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A seismic ray is a trajectory describing high-frequency seismic energy propagation in a medium whose properties vary slowly compared with the wavelength. In an isotropic medium it is normal to local travel-time wavefronts, and its travel time is $T=\int ds/v$ along the path.

Changes in wave speed bend rays, while interfaces can reflect or refract them. The approximation is useful for interpreting arrival times, but diffraction, interference and caustics require more than a single-ray description. In anisotropic media, energy propagation and the wavefront-normal direction need not coincide.

A resting shallow water layer has depth $h_0$ and wave speed $c_0=\sqrt{g{h}_0}$. A vertical wall receding into negative $x$ at speed $0<V<2c_0$ creates a simple wave with invariant $u-2c=-2c_0$. Inside the expansion fan $x/t=u+c$, proving the displayed profiles. The wall-adjacent state is $u=-V$, $c=c_0-V/2$; the fan occupies $c_0-3V/2<x/t<c_0$, with the resting state ahead. The wall lies at $x/t=-V$, strictly to the left of the fan tail because $c_0-V/2>0$. Thus the assumption on wall speed is exactly the condition for positive depth and continued contact between fluid and wall. The depth follows from $h=c^2/g$.

A shear wave in an isotropic elastic material has displacement perpendicular to its wavevector, with zero divergence for a plane wave. The Navier-Cauchy equation with inertia then reduces to $\rho {\mathbf{u}}_{tt}=\mu \Delta \mathbf{u}$, giving the displayed wave speed. At a boundary it can be generated by mode conversion from a compressional wave, even when the incident displacement is purely longitudinal.

For $p=P_0+\rho c_0^2{\alpha }^2/2$ and constant flux $c_0{A}_{0}q$, a downhill collapsible tube flow obeys the displayed steady equation. Uniform area satisfies $qR({\alpha }_1)=\beta$. Since $R^{\prime }<0$, linearizing about this root gives downstream exponential decay precisely when ${\alpha }_1^2<q$. This is spatial stability of a steady inlet-value problem, not a guarantee of temporal stability of the driven flow. The wave speed is $c_0\alpha$, so the stable spatial branch is supercritical.

For an isotropic radial wave speed $\alpha (r)$, the elastic ray equations imply conservation of $\mathbf{x}\times \mathbf{y}$, where $\mathbf{y}$ is the slowness vector. Indeed, its derivative is $\alpha \mathbf{y}\times \mathbf{y}+\mathbf{x}\times \nabla (1/\alpha )=0$. Thus each nonradial elastic ray lies in a fixed plane through the centre and $p=|\mathbf{x}\times \mathbf{y}|$ is constant. The angle $i$ is measured from the radial direction; tangential slowness continuity preserves $p$ across spherical interfaces.

A standing wave results from the superposition principle applied to equal-frequency, equal-amplitude waves travelling in opposite directions. Its spatial nodes of a standing wave remain fixed while its displacement oscillates in time.

For a pure standing wave, the two travelling waves also have equal amplitudes: their interference leaves a fixed mode shape, rather than a profile moving along the string. Choose the position origin and time origin so that their transverse displacements areHere $a>0$ is each constituent's amplitude, $k=2\pi /\lambda$ is the wavenumber, $\lambda$ is the wavelength, and $\omega =2\pi f$ is the angular frequency corresponding to frequency $f$. Their wave speed is $c=\omega /k$. The superposition principle and the sine addition formula giveThe spatial factor stays put; the temporal factor changes its size and sign. Other constant wave phases shift the position or time origin. Unequal constituent amplitudes generally leave moving displacement at the would-be nodes of a standing wave, rather than this exact factorization.^[ref]

For an original ideal-string example, set $L=1\mathrm{m}$, $c=20\mathrm{m}{\mathrm{s}}^{-1}$, and choose the third normal mode. Then $k=3\pi {\mathrm{m}}^{-1}$, $\omega =60\pi {\mathrm{s}}^{-1}$, and $f=30\mathrm{Hz}$ by fixed-end standing wave frequencies. With each constituent amplitude $a=5\mathrm{mm}$,Its nodes of a standing wave are at $x=0,1/3,2/3,1\mathrm{m}$ and its antinodes at $x=1/6,1/2,5/6\mathrm{m}$. These are constructed model parameters, not experimental readings.

