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Surface wave

A surface wave is a wave that propagates along the interface between two distinct media, such as air and water or a solid and vacuum, with its energy primarily confined to a region near that boundary, typically within a depth of about one wavelength.^[1] Unlike body waves that travel through the bulk of a medium, surface waves exhibit elliptical particle motion and are dispersive in layered or inhomogeneous materials, meaning their speed depends on frequency or wavelength.^[1] These waves arise from various physical mechanisms, including gravity, elasticity, surface tension, and electromagnetic forces, and they play crucial roles in natural phenomena and technological applications.^[1] In mechanical contexts, surface waves are common in fluids and solids; for instance, ocean waves are primarily gravity-driven surface waves where particles move in circular orbits, combining longitudinal and transverse components, with energy propagating horizontally while the water surface deforms vertically.^[2]^[3] Seismic surface waves, generated by earthquakes, include Rayleigh waves, which produce retrograde elliptical motion causing vertical ground roll at about 90% the speed of shear waves, and Love waves, which induce horizontal shear motion perpendicular to propagation and travel slightly faster than Rayleigh waves.^[4] These seismic waves are slower than body waves but often cause the most damage due to their larger amplitudes and concentration of energy near Earth's surface.^[4] Additionally, surface acoustic waves in solids, first theoretically described by Lord Rayleigh in 1885, rely on elastic restoring forces and are used in devices like filters for signal processing.^[1] Electromagnetic surface waves, solutions to Maxwell's equations, occur at boundaries between dielectrics or conductors and free space, with fields decaying exponentially away from the interface; a classic example is the Zenneck wave, analyzed in 1907, which guides radio-frequency energy along conductive surfaces like Earth's ground for over-the-horizon propagation.^[5] These waves, with frequencies ranging from radio to optical, enable applications in plasmonics, waveguides, and remote sensing, where their confinement enhances field strengths near the surface.^[5] Overall, surface waves' boundary-localized nature distinguishes them across physics, influencing everything from coastal erosion and earthquake engineering to modern telecommunications.^[1]

Fundamentals

Definition and Classification

Surface waves are waves that occur at the interface between two media possessing different physical properties, such as density or permittivity, and propagate parallel to this boundary while their amplitude decays exponentially away from it in both directions perpendicular to the interface.^[6] This exponential decay ensures that the wave energy is primarily confined to a thin layer near the boundary, typically on the order of the wavelength.^[7] Unlike bulk waves, which propagate through the interior volume of a homogeneous medium with minimal localization, surface waves are inherently bound to the interface and do not extend significantly into the bulk of either medium.^[8] Surface waves are broadly classified into mechanical, acoustic, and electromagnetic categories, depending on the underlying physical mechanism and the type of media involved. Mechanical surface waves arise from elastic deformations or gravitational effects at fluid-fluid or fluid-solid interfaces; they often combine longitudinal and transverse particle motions, as seen in ocean surface gravity waves at the air-water boundary, where water particles trace elliptical paths.^[8] In solids, mechanical surface waves like Rayleigh waves travel along the solid-vacuum interface, featuring coupled compressional and shear components that decay evanescently into the solid. Acoustic surface waves, a subset primarily in elastic solids, represent high-frequency mechanical vibrations, such as those generated in piezoelectric materials for signal processing, and are characterized by their sensitivity to surface perturbations.^[9] Electromagnetic surface waves involve oscillations of electric and magnetic fields at interfaces between dielectrics and conductors or plasmas, exemplified by surface plasmon polaritons at metal-dielectric boundaries, where the wave couples light to collective electron oscillations.^[10] This classification highlights the transverse or mixed polarization typical of surface waves, contrasting with purely longitudinal bulk modes in fluids, though specific implementations may emphasize one component over others based on the interface geometry and material properties.^[6]

Mathematical Description

Surface waves are typically analyzed in the frequency domain, where the scalar potential or field components in each medium satisfy the Helmholtz equation,

