A modular experimental system for teaching fluid dynamics with Faraday waves

We describe a modular setup for the observation of Faraday waves on a vibrating bath. The setup will be used as a project exercise on fluid dynamics in a first-year course on experimental physics at Aarhus University as well as for future research on fluids. As a demonstration of the setup, the acceleration threshold for the onset of Faraday waves on a silicone oil bath as a function of the driving frequency is measured and compared to thresholds calculated using different existing models. The possibility to characterize surface waves with the system is demonstrated by recording and analyzing images of Faraday waves, e.g. showing explicitly that the Faraday waves in the present case are subharmonic and establishing the dispersion relation for the waves.

in addition to the constant gravitational acceleration g. The fluid is characterized by a density ρ, a kinematic viscosity ν (dynamic viscosity η = ν × ρ), and a surface tension σ, and another fluid can be added to the bath or be the atmospheric air (ρ 2 ∼ 1.2 kg m −3 , η 2 ∼ 1.8 × 10 −5 Pa s) defining an upper boundary of the fluid. Figure 1 shows a graphical illustration of this physical model system. For a given forcing frequency, the liquid surface becomes unstable once the applied acceleration exceeds a threshold value, i.e. a a c ( f d ), which also depends on the properties (ρ, ν, and σ) of the liquid as well as on the liquid height (h) in the bath (plus the fluid properties of a possible additional fluid layer). Just above this threshold, the liquid surface displays Faraday waves with wavelength λ F and frequency f F = 1/2f d (subharmonic) or f F = f d (harmonic) and their interdependence (dispersion relation) reflects the nature of the waves (gravity and capillary waves) and potentially the nature of the interaction between the liquid and the walls of the bath.
This model system allows for controlled studies of surface waves [1][2][3][4][5] and liquid surface interactions [6][7][8][9][10][11] and as such it is a prototype system to examine fluid dynamics [12]. The model system has a special role in the development of fluid mechanics as it in particular allows isolated investigations of the instabilities of the fluid interfaces as pioneered by Michael Faraday [13], and later was the exact model system investigated in the seminal papers by Benjamin and Ursell [14], who provided a linear stability analysis without viscosity, and by Kumar and Tuckermann [12], who gave a full linear stability analysis including viscosity. Several papers, e.g. [15][16][17] have reviewed both early and modern experimental and theoretical studies on the Faraday instability. A full exploration of Faraday waves, i.e. solving the Navier-Stokes equations with appropriate boundary conditions for the model system depicted in figure 1, especially above threshold has not been realized analytically and most likely new aspects of this system are still to be found experimentally [1] or through computational investigations [17][18][19]. Recently, vibrating baths have gained additional attention as it was discovered [20][21][22][23]] that a small droplet of the same liquid can bounce and walk on the oscillating surface in the region below the acceleration threshold (a < a c ) for generation of Faraday waves where the surface is still stable. An oscillating liquid with a bouncing droplet is a fascinating dynamical system, both as a research object and for teaching, which connects a particle, the droplet, and its wave field, i.e. the surface wave it introduces on the liquid surface by bouncing. While the fluid system is of course purely classical in nature, the droplet-liquid system in some sense resembles the de Broglie-Bohmian pilot-wave view of quantum mechanics where a particle is associated explicitly with a wave field, and several analogs to quantum theories and experiments have been pursued with the droplet-liquid system [24][25][26][27][28][29][30]. Evidently, despite its conceptual simplicity, the coupled motion of a droplet and its wave field is very challenging to describe and its scientific exploration is still ongoing both experimentally and theoretically.
The experimental system with an oscillating bath (figure 1) thus forms the core of more presently active research fields within fluid dynamics. The relative simplicity of the experimental system and its fundamental role as an interesting model system in fluid mechanics makes it attractive for teaching physics at undergraduate level, since direct illustrations of essential fluid dynamical concepts (e.g. the Faraday instability and dispersion relations) can be done quantitatively and, moreover, direct connections to fascinating modern research (e.g. on bouncing and walking droplets) can be made without complicated instrumentation.