At $t=1/120\mathrm{s}$, every point has zero displacement, but most have nonzero transverse velocity: the string is crossing its straight configuration. The black dots alone mark permanent nodes of a standing wave. At an antinode the greatest transverse speed is $2a\omega =0.6\pi \mathrm{m}{\mathrm{s}}^{-1}$, much smaller than the chosen $20\mathrm{m}{\mathrm{s}}^{-1}$ wave speed. One is motion of string material; the other is propagation of the constituent profiles.

The beauty of a standing wave is that motion reveals geometry: fixed nodes of a standing wave turn a moving disturbance into a spatial ruler. Their separation measures wavelength, and boundary conditions turn the continuous choice of wavenumber into a discrete set of normal modes. This is useful both for interpreting string vibrations and for understanding how a bounded wave equation selects its eigenfrequencies.

For a uniform string with tension $T>0$, linear mass density $\rho >0$, left end fixed, and a freely transverse-moving point mass $M\ge 0$ at $x=L$, the wave equation and endpoint law areThe last equation is Newton's second law: the string pulls the mass back toward the interior, giving the minus sign. With wave speed $c=\sqrt{T/\rho }$, a nonzero normal mode has spatial part $X(x)=\sin (\omega x/c)$ and satisfiesFor $M>0$, its dimensionless roots $q=\omega L/c$ occur once in each $(n\pi ,n\pi +\pi /2)$, $n\ge 0$, because $\cot q$ decreases from infinity to zero while $Mq/(\rho L)$ increases. For $M=0$, $q=(n+1/2)\pi$: the right end has a homogeneous Neumann boundary condition. There is no nonzero zero-frequency mode, since a linear spatial profile vanishing at the left end and with zero right slope must vanish everywhere.

Multiplying the spatial equation by $X$ and using integration by parts givesFor two different modes, the same calculation gives the weighted inner product orthogonality $\rho {\int }_0^L{X}_m{X}_{n}dx+M{X}_m(L)X_n(L)=0$. The endpoint mass therefore contributes to both inertia and mode orthogonality.

For a one-dimensional wave equation whose principal symbol is ${\xi }_t^2-c(y{)}^2{\xi }_y^2$, a travel-time coordinate isIts two families of characteristic curves are $s(y)-t=\text{constant}$ and $s(y)+t=\text{constant}$ wherever the wave speed $c$ is positive.

Ultrasound is sound above the conventional upper limit of human hearing, approximately $20\mathrm{kHz}$. In a crystal, ultrasonic travel-time measurements determine long-wavelength elastic wave speeds, which are the small-wavevector slopes of the corresponding acoustic phonon branches.

A description of seismic wave speed as a function of position and, if needed, direction. It converts travel times into geometric constraints and is distinct from a direct depth measurement.

A vibroscope measures the vibration of a tensioned material fibre. In fibre testing, the measured fundamental frequency, suspended length and applied tension can be used to estimate linear mass density; known mass density then permits an area estimate. The physical measurement complements the single-fibre tensile test, whose conversion from force to engineering stress requires cross-sectional area.

The 2024 study by Bo Madsen, Rajnish Kumar and Hans Lilholt compared area-analysis procedures using measured frequencies of tensioned glass and carbon strands.^[ref] A practical arrangement holds a known strand length, applies a known tension, excites small transverse oscillations and detects the frequency of its response. The supports and identification of the vibration mode enter the interpretation.

For an original idealization, treat the strand as a uniform flexible string with fixed endpoints, length $L$, tension $T$ and linear mass density $\mu$. Its wave speed is $\sqrt{T/\mu }$ and its fundamental wavelength is $2L$, giving the vibroscope area estimatorThe string wave-speed and fixed-end standing-wave relations supply this model.^[ref]

Choose $L=20\mathrm{mm}$, $T=1.00\mathrm{mN}$, $f_1=2000\mathrm{Hz}$ and $\rho =2500\mathrm{kg}{\mathrm{m}}^{-3}$. The model gives $\hat{\mu }=1.5625\times 10^{-7}\mathrm{kg}{\mathrm{m}}^{-1}$ and $\hat{A}=62.5\mu {\mathrm{m}}^2$. Only if a circular area is assumed does this correspond to diameter $\sqrt{4\hat{A}/\pi }\approx 8.92\mu \mathrm{m}$.

