Effect of finite amplitude of bottom corrugations on Fabry-Perot resonance of water waves

Recently, the mechanism of Fabry-Perot (F-P) resonance in optics was extended to monochromatic water waves propagating in a domain with two patches of sinusoidal corrugations on an otherwise flat bottom. Assuming small-amplitude surface waves, an asymptotic linear analytical solution (ALAS) was derived by L. A. Couston et al. Phys. Rev. E 92, 043015 (2015). When resonance conditions are met, the ALAS predicts large amplification of the incident waves in the resonator area between the two patches of corrugations. Based on the ALAS, the amplitude of these standing waves is expected to increase exponentially with the relative amplitude of bottom corrugations ($\delta =d/h$, where $d$ is the corrugation amplitude and $h$ the still water depth). In the present work, we examine the effects associated with the assumptions made in deriving the ALAS regarding the effect of a finite amplitude of bottom corrugation (i.e., finite value of $\delta$), still in the linear wave framework. F-P resonance is studied by means of highly accurate numerical simulations, considering either the exact linear water wave problem (system A) or an approximate problem with a first-order expansion of the bottom boundary condition (system B). The numerical model is first validated on a Bragg resonance case, through comparisons with the ALAS, experimental measurements, and existing numerical simulations, showing its ability to represent well the so-called wave-number downshift of Bragg resonance (i.e., the slight decrease in the incident wave number where maximum resonance is reached in comparison with the value predicted by the ALAS). We then analyze how this downshift affects the F-P resonance, especially when the corrugations are of finite amplitude, i.e., $\delta$ varying from 0.05 to 0.4. The wave-number downshift appears to have a strong effect on the F-P resonance for $\delta >0.1$: very low wave amplification manifests for the wave number predicted by the ALAS. However, when the incident wave number is slightly decreased (by an amount increasing with $\delta$) the F-P resonance case can be recovered, and the maximum amplification values are found to be close to the predictions from the ALAS (e.g., up to a factor of about 27 for $\delta =0.4$). The variations of the reflection coefficient and enhancement factor obtained from systems A and B as a function of the incident wave number are discussed and compared to ALAS predictions. In particular, it is found that the resonance peak is extremely narrow when $\delta =0.2$ and 0.4.

Figure 1

Sketch of the problem setup for F-P resonance.

Figure 2

Enhancement factor $E_{\mathrm{ALAS}}^{\mathrm{FP}}$ as a function of the nondimensional corrugation amplitude $\delta =d/h$. The water depth, setup of two patches, and length of the resonator are the same as those used in [4].

Figure 3

Relative error on the phase celerity of the linear version of the numerical model (with respect to the exact Airy celerity) for the relative water depth $\mu =0.82$ as a function of the maximum order of polynomials $N_T$.

Figure 4

The simulation results (system A) of Davies and Heathershaw's experiments [6] with 10 bars are compared with the experimental measurements. The ALAS prediction based on Mei's theory [8] is also given as a reference.

Figure 5

Computed envelope of the free surface elevation at the end of the simulations of systems A and B ($t=100T$) for the case $\delta =0.05$ with the wave number $k=k_B$, compared to the envelope from the ALAS.

Figure 6

Same as Fig. 5, for the case $\delta =0.1$. The simulation duration is $t=200T$.

Figure 7

Same as Fig. 5, for the case $\delta =0.2$. The simulation duration is $t=900T$.

Figure 8

Effect of the detuned wave number (normalized by $k_B$) on the reflection coefficient $R^{\mathrm{FP}}$ for systems A and B. The ALAS prediction based on Eq. (10a) is also superimposed.

Figure 9

Effect of the detuned wave number (normalized by $k_B$) on the enhancement factor $E^{\mathrm{FP}}$ for systems A and B. The ALAS prediction based on Eq. (10a) is also superimposed.

Figure 10

Computed envelope of the free surface elevation at the end of the simulations of systems A and B ($t=900T$) for the case $\delta =0.2$ with the wave number slightly smaller ($k^{\prime }=0.99263k_B$) than the F-P condition, compared to the ALAS envelope (calculated for $k=k_B$).

Figure 11

Effect of the detuned wave number (normalized by $k_B$) on the enhancement factor $E^{\mathrm{FP}}$ for system A. The ALAS prediction based on Eq. (10a) is also superimposed.

Figure 12

Computed envelope of the free surface elevation at the end of the simulations with system A ($t=10000T$) for the case $\delta =0.4$ with the wave number slightly smaller ($k^{\prime }=0.979313k_B$) than the F-P condition, compared to the ALAS envelope (calculated for $k=k_B$).