In this paper, we describe an experimental realization of a system for studying both surface waves on the oscillating liquid as well as bouncing and walking droplets. The setup is made modular with the possibility to add inserts to the bath, for example, to study the occurrence of surface waves under different boundary conditions, e.g. straight walls or brimful conditions, and droplet motion in designed structures. Further, the setup is realized with laboratory equipment available or realizable at low cost in standard teaching laboratories at the university. The setup is embedded as a project exercise in a course on experimental physics for first-year students at Aarhus University and will also find future use on more advanced levels of study and in student-driven research on fluid properties and droplet dynamics.
The focus in the present paper is on describing and characterizing the oscillating bath by showing results on Faraday waves obtained using it, while results demonstrating its ability to study bouncing and walking droplets will be described in a separate article.

Qualitative outline of the description of the Faraday instability
For an inviscid liquid (η = 0), the Faraday instability was first described with a linear stability analysis by Benjamin and Ursell [14] exploiting the fact that Mathieu equations for each Fourier amplitude of normal modes of the surface waves can be formulated in this limit. For viscous fluids, a complete linear stability analysis was given by Kumar and Tuckerman [12] for two superimposed fluid layers and by Kumar [31] for one fluid layer. They demonstrated how the acceleration threshold for the onset of Faraday waves at a given driving frequency and at a given wavenumber can be evaluated by solving an eigenvalue problem, and they demonstrated how, for a given driving frequency, the corresponding observable values of acceleration threshold and wavenumber of the Faraday waves can be obtained. They presented a full hydrodynamical model (FHM) [12,31] and a phenomenological model (PhM) [12] in which an estimated representation of the viscous damping rate was used. In the limit of weak dissipation, Müller et al [32] derived an explicit analytical model (AnM) for both the acceleration threshold and the corresponding wavenumber at a selected driving frequency.
The mathematical details of these evaluations [12,31,32] are somewhat complicated and actual computations can potentially be difficult to perform for undergraduate students. As supplementary information, we therefore provide example codes written for MATLAB [33] from which such calculations can be done. Below, we give a brief outline of the main physical arguments leading to the analysis of the Faraday instability and illustrate the properties of Faraday waves by explicit calculations for selected cases with a fluid similar to the Silicone oil used in the present work. We have decided to follow a notation similar to the one given in the paper by Kumar [31].
The description of Faraday waves on the surface of an incompressible (i.e. ρ is constant) and Newtonian (i.e. η is independent of stress and flow rate) fluid proceeds from the equation of continuity (mass conservation) is the velocity field of the fluid at position x = (x, y, z), and the corresponding Navier-Stokes equations for the fluid motion here p is the local pressure in the fluid, and the components of the stress tensor π can be written: ith j and k representing x, y, and z, and The fluid motion can then be further defined by (1) imposing a no-slip condition at the bottom of the bath, i.e. requiring the vertical fluid velocity to follow the velocity of the bath's bottom at all contact points, and (2) kinematic boundary conditions at the side walls, i.e. requiring the fluid velocities perpendicular to the boundaries to vanish at the boundaries. When considering a laterally infinite container [12,31], the second condition becomes irrelevant. At the surface, a kinematic surface condition, stating that fluid elements must remain within the fluid or similarly that the surface is adverted by the fluid motion, must be imposed. The lateral stability of the surface can be described by a dynamical boundary condition at the surface, namely that the tangential components of the stress tensor (π xz and π yz , equation (3)) are continuous [12], or vanish [31] if we neglect the influence of the fluid above the surface. Finally, the vertical destabilization of the surface, represented by the occurrence of Faraday waves, can be represented by a discontinuity in the normal component (π zz ) of the stress tensor corresponding to the stress imposed by the curvature of the surface with surface tension σ.
