Mapping the dispersion of water wave channels

Large classes of electronic, photonic, and acoustic crystals and quasi-crystals have been predicted to support topological wave-modes. Some of these modes are stabilized by certain symmetries but others occur as pure wave phenomena, hence they can be observed in many other media that support wave propagation. Surface water-waves are mechanical in nature but very different from the elastic waves, hence they can provide a new platform for studying topological wave-modes. Motivated by this perspective, we report theoretical and experimental characterizations of water-wave crystals obtained by periodic patterning of the water surface. In particular, we demonstrate the band structure of the spectra and existence of spectral gaps.

: 3 leading to linearized wave-equations. Different expansions are used 14,15 , depending on the range of values of the parameters A/λ, A/h and λ/h, such as: (1) the shallow water characterized by small A/h, (2) deep water characterized by small A/h and small λ/h and (3) the intermediate regime characterized by small Ursell parameter. A unified treatment of these regimes is provided in 16 .
In the following section we provide a unifying theory of the standing wave-modes in the limit of small amplitudes. It parallels the plasmon hybridization theory 18,19 , where the solutions of the coupled hydro-and electro-dynamic equations are expanded in elementary modes and an effective Lagrangian is constructed for the amplitudes of these modes. Using specific examples, we show that the method reproduces the traditional treatments of water-wave phenomena.
Theory of standing water-waves. We consider the liquid to be confined in a tank with flat bottom and with vertical walls but otherwise of arbitrary shape. Vertical pillars of arbitrary sections can be also incorporated. The working assumptions are that the wave amplitudes are small, the liquid is incompressible and the velocity field is irrotational (i.e. no vortices are present). A schematic of a rectangularly-shaped water tank is shown in Fig. 1, which will be used to introduce our notation. The density distribution of the fluid at the equilibrium configuration is denoted by ρ . It is given by: Throughout, ρ 0 represents the density of water. The density of the fluid in a dynamical configuration is denoted by ρ(x, y, z; t), which in the notations of Fig. 1 is given by: The mass current-density is, by definition, the flux of mass per unit area and it can be expressed as j = ρ 0 v in terms of the average velocity of water molecules. It has to obey the continuity equation and the irrotational condition (i.e. absence of vortices): Inside the liquid and away from the surface, ρ = ρ 0 , hence: The common solution of these two equations can be written as j = ∇η, where η is a scalar field satisfying the Laplace equation ∇ 2 η = 0. At the interface with the fixed walls, the velocity field must be tangent hence the Laplace equation needs to be complemented with the boundary condition w · ∇η = 0 along the fixed walls, where w is the normal vector to the walls. Since the lateral walls of the container and the pillars are vertical, the elementary solutions of the Laplace equation can be obtain by separation of variables η(x, y, z) = φ(x, y)χ(z), which leads to the equation: In other words, Λ must be among the discrete set of (positive!) eigenvalues  Λ ∈ { } n n of the 2-dimensional Laplace operator with von Neumann boundary conditions and φ must be a corresponding (normalized) eigenvector: n nn n 2 2 The corresponding χ must satisfy the equation: n n n n z h 2 2 whose unique solution (up to a constant) is: At this point, let us define the constants: n n n z n n 0 which will play an important role in the following. The conclusion so far is that the scalar field η can be always decomposed in elementary modes as: n n n n The amplitudes A n play now the role of effective degrees of freedom for the system. Our next task is to derive the Lagrangian and the equation of motion for these degrees of freedom. Near the moving surface of the liquid, there will be a non-zero density difference ρ ρ − , and in the limit of small amplitudes, we can define a superficial mass density: S S V 0 1 which can take positive and negative values. Here, δV is a volume element centered around the equilibrium surface and whose width is larger than the amplitudes of the waves, hence the free surface of the liquid never intersects the upper and the bottom facets of δV (see Fig. 1). If ζ(x, y; t) represents the height of the column of fluid at point (x, y) and moment of time t, then: S 0 0 0 0 The relation between σ and the scalar function η can be derived from the continuity equation: written over volume δV. It can be transformed into: ( ( , , ; ) ( , , ))d 1 ( , , ; ) d 0, (15) which in the limit δS → 0 becomes: where we took into account that, in