\nabla^2 \psi + k^2 \psi = 0,

with k = \omega / c denoting the wavenumber, \omega the angular frequency, and c the wave speed in the medium.^[11] This equation governs the propagation within homogeneous regions above and below the interface, typically taken as the plane z = 0. Solutions are constructed to represent waves confined to the interface, decaying exponentially away from it in the direction normal to the boundary. The boundary value problem for surface waves involves solving the Helmholtz equation subject to continuity conditions at the interface z = 0. These conditions generally require the continuity of appropriate physical quantities, such as pressure or stress and normal displacement for mechanical waves, or tangential field components for electromagnetic or acoustic waves. For instance, in planar geometry, the problem is solved using separation of variables, assuming a form \psi(x, z) = f(z) e^{i k_x x}, where k_x is the wavenumber parallel to the interface. This reduces the partial differential equation to an ordinary differential equation in z: f''(z) - \kappa^2 f(z) = 0, with \kappa = \sqrt{k_x^2 - k^2}, yielding exponential solutions that ensure evanescent behavior. Similar separation applies in cylindrical coordinates for curved interfaces, leading to Bessel functions for radial dependence combined with exponential decay perpendicular to the surface.^[11] Evanescent decay is a hallmark of surface waves, with the amplitude in the non-propagating medium (say, z > 0) varying as e^{-\kappa z}, where \kappa = \sqrt{k_x^2 - k_0^2} and k_0 = \omega / c is the free-space or bulk wavenumber in that medium. This decay ensures the wave is bound to the interface without radiating energy away, provided k_x > k_0.^[12] The phase matching condition at the interface conserves the parallel component of the wavevector: k_x = k_1 \sin \theta_1 = k_2 \sin \theta_2, analogous to Snell's law, but with imaginary \theta_2 in the evanescent region to maintain the bound mode.^[12] Applying these boundary conditions yields the dispersion relation, a generic functional form \omega = f(k) relating angular frequency to the parallel wavenumber k = k_x. This relation determines the allowed modes, with phase velocity v_p = \omega / k and group velocity v_g = d\omega / dk describing propagation and energy transport, respectively. For example, in many systems, the dispersion exhibits non-dispersive (\omega \propto k) or dispersive (\omega \propto \sqrt{k}) behavior depending on the dominant restoring forces at the interface.^[6]

Mechanical Surface Waves

In Fluids

Surface waves in fluids typically propagate along the interface between a liquid, such as water, and a gas, like air, where gravity and surface tension provide the primary restoring forces. These waves are prevalent in natural settings, including oceans, lakes, and rivers, and their behavior is governed by the interplay of fluid depth, wavelength, and physical properties of the medium. Unlike waves in solids, fluid surface waves assume an incompressible, inviscid fluid in linear theory, leading to distinct dispersion characteristics.^[13] The classification of surface waves in fluids depends on the dominant restoring force and the ratio of wavelength λ to water depth h. Deep-water gravity waves occur when λ << h, where the dispersion relation simplifies to ω² = gk, with ω the angular frequency, g the acceleration due to gravity, and k = 2π/λ the wavenumber; these waves are dispersive, with phase speed c = √(g/k) increasing with wavelength. In contrast, shallow-water waves arise when λ >> h, yielding a non-dispersive relation c = √(gh), where the wave speed depends solely on depth, as seen in tsunamis across oceans. Capillary waves, dominant for short wavelengths (typically λ < 1.7 cm on water), are restored by surface tension σ, with dispersion ω² = (σ/ρ) k³, where ρ is fluid density; these are highly dispersive and often appear as ripples on calm surfaces. The full dispersion relation combining gravity and capillary effects, applicable across depths, is ω² = [gk + (σ/ρ) k³] tanh(kh), where tanh(kh) ≈ 1 for deep water and ≈ kh for shallow water.^[14]^[15]^[16]^[17] Generation of these waves primarily occurs through wind stress at the air-sea interface, where shear from airflow transfers momentum to the water surface, initiating small capillary waves that grow into gravity waves under sustained winds. Atmospheric pressure gradients can also excite waves, as in meteotsunamis, by inducing rapid sea-level changes. Seismic activity, such as underwater earthquakes, displaces water vertically to generate long-period shallow-water waves like tsunamis. Particle motion in these waves follows orbital paths, but nonlinear effects produce a net forward drift known as Stokes' drift, where surface particles move faster in the wave direction than deeper ones, with velocity scaling as u_s ≈ a² ω k e^{2kz} for amplitude a and depth z ≤ 0.^[18]^[19]^[20]^[21] In linear theory, the total energy of surface waves is equally partitioned, with 50% kinetic energy from fluid motion and 50% potential energy from surface displacement relative to the mean level. Nonlinear effects become prominent for steep waves, where the waveform distorts through steepening—the front face sharpens while the rear flattens—eventually leading to breaking, which dissipates energy via turbulence and whitecaps. A notable historical observation occurred in 1834 when engineer John Scott Russell witnessed a solitary wave, a nonlinear, non-dispersive pulse maintaining its shape over long distances, while riding along a Scottish canal; this discovery laid groundwork for understanding nonlinear wave phenomena in fluids.^[22]^[23]^[24]

