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The experiment in A time of flight method to measure the speed of sound using a stereo sound card by Carlos C. Carvalho and others makes wave speed visible as a delay. Two powered microphones feed separate channels of one stereo line input. A clap beyond one end of their common axis reaches the nearer microphone first. The displayed signals show two separated onsets when enlarged; count their sample separation $N$ to obtain $\Delta t=N/f_s$ and infer speed of sound $c=d/\Delta t$ from the measured microphone spacing $d$. The reported sampling rate is $44100{\mathrm{s}}^{-1}$. Repeating at different spacings gives a distance-versus-delay plot whose slope estimates speed of sound; dependence on distance helps identify acoustic travel rather than a fixed electronic delay. These are the paper's apparatus and observed readout, not measurements made for this article.^[ref]

A controlled implementation also needs timing calibration for acoustic speed measurements. Put the microphones side by side to estimate the channel-delay offset ${\tau }_0$, swap channels to check its sign, and check the acquisition clock against an independent timing reference. Measure spacings between consistent points on the microphones, keep the source outside their span and on their axis, and fit $\Delta t=d/c+{\tau }_0$ by linear regression rather than forcing a zero intercept. Repeat pulses and compare the same onset criterion before reflections arrive. These are proposed calibration and control checks, not additional procedures claimed for the paper.

One sample corresponds to $1/f_s\simeq 22.7\mu \mathrm{s}$ for the reported sampling rate, so a delay of only a few samples has large relative measurement error. Background measurement noise, different microphone responses and threshold choice can move the apparent onset; reflections can supply a later competing pulse. Record temperature, keep air motion small, and compare several spacings. The result estimates speed of sound under those conditions; it cannot prove an exactly zero precursor below the instrument's sensitivity or establish finite propagation speed for every real medium. Both new diagrams in these sections show explicitly synthetic ideal wave equation solutions.^[ref]

A static lossless taper preserves frequency. If characteristic impedance varies slowly on the local wavelength scale, reflection is small and mean power $Z{\omega }^2{A}^2/2$ is conserved. At fixed tension this gives the displayed amplitude law, while wave speed and wavelength scale as ${\rho }^{-1/2}$. A long section containing an abrupt step is not automatically adiabatic.

A small pressure disturbance in an elastic artery propagates through the coupled inertia of blood and compliance of the wall. If $\mathcal{A}$ is the undisturbed area, $\rho$ the mass density and $c$ the wave speed, the linearized equations give $c^2=\mathcal{A}/\rho dp/d\mathcal{A}$. A travelling pressure amplitude $p_{+}$ carries flow amplitude $\mathcal{A}{p}_{+}/(\rho c)$; the oppositely travelling wave has the opposite flow sign.

For a free massless real scalar field with wave speed set to one and metric $(+,-)$, define $\mathcal{E}=({\varphi }_t^2+{\varphi }_x^2)/2$ and $p=-{\varphi }_t{\varphi }_x$. Its symmetric stress-energy tensor is $T^{\mu \nu }=\left(\begin{array}{cc}\mathcal{E}& p\\ p& \mathcal{E}\end{array}\right)$. The Lorentz current has density $tp-x\mathcal{E}$ and flux $t\mathcal{E}-xp$. Its conserved charge is the displayed expression when boundary flux vanishes. In one spatial dimension it generates boosts, not spatial rotations; there is no independent purely spatial angular-momentum component. Its conservation says that the center of energy moves uniformly. This is a symmetry of the wave-field model and need not survive boundaries or additional material effects.

For $c>0$, the backward causal cone at $(x_{\ast },t_{\ast })$ isIts boundaries are characteristic lines for the one-dimensional wave equation; its base on $t=0$ is the initial-data domain of dependence. The forward cone from $(s,0)$ is $t\ge 0$, $|x-s|\le ct$. Here the cone is set by the model's wave speed, which may be acoustic or a string wave speed; it does not assert propagation at the speed of light.