A small (infinitesimal) vertical oscillation (instability) of the position of the surface relative to a flat interface can be described by normal modes of the horizontal plane with wavenumbers k x and k y , e.g. for a surface of infinite lateral extension [12,31] as and with a corresponding vertical velocity here s represents the viscous decay rate of the surface waves (∼ −2νk 2 ) and α effectively acts to modify the resulting frequency of the generated instability. For α = 1/2, subharmonic waves with f F = 1/2f d , 3/2f d , 5/2f d ... are described, while for α = 0 harmonic waves with f F = f d , 2f d , 3f d , ... are described. Which type of waves, subharmonic or harmonic, and with which wavenumber they will occur in the physical (experimental) situation, depend on the actual acceleration threshold for the different types of waves. For inviscid fluids, the Fourier amplitudes ζ n each obey a differential equation of the Matheiu form [14].
Requiring both real and imaginary parts of equation (11) to be zero simultaneously, the angular frequency ω 0 of free surface waves and their decay rate s can be obtained. In the limit where ν ≈ 0, equation (11) reduces to an equation for the angular frequency only gk k kh tanh 15 with two terms corresponding to free gravity waves (gk) and free capillary waves (σ/ρk 3 ).

Dispersion relation and threshold acceleration for Faraday waves
Kumar and Tuckerman [12] and Kumar [31] devised a general method for evaluating the threshold acceleration for a given driving frequency and wavenumber as an eigenvalue problem. As mentioned, we will not repeat the derivation nor the explanation here, but we give as supplementary information some example code where these explicit methods are implemented. Figure 2 illustrates the predicted acceleration thresholds for the occurrence of Faraday waves from the full hydrodynamical model and the phenomenological model formulated by Kumar et al [12,31] for a liquid similar to the Silicone oil used in the experimental investigation. For a given driving frequency, the wavenumber with lowest acceleration threshold will evidently represent the observable Faraday wave. For a relatively deep liquid (h = 5 mm), as displayed in figures 2(a)-(c), the subharmonic waves with f F = 1/2f d generally display the lowest acceleration thresholds over the entire frequency range and hence the observable Faraday waves at threshold are subharmonic. For more shallow liquids of say h = 1 mm, figure 2(d) exemplifies how harmonic waves with f F = f d or subharmonic waves with f F = 3/2f d may display the lowest acceleration thresholds.
Thus, experimentally, Faraday waves will occur on the liquid surface at the threshold with the wavenumber that displays the lowest acceleration threshold for forming the instability. In this representation, the actual dispersion relation for Faraday waves is hence directly linked to the analysis of the onset of the Faraday instability. Müller et al [32] gave analytical expressions for the wavenumber and the acceleration threshold as a function of the driving frequency for Faraday waves, valid in the limit of weak dissipation. Finally, assuming a dispersion relation dominated by capillary waves, a surface wave decay rate of s = −2νk 2 , and an infinitely deep liquid bath (h = ∞), a simple analytical form of the acceleration threshold can be derived [9,22,34] [12,31] represents the most reliable model as it relies only on the assumption of infinitesimal waves (threshold). For a liquid height of h = 5 mm, the analytical model of Müller et al [32] is in excellent agreement with the full hydrodynamical model (FHM) [12,31] for subharmonic waves while the phenomenological model (PhM) [12] deviates significantly. Also for harmonic waves ( f F = f d ), the deviation of the phenomenological model and the analytical model relative to the full hydrodynamical model are significant. As seen in figure 3(d), the discrepancies between the models are evident for all types of waves in the case of a shallow liquid filling. Figure 4 shows a graphical illustration of the constructed experimental system with the oscillating bath and the instrumentation used for the observations made in this study. As supplementary information, we display in figure S1 more details on the actual construction of the vibrating system, and in figure 5 we illustrate some details of the bath and give examples of constructed inserts.

Mechanical setup
As illustrated in figure 4, the complete system is setup on a table with two vertically displaced levels. The mechanical oscillator is mounted on the lower level (Level 1) with a vertically oscillating rod guided through a hole to the upper level (Level 2) where the rod is attached to an aluminum base on which the bath is mounted.