In Solids

Surface waves in elastic solids, known as seismic surface waves, primarily include Rayleigh and Love waves, which propagate along the free surface of the Earth or other solid media. These waves are generated by sources such as earthquakes or artificial impacts, where the sudden release of elastic energy excites guided modes at interfaces between layers of different elastic properties. In homogeneous isotropic solids, Rayleigh waves exhibit non-dispersive propagation, meaning their phase velocity is independent of frequency, while Love waves require layered structures for existence and are inherently dispersive. Rayleigh waves feature elliptical particle motion in the vertical plane, with the major axis of the ellipse oriented vertically and retrograde motion at the surface—particles move in the direction opposite to wave propagation during the upward phase. The wave speed c_R is approximately $0.92 c_S, where c_S is the shear wave speed, for a Poisson solid with Poisson's ratio \nu = 0.25; this value arises from solving the characteristic equation derived from boundary conditions at a stress-free surface. Specifically, the conditions \sigma_{zz} = 0 and \sigma_{xz} = 0 (normal and shear stresses vanishing at the surface) lead to the Rayleigh secular equation:

\left(2 - \frac{c^2}{c_S^2}\right)^2 - 4 \sqrt{1 - \frac{c^2}{c_P^2}} \sqrt{1 - \frac{c^2}{c_S^2}} = 0,

where c_P is the P-wave speed and c is the Rayleigh wave speed; the real positive root less than c_S yields c_R. These waves are non-dispersive in uniform media because the solution to this cubic equation in terms of slownesses provides a frequency-independent velocity. Rayleigh waves attenuate through both scattering by heterogeneities, which redistributes energy, and intrinsic losses due to anelasticity in the material.^[25]^[26] Love waves, in contrast, involve horizontally polarized shear horizontal (SH) particle motion perpendicular to the propagation direction and parallel to the surface, behaving as guided SH waves confined by a low-velocity surface layer over a higher-velocity substrate, such as the Earth's crust overlying the mantle. Unlike Rayleigh waves, Love waves are dispersive, with phase velocities depending on frequency and exhibiting multiple higher-order modes, each with a cutoff frequency below which the mode cannot propagate. The fundamental mode has no cutoff and dominates at long periods, while higher modes appear at shorter periods corresponding to the layer's thickness and velocity contrast. These waves also attenuate via scattering from geological irregularities and intrinsic dissipation within the viscoelastic layers.^[26] In seismology, Rayleigh waves carry approximately 70% of the total seismic energy from distant earthquakes, making them the primary component observed on teleseismic records due to their slower geometric spreading compared to body waves. This dominance enables their use in surface-wave magnitude scales, such as the Ms scale, which measures earthquake size based on the amplitude of Rayleigh waves with periods around 20 seconds recorded at regional to teleseismic distances, extending the principles of the original Richter local magnitude scale. Love waves contribute the remainder of surface wave energy but are less prominent in vertical-component recordings.^[27]^[28]

Acoustic Surface Waves

Principles and Generation

Acoustic surface waves (SAWs) are elastic waves confined to the surface of a solid, with energy concentrated within approximately one wavelength depth from the surface and decaying exponentially into the bulk. These waves are typically of the Rayleigh type, involving coupled longitudinal and shear horizontal displacements, but their characteristics are modified by the material's elastic, piezoelectric, and anisotropic properties. In piezoelectric substrates such as quartz or lithium niobate (LiNbO₃), SAW velocities range from approximately 3000 to 5000 m/s, depending on the crystal orientation; for example, the Rayleigh wave velocity is about 3159 m/s in ST-cut quartz and 3992 m/s in 128° YX-cut LiNbO₃.^[29]^[30]^[29] The existence of such surface waves was first predicted mathematically by Lord Rayleigh in 1885, who described their propagation at stress-free boundaries in isotropic elastic solids. Practical realization of SAWs for technological applications emerged in the mid-1960s using quartz substrates in early oscillators and delay lines, driven by needs in pulse compression radar.^[30]^[31]^[29] Generation of SAWs relies on the piezoelectric effect, where an applied voltage across electrodes induces mechanical strain, launching acoustic waves along the surface. Interdigital transducers (IDTs), consisting of interleaved metallic fingers patterned on the substrate, efficiently excite SAWs by creating a spatially periodic electric field that matches the wave's wavelength. The IDT can be modeled using an equivalent circuit that includes electrostatic capacitance C_T (proportional to the number of finger pairs N and per-section capacitance C_s), motional inductance, and resistance to account for energy conversion losses.^[30]^[31]^[29] The planar IDT design, introduced by White and Voltmer in 1965, revolutionized SAW generation by enabling precise control of frequency, phase, and amplitude through photolithographic patterning. In this configuration, the SAW wavelength \lambda equals the IDT period p (the distance between adjacent finger centers), and the operating frequency is given by

