2.1. Governing equations

In the analysed case, the propagation of waves can be described by the Helmholtz equation along with the Neumann boundary condition at the rigid walls of the waveguide, at $y=\pm L/2$ , and the cylinder, at $x^2+y^2=D^2/4$

(2.1) \begin{equation} \Delta \eta (x,y) + k^2 \eta (x,y) = 0 ,\end{equation}

(2.2) \begin{equation} \frac {\partial \eta }{\partial \textbf {n}}\Big|_{{\rm {walls}}} = 0 ,\end{equation}

with $k$ being a wavenumber following the linear dispersion relation

(2.3) \begin{equation} \omega ^2 = gk\tanh (kh) ,\end{equation}

where $g$ represents the gravity constant, $\omega$ denotes frequency and $h$ is the water depth. We restrict our considerations to the low-frequency range $kL/\pi \lt 1$ , i.e. there is only one mode propagating. The solution in the far-field region of the cylinder can be expressed as

(2.4) \begin{equation} \eta (x)=\begin{cases} Ae^{ikx} + ARe^{-ikx}, & \text {in the region I,}\\ ATe^{ikx}, & \text {in the region II,} \end{cases} \end{equation}

where $A$ denotes the amplitude of the incident wave and $R$ and $T$ denote the reflection and transmission coefficients, respectively. Region I corresponds to the far-field region in front of the cylinder ( $x\lt 0$ ), and region II corresponds to the far-field region behind the cylinder ( $x\gt 0$ ).

2.2. Making defect invisible – framework

We perform numerical simulations with a cylinder diameter of $D=0.6L$ in a domain $x\in (-5,5)L$ . We solve the Helmholtz equation (2.1) using the finite element method implemented in MATLAB PDE Toolbox. We impose the Dirichlet boundary condition $\eta =1$ at $x=-5L$ and radiation condition $\partial _x\eta =ik\eta$ at $x=5L$ . The maximum edge length of the mesh element is $h_{max }/L=0.01$ . It is worth noting that this problem can be solved using other methods, such as the method proposed by Kirby (Reference Kirby2008) combining a wave-based modal solution with the finite element solution. The scattering of a cylinder within a parallel waveguide can be analysed using the multipole method proposed by Linton & Evans (Reference Linton and Evans1992). We verify the results obtained using the finite element method with the multipole method for this geometry. We also consider different domain sizes and confirm that the considered domain size $x\in (-5,5)L$ is sufficient to provide accurate results.

Figure 2. Examples of wall modifications considered in the optimization problem with the parameters defining the shape: (a) $h_g/L= 0.16, \gamma /L = 1.6, \delta /L=1.06, N = 2$ and (b) $h_g/L = 0.2, \gamma /L = 1.6, \delta /L = 0.29,$ $N = 60$ .

The presence of the cylinder in the analysed scenario leads to a significant reflection of incident wave energy. In the case of a perfectly hidden object, there is no reflection ( $R=0$ ), resulting in perfect transmission ( $T=1$ ). We aim to make the cylinder invisible to incident waves by creating narrow indentations $F$ in the waveguide walls near the obstacle. The walls are given by $y=\pm (L/2+F)$ . To determine the specific shape of the wall perturbation providing broadband reduction of the reflection coefficient, we minimize the cloaking factor $\chi$ in the frequency range $k\in (k_1,k_2)$ with $k_1=0.01\pi /L$ and $k_2=\pi /L$ . The cloaking factor is defined as

(2.5) \begin{equation} \chi = \frac {\int _{k_1}^{k_2} |R|^2{\rm d}k}{\int _{k_1}^{k_2} |R_0|^2 {\rm d}k}, \end{equation}

where $R_0$ denotes the reflection coefficient for the reference case with straight parallel walls. One has to notice that, due to energy conservation, minimizing the reflection coefficient results in the absolute value of the transmission coefficient being close to unity but does not guarantee that the phase shift in the transmitted wave is zero. As an alternative objective function, we consider minimization of the cloaking factor based on total scattered energy, which physically represents radiated power, defined in the following way:

(2.6) \begin{equation} \chi _{sc}=\frac {\int _{k_1}^{k_2} (|R|^2+|T-1|^2){\rm d}k}{\int _{k_1}^{k_2} (|R_0|^2+|T_0-1|^2){\rm d}k} .\end{equation}

To minimize $\chi$ and $\chi _{sc}$ , we consider wall deformations that are symmetrical with respect to the cylinder. The variability of the walls near the obstacle is achieved through the utilization of a cosine Fourier series