the limit of small wave amplitudes, the width of δV is infinitesimally small hence there is no flux of matter through the lateral sides. Also, the upper side of δV was chosen in such a way that it doesn't touches the fluid, hence the only flux of matter is through the bottom facet of ΔV. The conclusion is: x y t z x y t z x y t t x y t ( , ; ) ( , , 0; ) ( , , 0; ) ( , ; ) (17) 0 We now start the construction of the Lagrangian. The kinetic energy of the fluid is: Note that only the free surface needs to be considered in the above integral because along the fixed walls ∇η · dS = 0. Hence: The gravitational potential energy of the fluid, relative to its equilibrium configuration, is: 0 0 or: The conclusion at this step is that the Lagrangian L = T − U of the full body of fluid can be written entirely in terms of a surface integrals: is a complete orthonormal set of functions, hence we can expand: n n n Due to relation (17), the coefficients A n in (11) and C n from above are related as: n n z n n n 0 0 hence: n n n n 0 Plugging these expansions into (23) and using the orthonormality of φ n 's: The Euler-Lagrange equations of motions are: n nn leading to the frequency of the resonant modes: The resonant standing wave pattern corresponding to this frequency is φ n , defined by the eigenvalue equation (7). Example: The rectangular tank. We consider the case of a rectangular tank of length L and width W. In this case the solutions of (7) are given by: where k = (k x , k y ) is a two component vector which takes the quantized values: The corresponding frequencies are: = .
π f g k kh tanh( ) (32) k 1 2 In the infinite volume limit L, W → ∞ and for kh << 1, ≈ kh kh tanh( ) and the pulsation of the modes become ω = gh k k . This is a linear dispersion law which gives a constant speed of waves: k in full agreement with the shallow water theory [ 14 , p. 27]. Furthermore, in the limit h → ∞, ≈ kh tanh( ) 1) and the pulsation of the modes becomes ω = gk k , hence a non-linear dispersion law, leading to an anomalous wavelength:   In order to verify that the frequencies at which peaks appear in the acquired signal are resonant modes, they were manually inputted into the function generator to actuate the system. It was then possible to observe (either via imaging to a screen or directly viewing the water) that these frequencies were indeed resonant modes.
A comparison between the theory and experiment for the two channels is shown in Fig. 4. The data in Fig. 4a was acquired using a channel of length, L = 19.75 cm, and height of water, h = 1.4 cm. The black, square data points correspond to the theory and the red, circular points correspond to the experiment. The experimental points correspond to the lower x-axis only, while the theoretical points correspond solely to the upper x-axis. Both share a common y-axis. As shown, at frequencies where resonant modes exist, a signal with higher intensity is acquired by the photodiode. Figure 4b demonstrates a typical phonon spectrum for water waves in a flat-walled waveguide. In anticipation of the periodic modulations introduced next, the data is shown as function of the wavenumber k · a, with a the periodicity of the modulations. The wavenumber axis is folded into the interval [−π, π] but only the positive part is shown. Figure 4c demonstrates the theory and experimental data for a flat-walled channel of L = 26.5 cm, and height of water, h = 1.4 cm. The black, square points correspond to the theory and the upper-x axis only, while the red, circular points show the experimental data and correspond to the lower x-axis. The phonon spectrum for the longer channel is demonstrated in Fig. 4d. The spectra for both lengths are missing some experimental data points, particularly at higher frequencies. This can be explained as a consequence of the measuring system's resolution. Due to the length limitations of the waveguide, higher resonant modes are difficult to distinguish because the distance between nodes decreases. As this occurs, it becomes harder for the laser (with a set spot size) to distinguish a clear resonance.
Periodically patterned wave-channel. A periodically patterned water wave guide was designed by 3D printing a channel with teeth spaced 1.12 cm apart. In order to study the properties of water within the wave guide, two channels of length, L = 19.75 cm and L = 26.5 cm were fabricated. The channels were filled to a height of water, h = 1.4 cm and excited over frequencies of 0.01 Hz and 15 Hz with 0.01 Hz step sizes. The measured signals for a 26.5 cm long periodic channel over three separate trials are shown in Fig. 5. Slight shifts in resonant frequencies are visible between the trials. A comparison of periodically patterned channels of two lengths is shown in Fig. 6. The red circle points correspond to a channel with, L = 19.75 cm and the black square points to L = 26.5 cm.