f = \frac{v}{\lambda} = \frac{v}{p},

where v is the SAW phase velocity. Efficient transduction requires substrates with a high electromechanical coupling coefficient k^2 > 1\%, which quantifies the conversion efficiency between electrical and mechanical energy; for instance, k^2 reaches 5–11.3% in 128° YX-cut LiNbO₃, far exceeding the 0.03–0.14% in quartz.^[31]^[32]^[29]

Propagation Characteristics

Acoustic surface waves (SAWs) in uniform media exhibit minimal dispersion, with the phase velocity remaining nearly independent of frequency, allowing for straightforward propagation modeling in homogeneous substrates like quartz or lithium niobate.^[33] However, when propagating through periodic structures such as gratings or photonic crystals, dispersion becomes significant due to interactions like Bragg reflection, which occurs at a wavevector k = \pi / \Lambda, where \Lambda is the grating period, leading to bandgaps that prevent wave transmission at specific frequencies.^[34]^[35] Attenuation of SAWs arises from several mechanisms, including viscous damping in the substrate material, where the attenuation coefficient \alpha scales approximately as f^2 (with f the frequency), due to energy dissipation from internal friction.^[36] Radiative losses occur through coupling to bulk acoustic waves, particularly in leaky SAW modes on certain substrates, converting surface-bound energy into volume-propagating modes.^[37] Diffraction from finite transducer apertures also contributes to attenuation by spreading the beam, reducing on-axis intensity over propagation distance L. The resulting insertion loss, expressed in decibels, is given by $20 \log_{10} (e^{\alpha L}), accounting for the exponential decay of amplitude.^[38] SAWs enable key interactions for device applications, such as acousto-optic effects where propagating waves modulate light through the photoelastic effect, inducing refractive index changes that diffract or shift optical beams for applications like signal modulation.^[39] Nonlinear mixing processes, including parametric interactions between counter-propagating or co-propagating SAWs, facilitate signal processing tasks such as frequency conversion and convolution, leveraging the material's cubic nonlinearity.^[40] The power flow of SAWs is quantified by integrating the acoustic intensity over the surface, where the local intensity is \frac{1}{2} \operatorname{[Re](/page/Re)} \{\sigma_{ij} v_j^*\}, with \sigma_{ij} the stress tensor and v_j the particle velocity; beam spreading due to diffraction scales as \sqrt{\lambda L}, where \lambda is the wavelength, impacting efficiency in long-path devices.^[41]^[42] A prominent application of SAW propagation characteristics is in bandpass filters for mobile phones, where these devices have been integral since the late 1980s, offering fractional bandwidths \Delta f / f of approximately 1-10% with low insertion loss.^[43] Temperature stability is enhanced using ST-cut quartz substrates, which minimize frequency drift over operational ranges through optimized crystal orientation.^[44]^[45]

Electromagnetic Surface Waves

Theoretical Foundations

The theoretical foundations of electromagnetic surface waves at dielectric interfaces are rooted in Maxwell's equations and the associated boundary conditions that govern field continuity across material boundaries. At an interface separating two non-magnetic media with permittivities \epsilon_1 and \epsilon_2, the tangential components of the electric field \mathbf{E} and magnetic field \mathbf{H} must be continuous, while the normal components of the electric displacement \mathbf{D} = \epsilon \mathbf{E} and magnetic flux density \mathbf{B} = \mu_0 \mathbf{H} (with \mu = \mu_0) are also continuous in the absence of surface charges or currents./02%3A_Introduction_to_Electrodynamics/2.06%3A_Boundary_conditions_for_electromagnetic_fields) These conditions permit transverse magnetic (TM, or p-polarized) modes for surface waves, where the magnetic field is transverse to the direction of propagation and the interface normal, as transverse electric (TE) modes do not support bound surface solutions in isotropic dielectrics.^[46] Evanescent fields characterize these TM surface waves, decaying exponentially away from the interface on both sides, ensuring field confinement without radiation into the bulk media. For such bound modes to exist, the permittivities must satisfy \epsilon_1 + \epsilon_2 < 0 and \epsilon_1 \epsilon_2 < 0, typically requiring one medium to have negative permittivity (e.g., in plasmas below the plasma frequency).^[46] This condition arises from applying the boundary constraints to the wave solutions, yielding decay constants with positive real parts perpendicular to the interface. The general dispersion relation for TM surface waves at a planar interface derives from solving the wave equation under these boundary conditions, giving the propagation constant k = \frac{\omega}{c} \sqrt{\frac{\epsilon_1 \epsilon_2}{\epsilon_1 + \epsilon_2}}, where \omega is the angular frequency and c is the speed of light in vacuum.^[47] This relation indicates that surface waves propagate non-radiatively when k > \omega / c, positioning them below the light line in the dispersion diagram and preventing coupling to free-space modes.^[48] Early theoretical developments include Zenneck's 1907 analysis of plane electromagnetic waves propagating along a planar conducting surface, which provided an approximate solution for low-loss media relevant to wireless telegraphy. Sommerfeld's 1909 rigorous treatment extended this by solving the exact boundary value problem for a vertical dipole over a lossy half-space, eliminating approximations and confirming the surface wave's validity through integral representations of the fields.