(2.7) \begin{equation} F(h_g,\zeta ,n,x)= \sum _{n=1}^N \frac {2h_g}{\pi (2n-1)}\left \{ \cos \left (\frac {\zeta \pi }{\gamma } (2n-1)x \right )- \cos \left (\zeta \pi (2n-1)\right ) \right \}(-1)^{5n-1} ,\end{equation}

(2.8) \begin{equation} \zeta = \left\{\begin{array}{ll} \lfloor \frac {2\gamma }{\delta } \rfloor , & \mod ( \lfloor \frac {2\gamma }{\delta } \rfloor ,2) \neq 0,\\ 1, &\quad \textrm{otherwise.} \end{array} \right. \end{equation}

Figure 3. (a) Geometries providing cloaking phenomenon: Fourier series geometry (blue solid line, $h_g/L= 0.258, \delta /L = 1.93, \gamma /L = 0.97, N = 40$ ), trapezium (green solid line, $a/L=0.485, b/L=0.465,$ $h_t/L=0.26$ ) and rectangular geometry (red dashed line, $h_r/L=0.26, w_r/L=0.481$ ). (b) Reflection coefficient in the considered frequency range. The inset presents the reflection coefficient for three considered geometries in a logarithmic scale to show the difference in the results.

In the above relation, $h_g$ denotes the depth of the groove in the channel wall, $\gamma$ represents half of the width of the variable geometry part, $\delta$ denotes the width of a single groove and $\lfloor 2\gamma /\delta \rfloor$ denotes the floor function. We impose constraints on the parameters defining wall deformation: (i) $1\leq N\leq 60$ , (ii) $0.09\leq h_g/L\leq 1$ , (iii) $0.2\leq \delta /L\leq 4$ and (iv) $0.1\leq \gamma /L\leq 2$ . The constraint $\delta \lt 2\gamma$ is introduced to eliminate incorrect geometries. Figure 2 depicts examples of geometries considered in the optimization problem with a graphical representation of different parameters defining the geometry of waveguide indentations. The shape of indentations given by (2.7) is initially chosen due to the lack of knowledge about the expected optimal shape. Its definition is developed to ensure that a few parameters – such as the length of the region with variable geometry, depth, number of grooves and the shape of individual grooves – could unambiguously define the geometry of the grooved region. Additionally, continuity between the variable geometry region and the section with parallel waveguide walls is maintained.

The optimization process requires (i) mesh generation for a given set of parameters defining the wall shape perturbation, (ii) solving of the Helmholtz equation for the considered frequency range $kL/\pi \in (0.01,1)$ and (iii) calculation of the cloaking factor $\chi$ . Given the lack of precise knowledge regarding the function $\chi (F)$ and the absence of analytical or derivative information, we employ a black box approach (Audet & Hare Reference Audet and Hare2018). Considering the high computational cost of objective function evaluation, we use a surrogate-based method with a radial basis function approximation (Gutmann & Gutmann Reference Gutmann and Gutmann2001; Vu et al. Reference Vu, D’Ambrosio, Hamadi and Liberti2016). The initial guess is evaluated at 63 random points in the parametric space within the constrained values.

2.3. Numerical results

Minimizing the cloaking factor $\chi$ results in the geometry depicted in figure 3(a) by solid blue line (denoted as Fourier) with $\chi =5.44\times 10^{-5}$ . The optimized shape resembles a trapezium, so we verify if we can simplify the indentation geometry and keep a similar objective function value. Geometry simplification is beneficial considering the manufacturing aspects of the experimental counterpart. Therefore, we repeat the optimization process, but this time, we vary the geometry by changing the height $h_t$ and half of the lengths of both bases of an indentation $a$ and $b$ . As a result we obtain $\chi = 1.96 \cdot 10^{-4}$ for $a/L=0.485$ , $b/L=0.465$ and $h_t/L=0.26$ (figure 3 a green solid line). The optimized trapezium has almost the same length of bases, so we go one step further with simplification, i.e. we perform optimization for a rectangular wall modification. We vary the height $h_r$ and the half of the width $w_r$ of the indentation. The best result is obtained for $h_r/L=0.26$ and $w_r/L=0.481$ with $\chi = 1.77\cdot 10^{-4}$ (figure 3 a, red dashed line). The efficiency of the rectangular geometry is shown in figure 4, where we present wave fields for $kL/\pi =0.8253$ obtained for the reference case (parallel waveguide) and the waveguide with rectangular indentations. The comparison of the reflection coefficient in the considered frequency range for three considered wall modifications and the reference case is presented in figure 3(b), where a significant backscattering reduction is visible. The local minima of the reflection coefficient correspond to the resonant frequencies of a given system. In the case of rectangular indentations, they are observed for $kL/\pi \approx 0.5$ and $kL/\pi \approx 0.9$ .