The mechanical oscillator is made from a commercially available audio vibrator (Fischer Amps Buttkicker Mini Concert). The audio vibrator was modified as illustrated in figure S1, i.e. a hole was drilled in its top cover and a 10 mm diameter steel rod was mounted (glued) directly on the oscillating magnet. Additionally, plastic inserts (3D printed) were positioned on each side of the oscillating magnet to limit the stroke of the oscillator and plane springs where inserted below and above the magnet to obtain a resulting harmonic motion of the magnet and rod. Thus, hard collisions between the moving magnet and the top and bottom sides of the house of the vibrator are avoided. The oscillator is excited with a standard laboratory function generator (GF467F, Centrad) combined with a commercially available (non-expensive) audio amplifier (NS-03G SUB, Nobsound).
In panels (a) and (c) the dispersion relations for subharmonic Faraday waves as calculated from the full hydrodynamical model of Kumar et al [12,31] (see also figure 2) are compared to free surface waves for an inviscid liquid (Equation (15)) and a viscous fluid (equation (11)). In panels (b) and (d) different model calculations for the harmonic ( f F = f d ) and subharmonic ( f F = 1/2f d ) Faraday waves are compared. The models are the full hydrodynamical model (FHM) of Kumar et al [12,31], the phenomenological model (PhM) of Kumar and Tuckerman [12], and the Analytical model (AnM) of Müller et al [32].  The two iron screws that are attached to each structure fit into a pair of wells in the bath and fix the structure via the coupling to the permanent magnets below.
The oscillating rod is fixed to and supports the Al-base on the upper level (Level 2) of the table. To ensure a stable vertical motion of the Al-base, the rod is guided into the upper level via ceramic plain bearings and additionally, one side of the Al-base is similarly guided by a stationary vertical rod fixed on Level 2.
As seen in figure S1, the Al-base has 16 cylindrical wells for permanent magnets (diameter 10 mm and height 6 mm) below the liquid bath. The magnets inserted in these wells serve to fix structures that can be inserted into the bath. The bath itself is made of plexiglass with an outer footprint of 110 × 120 mm 2 , which fits into a recess in the Al-base. The inner dimension of the bath is 100 × 100 mm 2 and the bath has a total depth of 10 mm. Figure 5 shows more details of the bath construction and also exemplifies inserts that can be placed inside the bath. One type of insert is a frame, figure 5(b), that follows the edge of the bath to allow modification of the liquid-wall boundary condition, e.g. straight walls or brimful conditions, relevant when observing Faraday waves in the bath. Another type of insert includes dedicated structures, e.g. chicanes or reflective elements, exemplified in figure 5(c), that can affect the wave (and droplet) motion at specific regions of the bath.
The upper level (Level 2 in figure 4) is made as a breadboard so all diagnostic instruments (lasers, photodiode, distance sensors, and cameras (see figure 4) can be positioned and fixed around the vibrating bath using standard optic components.

Observation of bath vibration and onset of Faraday waves
The vibration of the bath is directly observed via a distance sensor (IR08.D03S, Baumer, 0-3 mm), fixed on the upper level (Level 2) of the table, that hence measures the vertical position of the bath. The output from the distance sensor is coupled to one channel of a digital oscilloscope (PicoScope 2207A, Pico Technology). An example of the signal measured with the distance sensor is shown in figure 6(a) for a driving frequency of f d = 38 Hz.
The occurrence of Faraday waves on the liquid surface is observed from the reflection of a laser (Laser 1 in figure 4) that illuminates an area of the liquid surface in the central part of bath with an inclination angle of approximately 45 degrees. The reflected light is detected with a photodiode (FD100S, Thorlabs) positioned oppositely to the illuminating Laser 1 at a position where the laser's transverse size approximately equals the size of the photodiode. The signal from the photodiode is coupled to a second channel of the digital oscilloscope. Figure 6(b) shows examples of signals observed with the photodiode for a stable and an unstable surface. For the stable surface, the photodiode signal displays a near sinusoidal pattern (blue line in figure 6(b)) corresponding to the vibration of the bath, while a distinct change occurs when the surface becomes unstable (red line in figure 6(b)). Hence, the onset of Faraday waves on the liquid surface can for instance be identified from the mean of the observed signal. This particular way of identifying the acceleration threshold for the onset of Faraday waves is illustrated in figure 6(c).