Surface Plasmon Polaritons

Surface plasmon polaritons (SPPs) represent a class of electromagnetic surface waves formed at the interface between a metal and a dielectric material, resulting from the coupling of photons to collective electron plasma oscillations within the metal. This coupling occurs when the wavevector of the incident light matches the resonance condition at the boundary, enabling energy propagation confined to the surface with evanescent decay into both media. The phenomenon is fundamental to nanophotonics, as it allows subwavelength confinement of light beyond the diffraction limit. The formation of SPPs relies on the dielectric response of the metal, modeled by the Drude-Lorentz expression for the permittivity:

\varepsilon_m(\omega) = \varepsilon_\infty - \frac{\omega_p^2}{\omega(\omega + i\gamma)},

where \varepsilon_\infty is the core permittivity, \omega_p is the bulk plasma frequency, and \gamma accounts for damping due to electron collisions. For propagation along the interface, the dispersion relation governing the SPP wavevector k_\mathrm{spp} is

k_\mathrm{spp} = k_0 \sqrt{\frac{\varepsilon_m \varepsilon_d}{\varepsilon_m + \varepsilon_d}},

with k_0 = \omega/c the free-space wavevector and \varepsilon_d the dielectric permittivity of the adjacent medium (often taken as 1 for air or vacuum). Resonance excitation requires \mathrm{Re}(\varepsilon_m) < -\varepsilon_d to ensure a bound mode, as the negative permittivity of the metal at optical frequencies compensates the positive \varepsilon_d. In the limit where |\varepsilon_m| \gg \varepsilon_d, an asymptotic approximation yields k_\mathrm{spp} \approx k_0 \left(1 + \frac{\varepsilon_d}{2|\varepsilon_m|}\right), illustrating how the dispersion curve closely hugs the light line in the frequency region of negative \varepsilon_m, asymptotically approaching the surface plasmon frequency \omega_\mathrm{sp} = \omega_p / \sqrt{1 + \varepsilon_d} at large k. The propagation of SPPs is inherently lossy, primarily due to ohmic dissipation in the metal, limiting the distance over which the mode can travel. The propagation length is quantified as L_\mathrm{spp} = 1 / (2 \mathrm{Im}\{k_\mathrm{spp}\}), typically on the order of micrometers at visible and near-infrared optical frequencies for noble metals like gold or silver. This finite length arises from the imaginary component of k_\mathrm{spp}, which stems from the damping term \gamma in the Drude model and interband transitions. SPPs exist in two primary configurations: propagating modes on extended flat metal-dielectric interfaces, which support long-range waveguiding, and localized modes on nanostructured features such as nanoparticles. The latter, often termed localized surface plasmon polaritons, are described using Mie scattering theory for spherical particles, where dipole and higher-order resonances lead to strong field enhancement in subwavelength volumes. Historically, SPPs were first indirectly observed as anomalies in the diffraction spectra of optical gratings by Robert W. Wood in 1902, manifesting as sudden drops in certain diffraction orders. These "Wood's anomalies" were theoretically interpreted by Ugo Fano in 1941 as arising from the excitation of surface electromagnetic waves at the grating interface. The explicit theoretical prediction of SPPs on smooth, planar surfaces came from Robert H. Ritchie in 1957, who derived their existence through calculations of electron energy losses in thin metal films, confirming the coupled light-matter nature of the modes.