We perform the same analysis in the case of cloaking factor $\chi _{sc}$ (2.6) for three different geometries of the indentations, i.e. Fourier, trapezium and rectangular. The minimum value of $\chi _{sc}$ is obtained for the geometry defined through the Fourier series (2.7) with $\chi _{sc}=0.239$ . In the trapezium case, we obtain $\chi _{sc}=0.24$ and for the rectangle $\chi _{sc}=0.319$ . Optimization of the cloaking factor based on total scattered energy provides slightly better results compared with optimization of $\chi$ in terms of the phase difference ( $\Delta \varphi =\arg T$ ), where the phase shift is below $0.15\pi$ . The optimization of $\chi$ results in a phase shift below $\Delta \varphi \approx 0.25\pi$ in the considered frequency range. Unfortunately, in the case of minimizing $\chi _{sc}$ , the backscattered energy is much higher (see Appendix A for details), so we decided to perform experiments with the geometry provided by the minimization of $\chi$ .

The presented results are based on the initial choice of the function $F$ , given by (2.7), which defines the shape of the indentations. It is worth noting that other choices of $F$ might yield even better results with respect to the cloaking factors $\chi$ and $\chi _{sc}$ .

Figure 4. Numerical simulation. Real part of the wave fields for $kL/\pi =0.8253$ : (a) for a waveguide with parallel walls (reference case) and (b) with rectangular indentations. Wave fields are normalized by the incident wave amplitude.

2.3.1. Wave drift force

For surface water waves, a cloaking device modifies not only scattered field but also the forces acting on an object. Cloaking an object should significantly reduce the force incident waves exert on the body. The force acting on a fixed body can be determined by integrating the pressure on the body’s surface

(2.9) \begin{equation} \boldsymbol{F}(t) = \int _{S} \rho \left ( \frac {\partial \Phi }{\partial t} +gz + \frac {1}{2} ( \nabla \Phi )^2 \right ) \boldsymbol{n}_S \rm d \it S, \end{equation}

with $\Phi$ being the velocity potential. We can decompose $\boldsymbol{F}(t)$ into a sum of first-order and second-order components. The second-order term has a fluctuating part at frequency $2\omega$ and a time-independent wave drift force $\overline {\boldsymbol{F}^{(2)}}$ , which can be expressed as

(2.10) \begin{equation} \overline {\boldsymbol{F}^{(2)}} = \frac {1}{T} \int _0^T \left( -\frac {1}{2} \rho g \int _{z=0} \big(\eta ^{(1)}\big)^2 \cdot \boldsymbol{n}_s {\rm d} \ell + \frac {1}{2} \rho \int _S \big( \nabla \Phi ^{(1)} \big)^2 \cdot \boldsymbol{n}_s {\rm d}S \right) {\rm d} t ,\end{equation}

where $S$ is the wetted body’s surface, $\boldsymbol{n}_s$ denotes the unit normal vector and ${\rm d} \ell$ represents an infinitesimal segment along the water contour at $z=0$ (Dupont et al. Reference Dupont, Guenneau, Kimmoun, Molin and Enoch2016). The first-order surface elevation is denoted as $\eta ^{(1)}(x,y,t) = \textrm {Re}\{\eta (x,y)\cdot e^{-i \omega t} \}$ and $\Phi ^{(1)} =\Phi ^{(1)}(x,y,z,t)$ denotes the first-order velocity potential expressed as

(2.11) \begin{equation} \Phi ^{(1)} (x,y,z,t) = \textrm {Re}\left (\frac {g}{i\cdot \omega } \cdot \eta (x,y) \cdot \frac {\textrm {cosh}(k(z+h))}{\textrm {cosh}(kh)} \cdot e^{-i\omega t}\right ) .\end{equation}

We evaluate the mean drift force acting on the cylinder in the presence of waveguide wall indentations providing invisibility and compare the results with the reference geometry. We analyse the influence of rectangular indentations provided by the minimization of the cloaking factor $\chi$ . The results concerning the horizontal component of the drift force are presented for the two cases in figure 5(a). The black and red solid lines correspond to the force acting on a cylinder in a parallel waveguide ( $\overline {F_X^R}$ ) and with rectangular indentations in the waveguide walls ( $\overline {F_X^C}$ ), respectively. Implementing our cloaking device significantly reduces the horizontal force acting on the surface-piercing cylinder in the entire considered frequency range, with the maximum force being less than $2\%$ of the force in the reference case (see figure 5 b).