Observation of wave properties
The oscillating bath is observed with two high-speed cameras (Chronos 2.1-HD, Kron Technologies inc.). The first camera, CAM1 in figure 4, is positioned 345 mm above the bath and observes the bath at normal incidence. The other camera (CAM2) is located to the side of the bath and observes the bath at 45 degrees from a distance of approximately 150 mm. The recording with the cameras reported here are all taken at full resolution (1920 × 1080 pixel) with a frame rate of 1000 Hz and with an exposure time of 500 μs. The area of the bath is supplied with light from two commercial fluorescent lamps (not shown in figure 4) allowing the camera gains to be set low. The air from the ventilation system of the cameras disturbs the air in front of the cameras and could therefore potentially disturb the observed liquid surface. Hence, this air flow is guided away from the liquid surface by sets of inserted plastic plates.
Further, another laser (Laser 2 in figure 4) is setup with an approximately 10 degree inclination angle towards horizontal and with a focus on an up-facing side wall of the bath where a piece of black colored paper is attached. As the bath vibrates vertically, this laser spot moves horizontally allowing for the state of the oscillation to be observed from above with CAM1.

Reconstructing surface waves from camera images
To observe and analyze the structure of the Faraday waves on the liquid surface using CAM1, we use the idea of a free surface synthetic Schlieren (FSSS) technique introduced by Moisy et al [36] and further developed for example by Wildeman [37] and Li et al [38]. We explicitly apply a simple FSSS surface reconstruction technique based on direct simulations of light rays through the used imaging system to reveal the parameters needed for surface reconstruction. In the present implementation of a FSSS technique, we position a regular pattern of points (dots) below the liquid, i.e. in practice we place a paper sheet with a dotted pattern between the Al-base and the bath as indicated in figure 4. When the liquid surface is flat ( figure 7(a)), the pattern of dots is seen unperturbed by CAM1, however, when the liquid surface is curved ( figure 7(b)), as induced by waves on the surface, the dots are displaced in the camera image in a manner that is related to the gradient of the surface at the point on the surface from which the light originates. As mentioned, the relation between the apparent displacements of the dots in the camera image and the surface gradients can be revealed by direct simulations of light rays through the complete imaging system.
To illustrate in detail the principle behind the presently applied simulations, an example of simulated light rays used to determine the parameters for the surface reconstruction for the applied imaging system (liquid, lens and camera) is shown in figure 8. The green lines in figure 8 thus illustrates light rays from an example point y P in the object plane (i.e. corresponding to a dot on the paper below the bath) to the image plane (the camera sensor). For a curved surface, the transmitted light (green lines in figure 8) from the example point y P below the liquid passes the liquid surface at the mean position y L  , and is imaged to the point y S  at the sensor. For a flat surface, the similar light rays (not shown in figure 8) from the same point y P passes the liquid surface at y L and are imaged at the position y S . With a complete description (simulation) of the imaging system, the experimental observation of the points y S (flat reference surface) and y S  (curved surface) with the camera sensor, allows the gradient of the surface y y L ( )  a (i.e. ∂z/∂y) at the point y L  as well as the point y L  itself to be determined. Using a full pattern of well-known dimension below the liquid, a map of the gradients of the surface can be obtained, and the actual surface structure can be reconstructed by integration.