Sommerfeld–Zenneck Surface Waves

Sommerfeld–Zenneck surface waves, also known as Zenneck waves, represent a class of slowly evanescent electromagnetic modes that propagate along the interface between free space and a lossy half-space, such as conductive ground. These waves are transverse magnetic (TM)-polarized, with the magnetic field primarily in the y-direction perpendicular to the direction of propagation and the interface normal. The model considers a planar boundary at z=0, where the upper half-space (z>0) is vacuum (permittivity ε₀, permeability μ₀), and the lower half-space (z<0) is a lossy medium characterized by complex permittivity ε = ε' + iε'' and conductivity σ, where ε'' incorporates losses via ε'' = σ/(ωε₀). The exact solution arises from the pole of the reflection coefficient in the complex wavevector plane, obtained by solving Maxwell's equations with boundary conditions of continuous tangential E and H fields at the interface.^[49] The dispersion relation for the propagation constant k_{zs} along the interface (x-direction) is given by k_{zs} = k_0 \sqrt{\epsilon / (1 + \epsilon)}, where k_0 = \omega \sqrt{\mu_0 \epsilon_0} is the free-space wavenumber. For highly conductive media where |ε| \gg 1, this approximates to a nearly non-dispersive wave with phase velocity close to c, but with small attenuation α ≈ k_0 / (2 \sqrt{\epsilon'}). The field components exhibit evanescent decay away from the interface; for the magnetic field in the upper half-space, H_y = A e^{i k_x x - \kappa z}, where \kappa = \sqrt{k_x^2 - k_0^2} ensures decay for z > 0 since k_x > k_0. In the lower half-space, the corresponding decay constant involves ε, with κ = \sqrt{k_x^2 - k_0^2 \epsilon}. These waves differ from those over perfect conductors, as bound modes require finite loss to satisfy the boundary conditions; over ideal conductors (ε → ∞ without loss), no such surface wave exists.^[50]^[5] In practical applications, Sommerfeld–Zenneck waves underpin ground wave propagation for amplitude modulation (AM) radio in the medium frequency (MF, 0.3–3 MHz) and high frequency (HF, 3–30 MHz) bands, enabling over-the-horizon communication distances up to 1000 km. Vertical monopoles efficiently excite these modes, as their TM polarization matches the wave's structure, with early experiments demonstrating enhanced signal range along conductive surfaces. Typical attenuation rates are 0.1–1 dB/km in these bands, depending on ground conductivity and frequency, allowing reliable coverage beyond line-of-sight. This mechanism shares conceptual similarities with surface plasmon polaritons in the low-frequency limit but applies to classical radio contexts over lossy dielectrics rather than optical metals.^[51] Historical controversies surrounded the existence and physical reality of these waves, particularly debates over their propagation over perfect versus lossy conductors, with Sommerfeld's 1909 analysis containing a sign error that was later corrected. Modern numerical validations, such as finite-difference time-domain (FDTD) simulations, confirm their presence and behavior over rough conductive surfaces, aligning with analytical predictions for attenuation and excitation by sources like line currents. These simulations demonstrate increased damping due to surface roughness but validate the core evanescent mode for flat interfaces.^[49]