Figure 5. The mean drift force as a function of frequency $kL/\pi$ : (a) the distribution of the mean horizontal force acting on the cylinder for the reference case and the cloaked case, respectively; (b) the ratio of the mean horizontal forces $\overline {F_x^C}/\overline {F_x^R}$ , where $\overline {F_x^C}$ represents force for the cloaked case and $\overline {F_x^R}$ refers to the reference case with parallel walls.

2.3.2. Limitations of the cloaking device

Any passive cloaking device designed to achieve invisibility within a specific frequency range does so at the cost of increased scattering outside this range (Monticone & Alù Reference Monticone and Alù2016). Our study focuses on optimizing waveguide wall indentations to reduce backscattering within the frequency range $ kL/\pi \in (0.01,1)$ . To evaluate the performance of this system, we analyse its scattering behaviour outside the optimized frequency band, specifically in the range $ kL/\pi \in (1,2)$ . Figure 6 compares the reflection coefficients of a cylinder in a waveguide with straight parallel walls (reference case, black solid line) and a cloaking geometry featuring rectangular wall indentations (red line). In the reference case, the reflection coefficient decreases initially, reaching resonance at $kL/\pi =1.63$ , before increasing with frequency. For the cloaking geometry, the reflection coefficient is significantly reduced up to $kL/\pi =1.57$ . Beyond this frequency, however, the reflection coefficient rises sharply, exceeding that of the reference case.

Extending the frequency range over which a cloaking device effectively minimizes scattering is a possibility that requires further investigation. Achieving a broader cloaking bandwidth would likely come at the cost of an increased cloaking factor $\chi$ (Monticone & Alù Reference Monticone and Alù2016).

Our cloaking device, consisting of indentations in the waveguide walls, is obtained through an optimization procedure based on the model of linear water waves. It is, therefore, important to consider the potential response of this system to nonlinear waves. In the weakly nonlinear regime, neglecting the forcing by the fundamental frequency ( $\omega$ ), the harmonics satisfy the Helmholtz equations (Belibassakis & Athanassoulis Reference Belibassakis and Athanassoulis2011)

(2.12) \begin{equation} \left (\Delta +k_n^2\right )\hat {\eta }_n(x,y)=0 ,\end{equation}

where $k_n$ is associated with $n\omega$ , i.e. $D(n\omega ,k_n)=0$ , where $D(\omega ,k)=\omega ^2-gk\tanh (kh)$ denotes the water wave dispersion relation. A harmonic that satisfies (2.12) is called a free wave, whereas a non-dispersive term, with a wavenumber $nk_1$ , being slaved to fundamental frequency is called a bound wave (Monsalve et al. Reference Monsalve, Maurel, Pagneux and Petitjeans2022). For a cloaking geometry and waves with a fundamental frequency $\omega$ such that associated $k$ is in the range $kL/\pi \in (0,1)$ , the first-order reflection coefficient is very low. Based on the results presented in figure 6, we expect that backscattering of the free wave component $k_2$ (associated with $2\omega$ ) will also be low for $k_2L/\pi \lt 1.57$ with the cloaking geometry, but for $k_2L/\pi \gt 1.57$ the reflection of this component would significantly increase. Given that the energy content of the fundamental frequency typically dominates the total energy, our cloaking device is expected to perform very well even for nonlinear waves within the considered frequency band. Nevertheless, the increased scattering of harmonic components outside this band highlights a limitation that could be addressed in future designs.

Figure 6. Reflection coefficient as a function of frequency for a cylinder in the waveguide with straight parallel walls (reference case) and cloaked case (with rectangular indentations).

3.1. Measurement method

We perform the measurements using a combination of the optical flow (OF) technique (Farnebäck Reference Farnebäck2003) and the synthetic schlieren (SS) technique (Moisy, Rabaud & Salsac Reference Moisy, Rabaud and Salsac2009). The OF method enables the extraction of displacements from a reference image (e.g., dots), whereas the SS method transforms these displacements into deformations of the free water surface (Bheeroo & Mandel Reference Bheeroo and Mandel2023). The SS method assumes certain geometrical approximations and, as a result, has a few limitations (Metzmacher et al. Reference Metzmacher, Lagubeau, Poty and Vandewalle2022): (i) weak amplitudes, (ii) weak slopes and (iii) weak paraxial angles. The distance between our cameras and the dot pattern placed on the bottom of the channel is $2.39\, \rm m$ , resulting in a maximum parallax angle of $\beta _{max}=0.22$ . In the considered frequency range, the amplitude of the generated waves is in the range of $A\in (0.39,0.94)\, \rm mm$ with the corresponding wave steepness in the range of $kA\in (4.76 \times 10^{-3},13.85\times 10^{-3})$ . The error in free-surface deformation reconstructions can be estimated as (Moisy et al. Reference Moisy, Rabaud and Salsac2009)

