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Faraday wave
Faraday wave
Faraday wave
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Faraday waves observed in water in a Petri dish, vibrated at a frequency of about 50 hertz.
Faraday waves in a singing bowl

Faraday waves, also known as Faraday ripples, named after Michael Faraday (1791–1867), are nonlinear standing waves that appear on liquids enclosed by a vibrating receptacle. When the vibration frequency exceeds a critical value, the flat hydrostatic surface becomes unstable. This is known as the Faraday instability. Faraday first described them in an appendix to an article in the Philosophical Transactions of the Royal Society of London in 1831.[1][2]

If a layer of liquid is placed on top of a vertically oscillating piston, a pattern of standing waves appears which oscillates at half the driving frequency, given certain criteria of instability.[3] This relates to the problem of parametric resonance. The waves can take the form of stripes, close-packed hexagons, or even squares or quasiperiodic patterns. Faraday waves are commonly observed as fine stripes on the surface of wine in a wine glass that is ringing like a bell. Faraday waves also explain the 'fountain' phenomenon on a singing bowl.

The Faraday wave and its wavelength is analogous to the de Broglie wave with the de Broglie wavelength in de Broglie–Bohm theory in the field of quantum mechanics.[4]

Application

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Assembly of microscale materials on Faraday waves.[5]

Faraday waves are used as a liquid-based template for directed assembly of microscale materials including soft matter, rigid bodies, biological entities (e.g., individual cells, cell spheroids and cell-seeded microcarrier beads).[5] Unlike solid-based template, this liquid-based template can be dynamically changed by tuning vibrational frequency and acceleration and generate diverse sets of symmetrical and periodic patterns.

This phenomenon is also used by alligators to call mates. They vibrate their lungs at low frequencies slightly below the surface, causing their spikes to move and induce surface waves. These surface waves are basically Faraday waves and one can observe the splashing effect characteristic of certain resonances.[6][7]

This effect can also be used for mixing two liquids acoustically. Faraday waves form on the interface between the two liquids, which increases the surface area between the two, rapidly and thoroughly mixing the liquids.[8]

See also

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References

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Faraday wave

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from Grokipedia
Faraday waves, also known as Faraday ripples, are nonlinear standing waves that appear on the surface of a enclosed in a vibrating container, arising from a parametric instability when the vertical vibration amplitude exceeds a critical threshold. These waves were first observed and described by the English physicist in 1831, who noted their formation as "crispations" on a layer during experiments with vibrating plates, initially in the context of granular materials but extending to liquids in an appendix to his publication. Faraday's discovery highlighted the waves' subharmonic response, where the wave frequency is half that of the driving vibration, a key feature confirmed later by Lord Rayleigh in 1883 through theoretical analysis. The underlying mechanism of Faraday waves involves a parametric excitation process, where the vertical of the container modulates the effective acting on the surface, leading to and the growth of surface perturbations. This is mathematically modeled using the Mathieu , which describes the parametric of the system, with the onset of waves occurring along "resonance tongues" in the parameter space of driving and . The wave characteristics depend critically on properties such as , , and , as well as external parameters including vibration (typically 10–30 Hz), , and layer depth (often 3 mm to 2 cm). In viscous fluids, subharmonic modes dominate due to lower energy dissipation compared to harmonic modes, resulting in patterns that emerge above the threshold. Faraday waves exhibit a rich variety of spatiotemporal patterns, including stripes, squares, hexagons, and more complex quasi-crystalline structures with N-fold symmetries for N > 3, which form due to nonlinear interactions and three-wave resonances in the . These patterns are influenced by boundary conditions, such as container geometry (e.g., annular or baffled cells), and can transition to chaotic or turbulent states at higher amplitudes, generating three-dimensional vortices and flows. The waves have been observed not only in classical fluids but also in exotic systems, such as superfluids and Bose-Einstein condensates, where quantum effects modify their behavior. In terms of significance, Faraday waves serve as a paradigmatic example of and nonlinear dynamics in far-from-equilibrium systems, providing insights into , bifurcations, and . They have practical applications, including the measurement of interfacial tension between immiscible fluids by analyzing wave thresholds, nonlinear of vibrating structures using viscous layers, and modeling processes in management, mixing, and biological . Ongoing research continues to explore their role in walking droplets and analogs, underscoring their enduring relevance in physics.

Fundamentals

Definition and characteristics

Faraday waves, also known as Faraday ripples, are nonlinear standing waves that emerge on the free surface of a fluid layer when the containing vessel undergoes vertical oscillations at a constant frequency and amplitude. These waves arise through a parametric instability, where the periodic forcing modulates the effective gravity experienced by the fluid, leading to the spontaneous formation of organized surface patterns above a critical threshold of vibration amplitude. First described by Michael Faraday in 1831 during experiments on vibrating surfaces in contact with fluids, the phenomenon involves energy transfer from the mechanical vibration of the container to the fluid interface, resulting in persistent wave structures that counteract dissipative effects. A defining feature of Faraday waves is their subharmonic response, oscillating at half the of the imposed , which distinguishes them from harmonically driven waves. The onset requires the to exceed a threshold that depends on factors such as the driving , fluid viscosity, layer depth, and ; below this threshold, the flat surface remains stable, while supercritical forcing excites the . These waves exhibit nonlinear behavior, as the parametric excitation couples the fluid's inertial and restorative forces, enabling the development of complex spatial structures like hexagonal, square, or stripe patterns, which represent different modes of the surface deformation. Typically observed in low-viscosity Newtonian fluids such as or , the patterns' is governed by the balance between (or capillarity in shallow layers) and the forcing parameters. To an observer in the reference frame of the vibrating container, Faraday waves manifest as stationary crests and troughs, creating visually striking, time-independent undulations despite the underlying oscillatory motion. This apparent stationarity highlights their standing-wave nature, where antinodes correspond to regions of maximal surface variation. The characteristics underscore the waves' sensitivity to boundary conditions and forcing details, with pattern selection often favoring rolls (stripes) near onset and transitioning to more symmetric arrays like hexagons at higher amplitudes, providing a canonical example of in driven dissipative systems.