References

[1]
[PDF] Surface waves take many forms in nature, science, and ... - MIT
Surface waves take many forms in nature, science, and technology. They include, for example, water waves propagating along the interface.
[2]
Waves - Physics
Mar 1, 1999 · Surface waves, such as water waves, are generally a combination of a transverse and a longitudinal wave. The particles on the surface of the ...
[3]
Water waves
Water waves are surface waves, a mixture of longitudinal and transverse waves. Surface waves in oceanography are deformations of the sea surface.
[4]
Seismic Waves - HyperPhysics
Earthquakes also produce surface waves which may cause motion perpendicular to the surface or parallel to the surface. The waves which move the surface up and ...Missing: definition | Show results with:definition
[5]
Electromagnetic Surface Waves - MIT
Jun 15, 1996 · The term surface wave conjures up an image of energy flow that is confined to a region that is localized at or near the surface.Missing: definition | Show results with:definition
[6]
Electromagnetic surface waves supported by a resistive metasurface ...
Sep 23, 2020 · Field profiles of surface waves Electromagnetic surface waves are defined as waves that propagate along the interface and decay exponentially ...
[7]
Surface Wave Methods | US EPA
Apr 18, 2025 · A wide variety of seismic waves propagate along the surface of the earth. They are called surface waves because their amplitude decreases exponentially with ...
[8]
13.1 Types of Waves - Physics | OpenStax
Mar 26, 2020 · No, mechanical waves do not require any medium to propagate. Yes, both mechanical and electromagnetic waves require a medium to propagate.
[9]
[PDF] Excitation of Acoustic Surface Waves by Turbulence - VTechWorks
Jul 28, 2021 · Acoustic surface waves are generated by turbulent flow using a metamaterial and a Kevlar-covered cavity, linking acoustics and turbulent flows.
[10]
http://large.stanford.edu/courses/2007/ap272/white1/
[11]
https://ethz.ch/content/dam/ethz/special-interest/itet/photonics-dam/documents/lectures/EandM/WaveEquation.pdf
[12]
[PDF] 4 Waves - DAMTP
An interface between two fluids. 4.1.1 Free Boundary Conditions. The surface of a fluid is best viewed as the interface between two different fluids. In.
[13]
[PDF] Chapter 16: Waves [version 1216.1.K] - Caltech PMA
We now consider two limiting cases: deep water and shallow water. dispersion relation (16.9), up to a dimensionless multiplicative constant, by dimensional ...
[14]
[PDF] Chapter 2 - Deep water gravity waves
Returning now to surface waves, and the dispersion relation (2.19) shown in Fig. 2.4, we can see from Fig. 2.9 that c≤c for all wavenumbers (the slope of line B ...
[15]
[PDF] Lecture 2: Concepts from Linear Theory - WHOI GFD
Since the expression obtained for ω depends on the wave number, it is called a “dispersion” relation. If a system of linear evolution equations has a dispersion ...
[16]
Surface gravity waves | Applied Mathematics | University of Waterloo
The dispersion relation and the phase speed can be written as follows, w2 = gk and c = sqrt (g/k) The propagation speed of the waves in this case depends on ...
[17]
[PDF] Dynamics of Winds and Currents Coupled to Surface Waves
Aug 4, 2009 · Surface waves grow primarily by pressure-form stress from airflow over the waveforms, and they dissipate in the open sea by wave breaking that ...Missing: seismic | Show results with:seismic<|control11|><|separator|>
[18]
Tsunami Generation: Earthquakes - NOAA
Sep 27, 2023 · Most tsunamis are generated by earthquakes with magnitudes over 7.0 that occur under or very near the ocean and less than 100 kilometers (62 ...
[19]
[PDF] Surface Waves - UNOLS |
Jan 20, 2012 · • Wind acts via friction at the surface: wind stress τ ... Ocean waves so energetic that input from wind equals losses to wave breaking.<|separator|>
[20]
Stokes drift - PMC - NIH
A particle floating at the free surface of a water wave experiences a net drift velocity in the direction of wave propagation, known as the Stokes drift.
[21]
[PDF] Chapter 2 Lecture: Mean Properties of Linear Surface Gravity Waves ...
... kinetic and potential energy are the same (KE = PE), that is the wave energy is equipartiioned. This is a fundamental principle in also sort of linear wave.
[22]
Nonlinear wave evolution with data-driven breaking - Nature
Apr 29, 2022 · In this paper, we develop a blended machine learning framework to model wave breaking and its effects on the nonlinear evolution of ocean waves.
[23]
The Early History of Solitons (Solitary Waves) - IOPscience
The early history of solitons or solitary waves began in August 1834 ... John Scott Russell observed a solitary wave travelling along a Scottish canal.
[24]
[PDF] 8 Surface waves and normal modes - SOEST Hawaii
Surface waves are generally the strongest arrivals recorded at teleseismic distances. and they provide some of the best constraints on Earth's shallow ...
[25]
Separation of intrinsic and scattering seismic wave attenuation in ...
Sep 28, 2013 · The attenuation of seismic waves is a reduction in amplitude or energy caused by scattering from heterogeneity or intrinsic absorption due to anelasticity or ...Missing: generation | Show results with:generation
[26]
[PDF] Multichannel Analysis of Surface Waves (MASW) - An Overview