(3.1) \begin{equation} \frac {\Delta \eta }{\eta _{{rms}}}=(5.0\pm 0.2)\frac {L}{\lambda }\frac {\epsilon }{N_p} ,\end{equation}

with $\epsilon$ being relative Gaussian noise, $\lambda$ being wavelength and $\eta _{{rms}}=A/\sqrt {2}$ for a sinusoidal wave, where $rms$ is the root-mean-square. In our system, the error for the first camera is in the range $0.07\times 10^{-3}$ to $0.23\times 10^{-3}$ , while for the second camera, the error ranges from $0.08 \times 10^{-3}$ to $0.26\times 10^{-3}$ .

3.2. Experimental results

Based on the obtained temporal evolution of the free-surface deformations, we extract wave fields using the Fourier transform in the time domain

(3.2) \begin{equation} \hat {\eta }_n(x,y,\omega ) = \frac {2}{T_p} \int _{T_p} \eta (x,y,t) e^{i n \omega t} {\rm d}t ,\end{equation}

with $\omega$ being the fundamental frequency with $T_p=2\pi /\omega$ . An example of a wave field ${Re}(\hat { \eta }_1(x,y) )$ is shown in figure 8 where, for the reference case (figure 8 a,c), strong reflection is observed in front of the cylinder and a decrease in the wave amplitude behind the obstacle. The corresponding wave fields obtained in the case of the rectangular cloaking geometry are presented in figure 8(b,d). Based on wave field $\hat {\eta }_n(x,y)$ , we extract transverse modes by projecting it onto $\phi _n(y)=\sqrt {(2-\delta _{0n})/L}\cos (n\pi y/L)$

(3.3) \begin{equation} \eta _n(x)=\int _0^L\hat {\eta }_n(x,y)\sqrt {(2-\delta _{0n})/L}\cos (n\pi y/L)\text {d}y. \end{equation}

Figure 8. Real part of the experimental fields $\hat {\eta }_1(x,y)$ for parallel waveguide (a for $kL/\pi \approx 0.79$ and c for $kL/\pi \approx 0.94$ ) and rectangular cloaking geometry (b for $kL/\pi \approx 0.79$ and d for $kL/\pi \approx 0.94$ ). Wave fields are normalized by the incident wave amplitude.

Figure 9. Comparison of numerical and experimental results. Reflection $|R|$ and transmission $|T|$ coefficients as a function of frequency $kL/\pi$ . Black markers (experiment) and solid magenta lines (numerical simulation) correspond to the reference waveguide with straight walls. Blue markers (experiment) and red solid lines (numerical simulation) correspond to the waveguide with modified walls.

In the considered frequency range, only the plane mode propagates. This fact allows us to extract reflection $R$ and transmission $T$ coefficients using (2.4) (taking into account reflection from the end of the channel). In figure 9, we present a comparison of both coefficients as a function of $kL/\pi$ for numerical simulations and experimental values. The error bars for the coefficients were estimated based on four realizations of the experiment at a $95\,\%$ confidence level and expressed as $\overline {|R|} \pm t_{(n-1,1-\alpha /2)}{s}/{\sqrt {n}}$ (similarly $\overline {|T|}$ ) with $n$ , $\overline {|R|},s$ being the number of realizations, sample mean and sample standard deviation of the results, respectively (Martin Reference Martin2012). The $t_{(n-1,1-\alpha /2)}$ parameter is the critical value of the Student’s t distribution for significance level $\alpha =0.05$ . We observe a significant reduction of the reflection coefficient in the entire frequency range. In the inset of figure 9, we present the same reflection coefficient in logarithmic scale to demonstrate the extremely high efficiency of the proposed cloaking device. Although the experimental values are not as good as in numerical simulations, one can notice that the reflection coefficient decreases by at least ten times the reference value corresponding to the straight parallel waveguide, meaning that, in the investigated frequency range, the energy of the reflected wave, proportional to $|R|^2$ , is at least a hundred times lower than initially. The best experimental result is obtained for $kL/\pi \approx 0.64$ , where backscattered energy is around 5000 times lower than in the reference case.