To actually reveal the parameters for the surface reconstruction, simulations as exemplified in figure 8 are performed. Thus, the light rays (green lines in figure 8 originating from points y P on the paper surface below the liquid are propagated numerically as simple rays (straight line trajectories) (1) through the liquid, (2) across the flat or curved liquid surface where the refractive index changes from n r = 1.403 to n air = 1 (using Snell's law), (3) through air from Figure 8. Illustration of simulations of light rays from the bottom of the liquid bath to the camera sensor as used to derive the parameters needed for the surface reconstruction with the present imaging system. The green lines show simulated light rays propagating from a point (y P ) on the paper below the bath through the liquid and its curved surface, through air, through the lens, and finally to the image sensor of the camera. The inserts show details of the wave propagation around the liquid surface and at the imaging sensor, illustrating how the position (y P ) of the dot on the paper is displaced from y S for a flat surface to y S  for a surface curve disturbed by the wave. Note for the lower insert that the asymmetrical distance scales of the vertical and horizontal axes chosen to emphasize the positions of the points (y P , y S , and y S  ) distort the display of the light rays (green lines). For the simulation, the refractive index of the liquid and air was set to n r = 1.403 and n air = 1.000, the height of the liquid in the bath was h = 5.00 mm and the surface wave was simulated with a wavelength λ = 5. From ray simulations as exemplified in figure 8, the parameters needed for the evaluation of the surface gradients from the experimental observations of positions y S and y S  can now be determined explicitly for the applied imaging system. For a flat surface, figure 9(a) shows the direct linear imaging of points y P from the object plane to points y S on the image plane, and figure 9(b) shows the displacements (y L − y P ) of the probed points on the liquid surface relative to points on the paper. As expected, both these relations are perfectly linear, and the value of the coefficient a SP (= −0.130) reflects the de-magnification of the object (dotted paper) to the image plane (camera sensor), while the coefficient a LP (= −0.012) mainly reflects the light acceptance of the imaging system and the refractive index of the liquid. The simulated value of a SP can be compared directly to the observed images since the distance between points on the paper sheet is well known (1 mm), and hence the correspondence between the simulations and actual physical system can be directly verified through the value of a SP . For a curved surface, , the actual surface can be reconstructed by integration. In this particular case, we used the grad2Surf software [39] developed for MATLAB [33]. Figure 7(c) shows the final reconstructed surface based on the images in figures 7(a)-(b) using the described technique. Figure 10 displays the measured acceleration thresholds for the onset of Faraday waves as determined in the present experiment (red dots) as a function of the driving frequency for a situation where brimful boundary conditions are used, i.e. with the insert displayed in figure 5

Acceleration threshold for the onset of Faraday waves
Using tabulated values for the kinematic viscosity (ν) and surface tension (σ) for the used Silicone oil, the experimental data are compared to the full hydrodynamical model (dashed blue curve) [12,31], the phenomological model (full green curve) [12], the analytical model  (16)) in a bath of infinite depth. As expected, the model for capillary waves fails to make a quantitative prediction of the acceleration threshold. With the tabulated values for the used oil, the other models also fail to account for the observed acceleration thresholds. The phenomenological model in particular fails to account for the frequency dependency at low frequencies. The full hydrodynamical model and the analytical model essentially give identical predictions and qualitatively predicts the correct shape of the frequency dependence, however, generally underestimating the acceleration threshold at higher frequencies.
Allowing the viscosity and the surface tension to vary as free parameters, a good fit to the experimental data can be obtained with both the full hydrodynamical model and the analytical model. Figure 11 shows examples of measured and reconstructed surfaces for three different forcing frequencies at accelerations slightly above the Faraday thresholds at these frequencies. It is clearly seen how the standing wave pattern depends strongly on the excitation frequency reflecting the dispersion relation for the waves and the conditions for standing waves (equation (7)- (8)). The contribution from two components of standing waves can be seen in the formation of square structures, most visible for 30 Hz and 40 Hz. In figure 12, the time evolution of selected points on the waves corresponding to figure 11(a), (c) and (e) are illustrated. It is evident that the waves observed are all subharmonic, f F = 1/2f d , to a very high degree, i.e. a deviation from this exact correspondence cannot be seen. The bath position as monitored with Laser 2 (figure 4) is seen to be well described by a harmonic function, while small disturbances are seen for the Faraday waves, i.e. most clearly seen for the lower frequency (highest amplitude). The phase shift is close to zero for 15 Hz and 30 Hz, but has a finite value for the data displayed for 40 Hz, showing that the applied acceleration is higher than the threshold value. The actual accuracy of the selectable acceleration is presently determined by the voltage resolution of our function generator and in the present case this limited a more accurate match of the Faraday threshold for 40 Hz.