A comparison of the experimental resonant frequencies of a 26.5 cm long periodically patterned channel and theoretical values (Eq. (35)) is show in Fig. 7. When the water is in resonance, the amplitude of the signal is greater and a peak appears. This behavior is similar to that of the flat-walled channel experiments. The experimental data corresponds to the lower x-axis and y-axis, while the theoretical points correspond to the upper x-axis and y-axis. Gray dashed lines have been added to demonstrate where expected resonances should occur. Figure 7b shows the phonon spectrum of the periodic channel with the wavenumber axis folded as previously explained.

Discussion
In all reported experiments, the resonant modes are dependent on the height of the water within the channel as governed by Eq. (32). Maintaining the height of the water is extremely important. Due to the small step size in frequency (0.01 Hz), the experiments took approximately 5 hours to perform. Throughout the duration of that time, it is possible that some of the water in the channel evaporated. As a result, it is possible that a measurable shift in resonant frequencies occurred. However, it is believed that throughout the course of the reported experiment significant evaporation did not occur. The repeatability of the experiment for a 19.75 cm long flat-walled channel is show in Fig. 2. The data suggests that the experiment is repeatable with small deviations between trials. This behavior can be accounted for by considering the inaccuracy in the height of the water in the waveguide. Since the channels are filled by volume, if slightly off, the resonant frequencies will be different. In addition, if any evaporation occurs, shifts in resonant frequencies are possible. Also, the signal is highly dependent on the initial intensity of the transmitted light; therefore, if any change in it occurs, the overall amplitude varies. At higher resonant modes (>7), the intensity of the transmitted signal is greater. This can be explained by the Beer-Lambert law, which states that as the height of the fluid in the channel decreases, the intensity of the transmitted signal increases. Thus, as higher resonant modes form and the peaks for the waves are not as high as at lower ones, the height of the fluid, is smaller. Thus, the intensity of the transmitted signal is larger and an increase is reflected in the acquired data.
Similarly, Fig. 2 demonstrates the repeatability of a 26.5 cm longer periodically patterned waveguide. Once again, the data suggests the experiment is repeatable with small deviations between trials. The same reasons the variability exists in the flat-walled channels can be applied to the periodic channels. A comparison of two different lengths of periodic waveguides is shown in Fig. 6. As in the case of the flat-walled channel, the signal is clearly different between the two lengths. A difference in signal is expected since length is one of the main factors that governs the resonant frequencies of the water within the channels. For periodic systems, spectral gaps are expected. It is difficult to identify the location of said gaps solely from the RMS signal, but they can be seen when plotted against the theoretical values.
The comparison of experimental and theoretical data for a flat-walled channel shown in Fig. 4 is a representative plot that demonstrates how the theory aligns with the experiment for two different lengths. Raw data compared with theory is show in (a) and (c). On the right, (b) and (d) demonstrate the spectrum where the x-axis corresponds to the situation when the k vector is modified. For a flat-walled channel, we assume no gaps open and  Fig. 7. The data suggests that a resonance is not measured at every theoretical point, particularly those where the spectrum winds back on itself. This behavior can be explained by the presence of gaps due to the periodic spacing of the teeth.