These are stipulated model values, not published fibre data. Checking several tensions tests the predicted proportionality of $f_1^2$ to $T$. Finite bending resistance, uncertain effective length and changes in strand dimensions under tension can invalidate the flexible-string estimate. A resonance peak alone does not establish that all of its assumptions hold.

A semi-infinite wave equation on a string with linear mass density $\rho$, wave speed $c$, and tension $T=\rho c^2$ has positive characteristic mechanical impedance $Z=\rho c$. At an end attached to a massless viscous damper of coefficient $\alpha >0$, displacement and transverse-force balance give the displacement reflection coefficientHere $\mathcal{A}$ is the fraction of mean incident power dissipated. A prescribed $0<\mathcal{R}<1$ gives two positive damper coefficients, $\alpha /Z=(1\pm \sqrt{\mathcal{R}})/(1\mp \sqrt{\mathcal{R}})$, with opposite reflected phases. Matching $\alpha =Z$ removes reflection.

A wave is a spatially organized disturbance whose evolution can transmit a pattern and energy through space. Shake one end of a stretched mechanical string: the displacement travels along it while each small piece mainly moves up and down. In a small-amplitude acoustic wave, nearby fluid elements similarly oscillate about their mean positions while the pressure disturbance travels onward. Transport of a wave therefore need not mean bulk transport of the material carrying it.

For a one-dimensional nondispersive travelling wave, $u(x,t)=f(x-ct)$ keeps its shape while moving at wave speed $c$. A sinusoidal example has wavelength $\lambda$ and frequency $f=c/\lambda$. Oppositely travelling disturbances can instead form a standing wave. Interference follows from the superposition principle when the governing equations are linear; nonlinear waves can change shape and interact more strongly. Measuring arrival time, frequency and wave amplitude makes waves useful probes of both a source and the medium between source and receiver.

The wave equation is a second-order linear partial differential equation describing waves with finite propagation speed. In one spatial dimension its homogeneous, constant-coefficient form isHere $x$ is position, $t$ is time, $u(x,t)$ is the disturbance, and subscripts denote partial derivatives. For a wave equation on a string, $u$ is transverse displacement; in linear acoustics, it can be a pressure perturbation. In several spatial dimensions, $u_{xx}$ becomes the Laplacian $\Delta u$. The same mathematical structure connects vibrating strings and acoustic waves, although their physical assumptions and wave speeds differ. ^[ref]

For the whole-line Cauchy problem, prescribe both displacement $u(x,0)=f(x)$ and velocity $u_t(x,0)=g(x)$. A classical solution has continuous partial derivatives through order two; $f\in C^2(\mathbb{R})$ and $g\in C^1(\mathbb{R})$ suffice for D'Alembert's formula to produce one. A finite string additionally needs boundary conditions. These statements concern the unforced model with constant $c$; damping, spatially varying coefficients and nonlinear terms change the solution formula.

A travelling wave $F(x-ct)$ carries its entire profile to the right at wave speed $c$; $G(x+ct)$ travels left. Their sum is again a solution by the superposition principle. A standing wave instead has a fixed mode shape whose size and sign oscillate: in $\sin (kx)\cos (ckt)$, each node of a standing wave stays stationary. The distinction is especially clear in the pulse and string diagrams below. For this ideal model, the dispersion relation is ${\omega }^2=c^2{k}^2$: each propagation branch is nondispersive, so a localized profile translates without spreading. This makes the equation a useful reference against which to understand wave dispersion and damping.

The appealing combination is local interaction with global predictability: neighbouring displacements determine acceleration through $u_{xx}$, yet D'Alembert's formula explicitly identifies exactly which initial locations can influence a later observation. Finite propagation speed turns that observation into a domain of dependence, rather than requiring knowledge of the entire initial string.