Historical background

Michael Faraday first observed the phenomenon of surface waves on a vertically vibrated fluid in 1831 while experimenting with a glass vessel containing water shaken at its . He described these standing waves, which oscillate at half the driving , in his seminal paper "On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces," noting their similarity to patterns formed on vibrating solids. This discovery extended earlier 19th-century investigations into and , particularly Ernst Chladni's 1787 demonstrations of nodal patterns on vibrating plates dusted with powder, which visualized acoustic modes. In 1883, Lord Rayleigh provided the first theoretical framework for Faraday's observations through experiments and analysis that confirmed the subharmonic response and linked it to parametric instability in fluids. His work, detailed in papers such as "On the crispations of resting upon a vibrating support," resolved discrepancies in earlier measurements and established the waves' dependence on and forcing . Despite this progress, interest waned until the mid-20th century, when T. Brooke Benjamin and F. Ursell formalized the analysis of the surface under vertical in their 1954 study, deriving the threshold for instability and emphasizing the parametric excitation mechanism. The 1980s marked a revival through experimental studies on pattern formation, with S. Douady and S. Fauve demonstrating the selection of standing wave modes in finite containers and the role of container geometry in stabilizing squares or rolls near onset. Their 1988 experiments highlighted the transition from linear instability to nonlinear patterns, bridging Faraday's original observations with modern nonlinear dynamics. Key milestones include Faraday's 1831 discovery, Rayleigh's 1883 theory, Benjamin and Ursell's 1954 analysis, and the 1980s experimental resurgence. Post-1990s developments introduced computational modeling to simulate complex pattern evolution, enabling predictions of quasiperiodic and regimes beyond analytical reach, as seen in numerical studies of viscous Faraday waves in three dimensions.

Theoretical framework

Parametric excitation mechanism

The parametric excitation mechanism underlying Faraday waves relies on a phenomenon where surface waves are driven at half the frequency of the imposed vertical vibration, known as subharmonic response. This process is analogous to the of a parametrically driven , where periodic modulation of a system parameter leads to and energy transfer to the oscillating mode. In Faraday waves, the container's vertical oscillation with amplitude AA and angular frequency ω\omega imposes a time-varying acceleration Aω2cos⁡(ωt)A \omega^2 \cos(\omega t), which modulates the effective gravitational acceleration as geff=g+Aω2cos⁡(ωt)g_\text{eff} = g + A \omega^2 \cos(\omega t). This modulation periodically alters the restoring force for surface displacements, destabilizing the flat interface when the vibration amplitude exceeds a critical threshold. Energy is thereby pumped into specific surface wave modes through this parametric forcing, leading to exponential growth of perturbations until nonlinear effects limit the amplitude. The dynamics of individual modes can be captured by the Mathieu equation for the surface displacement η\eta: d2ηdt2+(ω02+εcos⁡(ωt))η=0,\frac{d^2 \eta}{dt^2} + \left( \omega_0^2 + \varepsilon \cos(\omega t) \right) \eta = 0, where ω0\omega_0 is the natural frequency of the unforced mode, and ε\varepsilon represents the strength of the parametric modulation, proportional to Aω2A \omega^2. Instability occurs within certain parameter tongues in the Mathieu stability chart, with the subharmonic response dominating near ω0=ω/2\omega_0 = \omega / 2. The minimal vibration amplitude required for onset, AcA_c, depends on fluid properties such as surface tension σ\sigma, density ρ\rho, and viscosity, as well as driving parameters including frequency ω\omega and fluid depth; this reflects the balance between gravitational and capillary restoration in the dispersion relation that selects the most unstable mode.

Linear stability analysis

The linear stability analysis of Faraday waves examines the onset of in a layer subjected to vertical oscillatory forcing, focusing on small-amplitude surface perturbations. Surface displacements are modeled as perturbations to the flat interface, and the governing equations are the incompressible Navier-Stokes equations linearized around the base state of uniform oscillation. Due to the time-periodic nature of the forcing, is applied to analyze the stability, where solutions are sought in the form of quasi-periodic functions with a Floquet multiplier determining growth or decay. In the inviscid limit, the dispersion relation for gravity-capillary waves on a fluid of depth hh governs the unforced modes: ω2=gk+σρk3tanh⁡(kh),\omega^2 = g k + \frac{\sigma}{\rho} k^3 \tanh(k h), where ω\omega is the wave frequency, kk is the wavenumber, gg is gravity, σ\sigma is surface tension, and ρ\rho is density. Under vertical forcing with amplitude aa and frequency Ω\Omega, the effective gravitational acceleration becomes modulated as geff(t)=g(1+acos⁡Ωt)g_{\text{eff}}(t) = g (1 + a \cos \Omega t), leading to parametric resonance. Instability arises subharmonically (at frequency Ω/2\Omega/2) when the forcing amplitude aa exceeds a threshold, effectively rendering geffg_{\text{eff}} negative on average for certain kk, opening instability tongues in the aa-kk plane. The neutral curve delineates the onset of instability, marking the minimum aa (or acceleration Γ=aΩ2/g\Gamma = a \Omega^2 / g) required for zero growth rate at each kk. For finite depth without surface tension, the dominant (most unstable) wavenumber near onset approximates kc≈Ω/(2gh)k_c \approx \Omega / (2 \sqrt{g h})
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