Rayleigh waves which has more than 70% of the total seismic energy is the principal component of ground roll. Frequency component of a surface wave has a ...
[27]
Moment magnitude, Richter scale | U.S. Geological Survey
... scales that are an extension of Richter's original idea were developed. These include body wave magnitude (Mb) and surface wave magnitude (Ms). Each is ...
[28]
Surface Acoustic Wave (SAW) Sensors: Physics, Materials, and ...
This review article presents the physics of guided surface acoustic waves and the piezoelectric materials used for designing SAW sensors.
[29]
[PDF] Chapter 1: - MAE Center
Lord Rayleigh conducted the first study on the propagation of surface waves at stress free surfaces in 1885, and the waves he discovered are now known as ...<|separator|>
[30]
A HISTORY OF SURFACE ACOUSTIC WAVE DEVICES
Aug 6, 2025 · This paper gives a historical account of the development of Rayleigh-wave, or surface-acoustic-wave (SAW), devices for applications in ...Missing: 1950s | Show results with:1950s
[31]
[PDF] An Introduction to the Design of Surface Acoustic Wave Devices.
Thus the interdigital transducer allows precise control of frequency, phase and amplitude of signals on a surface wave delay line, or equivalently there is a ...
[32]
https://apps.dtic.mil/sti/tr/pdf/ADA073425.pdf
[33]
[PDF] Propagation of acoustic surface waves in periodic structures
Jan 23, 2020 · Bragg reflection of ASW (Ax2l). 61. 5.1. Bragg reflection of Rayleigh waves. 5.2. Modulation of groove depth. 5.3. Oblique incidence of ASW onto ...Missing: π/ Λ
[34]
Resonant scattering of surface acoustic waves by arrays of magnetic ...
Dec 19, 2023 · Then, the assumed array period Λ = 333 nm leads to Bragg reflection of acoustic waves with wavelength λ = 2Λ = 667 nm and an acoustic bandgap ...
[35]
[PDF] Measurement of surface acoustic wave attenuation in ... - HAL
Jul 8, 2025 · Attenuation is strongly related to multiple mechanisms such as material viscosity (intrinsic attenuation) and various surface con- ditions.
[36]
Bulk-acoustic waves radiated from low-loss surface-acoustic-wave ...
Mar 1, 2004 · Prior to this work, BAW radiation in SAW devices has been analyzed with the help of numerical methods4 including models for leaky surface waves.
[37]
Attenuation of 7 GHz surface acoustic waves on silicon | Phys. Rev. B
Sep 26, 2016 · The amplitude of SAWs is attenuated by coupling to bulk waves created by the Al grating, diffraction due to the finite size of the source, and ...
[38]
[PDF] Acousto-Optic Modulator Driven by Surface Acoustic Waves
A typical AOM is a bulk-type AOM, in which the inci- dent light is diffracted by a bulk acoustic wave through a photoelastic effect. In this case, the ...
[39]
Nonlinear mixing of surface acoustic waves propagating in opposite ...
The parametric mixing of two modulated surface acoustic waves propagating in opposite directions is studied with reference to nonlinear signal processing ...
[40]
[PDF] The Power Flow Angle of Acoustic Waves in Thin Piezoelectric Plates
Sep 1, 2008 · As is well known, the angle between propagation direction of wave and power flow (PFA) is a very important parameter for development of various ...Missing: spreading | Show results with:spreading
[41]
History of SAW devices - IEEE Xplore
This paper gives a historical account of the development of Rayleigh-wave, or surface acoustic wave (SAW), devices for applications in electronics.
[42]
Temperature Stability of Surface Acoustic Wave Resonators on In ...
However, ST-cut SAW devices are still not as temperature stable as AT-cut quartz. Hence new SAW resonators with better temperature stability are needed.
[43]
[PDF] SAW Filters: Performance Characteristics (Part I) - Abracon
Aug 16, 2021 · The operating bandwidths of a SAW filter can range from a very narrow bandwidth (about 2 MHz to 10. MHz) to designs with wide bandwidths (up to ...Missing: Δf/ f
[44]
Surface waves at metal-dielectric interfaces: Material science ...
Surface waves (SWs) are waves that exist at the interface of two different materials with their field decaying evanescently away from the interface (Fig. 1(a)).Missing: formula | Show results with:formula
[45]
(PDF) Surface electromagnetic waves: A review - ResearchGate
Aug 6, 2025 · Researchers have been interested in surface electromagnetic waves guided by planar interfaces of dissimilar mediums.
[46]
[PDF] Surface Electromagnetic Waves on Structured Perfectly Conducting ...
The solutions can then be sought for values of k inside the first Brillouin zone, and inside the non-radiative region defined by k > (ω/c). Dispersion curves ...<|control11|><|separator|>
[47]
(PDF) The Sommerfeld half-space problem revisited: From radio ...
Aug 6, 2025 · The classical Sommerfeld half-space problem is revisited, with generalizations to multilayer and plasmonic media and focus on the surface ...
[48]
[PDF] Lecture-Surface-Waves.pdf - EMPossible
Jan 30, 2020 · An expression for the “surface plasma frequency” can be derived by substituting this equation into the dispersion relation, letting ω=ωsp, and ...
[49]
On the Far‐Zone Electromagnetic Field of a Horizontal Electric ...
Dec 22, 2017 · The Sommerfeld half-space problem revisited: From radio frequencies and Zenneck waves to visible light and Fano modes (Invited). Journal of ...