Ai(t) is governed by a Mathieu equation,
Here, ωf is the forcing frequency, f is the forcing amplitude. Subsequently, we use 1/k=λ/2π as the length scale and as the time scale, where λ is the characteristic wave length and g is the gravitational acceleration. In equation (3), ωi is the natural frequency for the ith spatial mode with wavenumber ki=i and dimensionless capillary number κ=σk2/ρg. The subharmonic resonance corresponds to p ≈ 1, i.e. ωf ≈2ωi. To excite the fundamental mode with one wavelength in the tank (k=k1=1), the forcing frequency must satisfy ωf ≈2ω1=2ωN. This forcing frequency eliminates the sloshing mode, thereby preserving the spatial symmetry about the tank centerline.
One notable result of Jiang et al. (1996) is that for larger wave steepness, unexpected flat and dimpled crests appear in the physical experiments while the numerical simulation presents a wave with much sharper crest. (The dimpled crest feature is seen in the numerics but for a different phase.) These intricate wave forms are not described by any standing-wave model. They are strongly asymmetric about the peak in time and only emerge at finite amplitudes.
The main theme of this paper is that the new standing wave leads eventually to breaking with period tripling in the physical experiments. Steep and breaking wave profiles are quantified by a non-intrusive optical system. At sufficiently large forcing amplitude, a steep wave with a dimpled crest generates two plungers breaking to each side of the tank centerline. Increasing the forcing amplitude further leads to period-tripling: breaking every two of three waves in a three-wave cycle. The dynamics of both steep and breaking standing waves are related to the interaction between the fundamental mode and it temporal second harmonic. To estimate the viscous dissipation and dissipation due to breaking, we present the first direct measurements based on the periodicity of the breaking events and accurate force measurements. Before presenting some preliminary results, we first describe briefly the method we use to measure wave dissipation.
For nonbreaking waves, experiments with a fixed forcing frequency and forcing amplitude can be divided into three stages: (1) initial state when the tank is oscillating but no wave growth is observed; (2) intermediate state when the wave amplitude grows to a maximum and then slowly decays to a finite amplitude; (3) the final periodic state when the limit-cycle amplitude of the Faraday wave remains constant. Using a load cell located between the shaker and the tank, we can measure the time history of the support force during these three stages. The measured force is integrated with respect to the tank displacement to yield the work done by the shaker on the system. The wave dissipation can then be estimated as described below.
The total energy (work) input to the system is transformed into wave energy, wave dissipation and mechanical work. Here, mechanical work includes bearing dissipation, friction and other energy losses that are unrelated to the fluid motion in the tank. During stage 1, the wave surface remains horizontal so both wave energy and wave dissipation are negligible. The measured work per unit wave period represents mechanical work only. During stage 2, part of the work done by the shaker is converted into wave energy. The measured work in stage 3 should equal the mechanical work plus the wave dissipation as a periodic wave field exists. As both forcing frequency and amplitude are constant in the experiments, and the center of the water body is fixed relative to the oscillating tank, the mechanical work in stage 3 is the same as that determined in stage 1. The wave dissipation ΔE is then determined by ΔE=Wtotal–Wmech.
Since this technique is based on the balance between energy input and energy dissipation, it only applies to periodic wave fields. Breaking-wave dissipation would be difficult to estimate because of the irreversible breaking process. Fortunately (also unexpectedly), the breaking waves in our experiments are periodic! At a forcing amplitude that first initiates breaking, double plungers appear (stage 4), but the waveform, once established, remains temporally and spatially periodic. At larger wave steepness (i.e. larger forcing), the dimpled waveform evolves into three different, but three-wave periodic breaking modes (stage 5): the most violent breaking mode, a sharp-crest wave with upward jet (A), occurs every three periods, followed repeatedly by a dimpled waveform with plunging breaking (B), and a round-crest mode (C). The energy dissipation during the three-wave periodic breaking is then estimated by calculating the average work every three periods in stage 5.
As in Jiang et al. (1996), we use the Cauchy integral method to simulate two-dimensional standing waves. In potential flow, the periodic free surface on the deep water can be conformally mapped to an approximate unit circle. The Lagrangian form of the kinematic and dynamic conditions are applied on the free surface:
Here, D/Dt represents the material derivative. The complex potential w(ξ)=+iψ is solved on the free surface ξ=x+iy and w* is the complex conjugate of w(ξ).