the superharmonic instability (i.e., a perturbation with the same period as the solution). The threshold between superharmonic stability and instability occurs at the maxima of the Hamiltonian and L2-norm. We examine the onset of wave breaking in traveling-wave solutions subject to the modulational and superharmonic instabilities. We present new instability results for the Euler equations in finite depth and compare them with the Whitham results. We show that the Whitham equation more accurately approximates the wave steepness threshold for the superharmonic instability of the Euler equations than does the cubic Whitham equation. However, the cubic Whitham equation more accurately approximates the wave steepness threshold for the modulational instability of the Euler equations than does the Whitham equation.

Instability of near‐extreme solutions to the Whitham equation

“…The modulational instability onset for the Whitham equation occurs at $$s=0.062$$, while the onset in the Euler equations with dimensionless depth $$h=1$$ occurs at $$s=0.085$$ (see Ref. [33]). 5.Solutions to both the Euler and Whitham equations with sufficiently large steepness are unstable with respect to superharmonic instabilities.…”

“…The superharmonic instability onset for the Whitham equation occurs at $$s=0.1045$$ (see Ref. [33]), while the onset in the Euler equations with dimensionless depth $$h=1$$ occurs at $$s=0.099$$ (see Ref. [17]). …”

“…Carter and Rozman 32 studied the stability of large-amplitude solutions to the Whitham equation with and without surface tension. Carter et al 33 numerically studied the superharmonic instability in the Whitham equation and made comparisons with results from the finite-depth Euler equations. They found that Whitham solutions steeper than 𝑠 = 0.062 are unstable with respect to the modulational instability and that solutions steeper than 𝑠 = 0.104 are unstable with respect to the superharmonic instability.…”

“…Carter and Rozman 32 studied the stability of large‐amplitude solutions to the Whitham equation with and without surface tension. Carter et al 33 . numerically studied the superharmonic instability in the Whitham equation and made comparisons with results from the finite‐depth Euler equations.…”