Properties of Faraday waves
To analyze the wavelength composition of the Faraday waves in more detail, a Fourier transform analysis is applied, i.e. Fourier amplitudes are explicitly calculated as is the (complex) amplitude of the surface as a function of position and k x = 2π/λ x and k y = 2π/λ y are wavenumbers in the two directions corresponding to wavelengths λ x and λ y . Panels (b), (d), and (f) of figure 11 display the actual real amplitude given by or the corresponding surfaces in panels (a), (c), and (e). The direct square symmetry is still evident (especially for 15 Hz and 30 Hz), but also shows preference for linear combinations as most clearly seen for 40 Hz.
The time domain variations ( figure 12), showing f F = 1/2f d , and the spatial variations (wavelength) can be combined to obtain a direct measurement of the dispersion relation for   (15)) as well as in comparison to the dispersion relation obtained with the full hydrodynamical model of Kumar et al [12,31]. For illustration, the gravity and capillary components of the dispersion relation for free surface waves are plotted individually showing for example that the total dispersion relation has about equal contributions from gravity and capillary waves at 12-13 Hz.

Experimental improvements
Our setup is constructed with an awareness of keeping the cost affordable for teaching laboratories at university or high school, and clearly improved results could be obtained with other choices of instrumentation and design for certain parts of the setup. With a function generator with better resolution in the setting of the amplitude level (at present the resolution is 0.01 V below 1 V and 0.1 V above), a finer variation of the bath's acceleration would be possible and hence, the Faraday threshold as a function of the driving frequency could be mapped out with better accuracy. Also, the development of the waves just above the Faraday threshold could be investigated in higher detail. This concerns for example the phase shift of the oscillating waves relative to the bath's oscillation. In a future upgrade of the experiment, we would indeed foresee the addition of a better function generator. Alternatively, a high-quality sound card from a personal computer could also be applied instead of a function generator. With a larger bath, we could obtain a higher resolution in the Fourier transforms (figures 13(a)-(c)) allowing for better accuracy in the measurement of the dispersion relation, however, this would also require cameras of higher resolution, and would in general be associated with a relatively large cost.
The applied oscillator is made with a commercially available audio vibrator (Buttkicker), modified with two plane springs to make the resulting oscillation harmonic, which reduces significantly the price tag for the oscillating system. With a stronger vibrator and possibly with an improved design of the vibrating system [40], we could examine other regions of  [12,31] and to the dispersion relation (equation (15)) for free surface waves on an inviscid fluid (dashed red line). For illustration, the individual contributions from free gravity (dashed green line) and capillary (dashed blue line) in equation (15) are displayed.
Faraday waves, for instance at higher frequencies, however, for the investigation of Faraday waves in a teaching perspective, the accessible (15-60 Hz) region of excitation is sufficient where the transition between gravity waves and capillary waves is highlighted.
For the observation of the dispersion relation, good results could also have been obtained with cameras from modern smartphones. As our system will be used as an integrated stationary exercise in a laboratory course as well as for research, we decided to use (still affordable) permanently installed high-speed cameras.

Surface reconstruction
In the present paper, we implemented a FSSS surface reconstruction method based on direct simulations of light rays from points in a regular dotted pattern below the liquid to the corresponding points registered by the imaging sensor. The simulations provide explicit information on all parameters of the specific imaging system (a SP , a LP , and a αS ) needed to reconstruct the gradient of the liquid surface (see equations (17)- (19) and figure 9) from measurements of the displacement of the dots in the pattern as compared to the dot-positions measured with a flat surface. An advantage of this approach using direct simulation of light rays, is that it allows a direct clarification of the properties (e.g. possible distortions or nonlinearity) of the applied imaging system. Moreover, the direct propagation of light rays is relatively easily implemented numerically and promote a deep understanding of the significance of the applied imaging system for students. The analytical implementation of the FSSS technique by Moisy et al [36] relied on the paraxial approximation, the weak slope approximation, and the weak amplitude approximation. Compared to the present case, these approximations are essentially equivalent to making the approximation y L  = y a S SP  , i.e. the relation in figure 9(d) would be flat, which in our case would lead to significant derivations in the reconstruction of the image.
