Perfect Resonant Absorption of Guided Water Waves by Autler-Townes Splitting

The control of guided water wave propagation based on the Autler-Townes splitting resonance concept is demonstrated experimentally, numerically, and theoretically. Complete wave absorption is achieved using an asymmetric pointlike scatterer made of two closely spaced resonant side channels connected to a guide and designed so that its energy leakage is in perfect balance with the inherent viscous losses in the system. We demonstrate that the nature of the resonators and guide junction completely controls the positions of the wave numbers at the reflection and transmission zeros on the real axis; the asymmetry of the resonators completely controls their positions on the imaginary axis. Thus, by adjusting these two independent parameters, we obtain a zero reflection and transmission.

Figure 1

(a) Conceptual view of the setting made of two closely spaced resonant channels (acting as a scatterer) connected to a guide where water waves propagate; the insets show pictures of the junctions made by 3D printing (b) a symmetric scatterer with $b_1=b_2$ and (c) an asymmetric scatterer ($b_1\ne b_2$) used for perfect absorption.

Figure 2

Critical curve $(d/a{)}_c$ against $b/a$ from direct numerics. (a)–(c) typical variations of ($R_0$, $T_0$) vs wave number $k$ with two perfect transmission zeros at real wave numbers ${\kappa }_0^{\pm }$ for $d/a>(d/a{)}_c$ and weak transmission dips for $d/a<(d/a{)}_c$. (d) Transmission spectrum for $b/a=4/3$ as used in the ATS experiment.

Figure 3

Experimental ATS; monopolar and dipolar resonances. (a) $\mathrm{Re}[\eta (x,y)]$, the real part of $\eta (x,y)$, measured at $f=3.3\mathrm{Hz}$. (b) Real part of the radiated field $\mathrm{Re}[{\eta }^{\mathrm{rad}}(x)]$ vs $f$, revealing a transition from monopolar to dipolar resonances. (c) Normalized symmetric $A_s$ and antisymmetric $A_a$ channel amplitudes vs $f$ ($A=|A_1|+|A_2|$).

Figure 4

Scattering coefficients vs nondimensional wave number, experiments (symbols), and numerics (lossy case, plain line; lossless case, dashed lines). The insets show $\mathrm{Re}[\eta (x,y)]$ at $f=2.7$ and $f=3.3\mathrm{Hz}$ (arrows).

Figure 5

Experimental measurements of the wave field for the perfect absorption. (a) 2D field of the surface elevation $\eta (x,y)$ and mean profile $\eta (x)$ (real part and amplitude) at $f=2.9\mathrm{Hz}$ with no reflection [no beating of $|\eta |(x)$ for $x<0$] and no transmission [$\eta (x)=0$ for $x>0$]. (b) Profiles of the incident ${\eta }^{\mathrm{inc}}(x)$ and radiated ${\eta }^{\mathrm{rad}}(x)$ waves vs frequency.

Figure 6

Experimental PA. (a) Reflection $|R{|}^2$, transmission $|T{|}^2$, and absorption $\alpha$ against nondimensional frequency $kb/\pi$. (b) modulus and phase of $A_s/A_a$.

Figure 7

Critical curves $(d/a{)}_c$ for junction 1 (ATS exp) and for junction 2 (PA exp) obtained from direct numerics. (a) The transmission spectrum for $b/a=4/3$ as used in the ATS experiment. (b) The transmission spectrum for $b/a=1.31$ as used in the PA experiment. (a),(b) We report the geometry of the junctions; $d/a=3$ and $d^{+}/a=4$ for junction 1, $d/a=2$ for junction 2. The arrows show the values of $d/a$ in the experiments.

Figure 8

PA for $ϵ={ϵ}_{\mathrm{PA}}$ against $b/d$ [dashed red line from (5)]. Outside the shaded region, ATS takes place, resulting in real-valued ${\kappa }^{\pm }$. For $b/d=0.57$, PA is achieved theoretically at circle (PA theo) and experimentally at square (PA exp). The inset shows the numerical PA with $\alpha \simeq 1$.