An ideal uniform, flexible string under constant tension $T>0$ has transverse displacement $y(x,t)$ satisfyingwhere $\mu >0$ is linear mass density, meaning mass per unit length of the straight reference string. The approximation assumes small slope $|y_x|\ll 1$, transverse motion, negligible bending stiffness, negligible damping, and no distributed external force. Constant tension and uniform linear mass density make $c$ constant. Some texts use $\rho$ for this same linear mass density, giving $c=\sqrt{T/\rho }$; here $\rho$ must have units $\mathrm{kg}{\mathrm{m}}^{-1}$, rather than units of volume mass density. ^[ref]

The units provide a useful check: $T/\mu$ has units $(\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-2})/(\mathrm{kg}{\mathrm{m}}^{-1})={\mathrm{m}}^2{\mathrm{s}}^{-2}$. For $T=100\mathrm{N}$ and $\mu =0.010\mathrm{kg}{\mathrm{m}}^{-1}$, the wave speed is $100\mathrm{m}{\mathrm{s}}^{-1}$, so a pulse crosses a one-metre string in $0.010\mathrm{s}$. Quadrupling tension doubles wave speed; quadrupling linear mass density halves it. These square-root laws explain how material and loading set the time scale of the vibration.

For a length-$L$ string with fixed endpoints, the homogeneous Dirichlet boundary conditions are $y(0,t)=y(L,t)=0$. Its nonzero normal modes haveThus $cn\pi /L$ is an angular frequency, in radians per second, whereas $cn/(2L)$ is the frequency in hertz. The wavelength is $2L/n$, and each node of a standing wave lies at $x=jL/n$ for some $j=0,\dots ,n$. For the numerical string above with $L=1\mathrm{m}$, the first three frequencies are $50$, $100$ and $150\mathrm{Hz}$, with angular frequencies $100\pi$, $200\pi$ and $300\pi {\mathrm{s}}^{-1}$. ^[ref]

Each normal mode is a standing wave, but it also has a travelling wave description:At a fixed endpoint an incident displacement pulse reflects with reversed sign, allowing incident and reflected displacements to cancel at the endpoint. Periodic reflection for a fixed-end string implements both endpoints through odd, $2L$-periodic extensions of the initial data. This is the bridge between moving pulses and the stationary Fourier sine basis. ^[ref]

A short derivation comes from Newton's second law. The transverse force on a small segment is approximately $T[y_x(x+\delta x,t)-y_x(x,t)]$, and its mass is $\mu \delta x$. Dividing the force balance by $\delta x$ and taking the limit gives $\mu y_{tt}=T{y}_{xx}$. Separation of variables then gives ${\varphi }^{\prime \prime }+k^2\varphi =0$; the fixed-end boundary conditions select $k=n\pi /L$, while the temporal equation is a harmonic oscillator equation with $\omega =ck$.

Wave propagation is the evolution of a disturbance through space and time under the medium's dynamical equations. In a uniform nondispersive one-dimensional model,the wave equation admits rightward and leftward profiles $u=f(x-ct)+g(x+ct)$. Each profile travels at wave speed $c$ without changing shape, so a measured delay over a known distance can determine $c$.

A real medium can alter this simple picture through dispersion, wave absorption, wave reflection and wave scattering. Phase velocity follows a fixed wave phase, while group velocity describes a narrowband wave packet's envelope in a regime where that approximation holds. Neither should be identified automatically with every signal's arrival speed. Specifying an initial disturbance and suitable boundary conditions determines which of the possible propagation patterns occurs.

For the homogeneous one-dimensional wave equation $u_{tt}=c^2{u}_{xx}$ with constant $c>0$, wave speed means the speed at which a travelling wave profile translates. In $F(x-ct)$, a feature initially at $x=s$ follows $x=s+ct$, giving signed velocity $+c$; in $G(x+ct)$ it follows $x=s-ct$, giving signed velocity $-c$. The characteristic lines for the one-dimensional wave equation therefore have slopes $dx/dt=\pm c$. The speed of either profile is $c$, even though the displacement at a fixed point can change sign or pass through zero.^[ref]

The same $c$ bounds the causal cone for the constant-speed wave equation. For a wave equation on a string, $c=\sqrt{T/\mu }$ depends on tension and linear mass density; for small acoustic waves in stationary uniform air it is the speed of sound. This interpretation assumes the stated ideal wave equation. In media with wave dispersion, phase velocity, group velocity and the earliest signal arrival have different meanings: a measured wave crest speed need not be the speed of a wave envelope, and neither alone establishes a causal cone for the constant-speed wave equation.