In other implementations of the FSSS method, randomly distributed points or regular patterns have been applied (see e.g. Wildeman [37] and Li et al [38] for overviews of techniques applied). For the purpose of surface reconstruction applied in this paper, a regular pattern (in the present case dots) makes the point identification and image reconstruction simple and easy to implement numerically. More advanced FSSS techniques [37,38] could evidently also be implemented for the present experimental system, however, we consider the relative simplicity of the concept of direct ray propagation and the simple linear transformations (equations (17)- (19)) as appealing in a teaching situation, and the obtained results (figures 11 and 13) are quite satisfactory.

Examples of future directions for student-driven research
The setup will be integrated as a project exercise into a course on experimental physics at the bachelor level at Aarhus University (see also [41] for some description of the course).
We believe that the aspects of the setup demonstrated in this paper, could for example enable students to develop an intuitive understanding of wave phenomena including in particular standing waves and dispersion relations. In this respect, it should be noted that the setup operates in a regime where both gravity and capillary waves are of equal importance ( figure 13).
The models established by Kumar et al [12,31] and Müller et al [32] are well suited for training numerical calculations at the university level. Additionally, several aspects, beyond the demonstrations of the Faraday instability and the mapping of the dispersion relation shown here, can be imagined with the present setup either for teaching purposes or as dedicated student-driven research projects. One possibility would be to explore further the idea of Kumar and Tuckermann [12] that the measurement of the acceleration threshold for Faraday waves as a function of the driving frequency can be used to actively determine the fluid viscosity (ν or η) and the surface tension σ simultaneously by a fit with the (exact) full hydrodynamical model to measured data. The potential of this method, which to the best of our knowledge has not been examined experimentally before, was indicated for the data set shown in figure 10, where a very good match of the model to the experimental data could be obtained by adjustment of the tabulated viscosity and surface tension for the used Silicone oil. It would thus be a very interesting line of advanced student research to validate this method in more detail. Moreover, upon validation, the method could constitute a universal way to examine fluid properties for example as a function of temperature. In the student laboratory in the first year of study, fluids like water, household oil, and syrup could be investigated.
Another interesting direction for future student-driven research could be the observation and investigation of harmonic waves ( f F = f d ) and higher subharmonic waves ( f F = 3/2f d ) and in particular the transitions between subharmonic and harmonic waves that could occur for low liquid depths and at low frequencies as demonstrated with the calculations displayed in figure 2(d) and figure 3(d).
The explicit importance of liquid-wall interactions [3] can also be examined with the present setup. In the demonstration reported here, we used the brimful conditions, but clearly the effect of straight walls or other types of boundaries can be realized with appropriate inserts into the bath. For example, the onset and significance of meniscus waves (emitted with frequency f d ) in competition with subharmonic waves at 1/2f d could be investigated. Also, the dependence of the standing waves on the bath geometry, i.e. square, circular, oval, etc can relatively easily be studied with simple inserts.
Finally, the setup is also meant to examine the dynamics of bouncing and walking droplets. In this respect, the description in this paper serves to demonstrate the general functioning of the setup, while actual results on droplet dynamics will be given in a separate manuscript.

Conclusion
We have described a setup for investigations of Faraday waves for laboratory teaching at university. In its present form, the setup allows for example for the examination of the acceleration threshold for Faraday waves as a function of driving frequency as well as for investigations of the nature for the Faraday waves. At a given driving frequency, the onset of the Faraday instability of the liquid surface can be directly deduced from the reflection of a laser from the liquid surface in a simple and objective way. Images acquired at high speed (∼1 kHz) of the bath taken from above, allows reconstruction of the surface using a free surface synthetic Schlieren technique. These surfaces allows the deduction of both the wave frequency and wavelength and can thus be used to determine the dispersion relation for Faraday waves of the investigated liquid. Similar setups can be used for teaching fluid dynamics at different levels at university, and, moreover, with the present setup, we foresee several interesting projects within fluid dynamics to be realized. The setup will also be used for student-driven investigations of bouncing and walking droplets.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.