Methods
The experimental apparatus. Flat-walled and periodic channels were 3D printed using acrylonitrile butadiene styrene (ABS) shown in Fig. 8. At frequencies much higher than those actuated in the present experiments, harder materials should be used for the walls. The waveguides were positioned atop an acrylic sheet suspended by 20 cm long pendulum rods to maintain small oscillations (that is, θ < 5°). In order to prevent leakage of fluid, a seal was created using silicone caulk between the channel and the acrylic. Adafruit Industries 5 mW 650 nm laser diodes were positioned above the channel and permitted to emit light through the water and acrylic onto an Opto Diode Corp 9.91 mm × 4.28 mm photodiode positioned underneath. The experimental apparatus can be seen in Fig. 9 (the channel can be interchanged depending on geometry of interest). The system was actuated by using an HP 33120A function generator connected to a Brüel & Kjaer Power Amplifier Type 2718 which drove a Brüel & Kjaer 4810 Mini-shaker. The actuator was attached to the acrylic sheet using two bolts. The function generator was computer-controlled by a previously reported 20 custom LabVIEW program via a USB GPIB cable and a signal    This system made it possible to take a measurement of the intensity of transmitted light utilizing laser/photodiode pairs, which identified resonant frequencies of the fluid within the channel. As the water approached a resonant frequency, the intensity of the transmitted light increased. This behavior occurs because when the fluid is off-resonance, the turbulent nature of the water refracts the light away from the photodiode, and decreases the intensity of the attenuated light. As the system becomes resonant, the light is allowed to focus on the photodiode and the highest intensity of the light is transmitted. The intensity of the transmitted signal is measured by the Beer-Lambert Law 21 : Where I is the intensity of the signal, I 0 is the intensity of the signal with no fluid present, α is the absorption coefficient, and h is the height of the fluid in the channel. Therefore, as the height of the water in the channel changes, the intensity of the signal varies. Recalling Eq. (32), it is clear that the resonant modes also depend on the height of the water in the channel. The experiments reported in this paper were performed with a height of water of h = 1.4 cm.
Previous methods. Prior to finalizing the above-mentioned suspended design of the actuating system, various methods were tested and deemed inadequate for testing. The first design utilized a circular tank of water which rested on a raised sheet of Poly(methyl methacrylate) (PMMA). The fluid was actuated by a cylindrical pin positioned off-center. In this configuration, waves propagated quite well at higher frequencies (>20 Hz), but poorly at lower ones. Resonant modes were determined by observation of the water within the system and images could be acquired on a screen positioned underneath the raised PMMA sheet. Standing waves formed at resonant modes and were relatively easy to observe. While this approach worked quite well with higher frequencies, below 20 Hz it was extremely difficult to distinguish resonant modes by eye. In order to measure the resonant modes at lower frequencies, the same apparatus was utilized, but a laser was positioned at a 45° angle such that the light hit the water at the center of the circular tank. A detector was placed opposite it to detect the reflected light from the surface of the water. Due to the multiple-interface system, it was extremely difficult to isolate the reflected laser light from the surface of the water onto the detector. The data was inconclusive at low frequencies and a new method was pursued. In addition, the circular tanks were replaced with flat-walled rectangular ones of varying lengths.
The second design used a glass tank of water 2.54 cm tall in which the rectangular channels were positioned. The fluid was actuated utilizing the same cylindrical pin as previously described. For the flat-walled circular tanks this method of actuating the fluid proved quite successful. However, upon switching to non-circular tanks, the waves did not propagate throughout the system as expected due to an inefficient amount of energy required to excite the wave across the full length of the channel. It was hypothesized that switching the small-tipped cylinder with a wider paddle would produce enough energy to excite the system. A tip was still desired, so the geometry was changed by angling the walls varying degrees. Paddles of various sizes were used, but proved inefficient at exciting the water. It was found that a paddle with angled walls of 45° could sufficiently excite the water across the channel length. This paddle was capable of exciting the water throughout the system for flat-walled rectangular tanks; however, could not do so for channels with periodicity. Therefore, a new method of exciting the water was pursued.
The next design utilized a glass sheet atop an actuator, similar to a Chladni plate. The channels were placed and sealed on the plate for excitation. A stroboscope was designed using a red LED that flashed at the same frequency of the excitation. A switch was incorporated to allow the user to select the strobe light or steady light for imaging. This method of excitation worked rather well, and resonant modes at low frequencies were acquired (see Fig. 10).
While this method of water excitation proved quite successful, it was not without flaw. Determination of the resonant modes solely relied on observation of the modes in the channel. Videos and images were acquired demonstrating these observations. For flat-walled channels, this method seemed adequate. Once a periodic channel was installed, this method no longer was viable since resonant modes were not as clearly defined. It became apparent that observing resonant modes in channels with periodicity would not be possible. In order to determine what was happening in said channels, the laser/photodiode apparatus described earlier was adapted.