For real horizontal wavenumber $k$, positive real wave speed $\beta$ and $\omega =u+iv$ with $v>0$, write $k_{\beta }=a+ib$ with $k_{\beta }^2={\omega }^2/{\beta }^2-k^2$. Its imaginary part gives $ab=uv/{\beta }^2$. The number $b$ cannot vanish, and $\mathrm{Re}(k_{\beta }/\omega )=v[b+u^2/({\beta }^{2}b)]/|\omega {|}^2$ has the sign of $b$. The branch $b>0$ makes $e^{i{k}_{\beta }z}$ decay towards increasing depth and analytically continues the outgoing propagating or evanescent branch to the upper frequency half-plane.

For the lossless small-slope wave equation on a string, tension $T$ and linear mass density $\mu$ give wave speed $c=\sqrt{T/\mu }$. With transverse force $F_y=-T{y}_x$ exerted across a cut toward increasing $x$, a rightgoing displacement $y=f(x-ct)$ has velocity $v_y=-c{y}_x$. Its positive mechanical impedance is therefore $Z=F_y/v_y=T/c=\sqrt{T\mu }$. A leftgoing component has $F_y/v_y=-Z$ under the same fixed orientation. A rightgoing time-harmonic wave with peak displacement wave amplitude $|A|$ carries mean power $Z{\omega }^2|A{|}^2/2$.

These characteristic curves are straight when wave speed $c$ is constant. A right-moving component is constant on $x-ct=s$, and a left-moving component on $x+ct=s$. The full superposition generally is not constant on either family, because the other component can vary along it. Their backwards intersections with the initial line select the endpoints used in D'Alembert's formula.

Work in the material's rest reference frame, with constant speed of sound $c>0$. For one-dimensional propagation, choose the positive $x$ direction along the travelling wave from its emission point toward the receiver. Define $u_s=d{x}_s/dt$ and $u_o=d{x}_o/dt$ as the signed source and receiver velocities along that same axis. Both use the same sign convention: positive $u_s$ moves the source toward the receiver ahead, whereas positive $u_o$ moves that receiver away from the incoming wave. Assume $|u_s|<c$ and $|u_o|<c$, constant velocities, and no source-receiver crossing during the interval considered. Then

To derive this, let $T_s=1/f_s$ be the emission wave period. During $T_s$, one wave crest travels $c{T}_s$ while the source advances $u_s{T}_s$. Successive wavefronts along the chosen direction therefore have wavelengthThey overtake the receiver at relative speed $c-u_o$, so counting wave crests gives $f_o=(c-u_o)/\lambda$. This separates the source's effect on wavelength from the receiver's effect on crossing rate. Even with a moving source, a stationary receiver obeys $f_o\lambda =c$ because the wave speed through the material remains $c$.

For a stationary receiver and a source approaching at speed $v$, set $u_o=0$ and $u_s=v$: $f_o=f_{s}c/(c-v)>f_s$. For recession choose the positive axis along the departing wave toward the receiver again, so the source has $u_s=-v$: $f_o=f_{s}c/(c+v)<f_s$. For the deliberately chosen values $c=343\mathrm{m}{\mathrm{s}}^{-1}$, $v=20\mathrm{m}{\mathrm{s}}^{-1}$ and $f_s=500\mathrm{Hz}$, the two predictions areThe corresponding wavelengths are $0.646\mathrm{m}$ and $0.726\mathrm{m}$, compared with $c/f_s=0.686\mathrm{m}$ for a stationary source. These are calculations for chosen parameters, not reported measurements.

A receiver approaching a stationary source at $20\mathrm{m}{\mathrm{s}}^{-1}$ instead has $u_s=0$, $u_o=-20\mathrm{m}{\mathrm{s}}^{-1}$, giving $f_o=500(363/343)\mathrm{Hz}\simeq 529.15\mathrm{Hz}$. Moving away gives $f_o=500(323/343)\mathrm{Hz}\simeq 470.85\mathrm{Hz}$. The equal closing speeds do not make these equal to the moving-source values. If both source and receiver move together with $u_s=u_o$, however, the numerator and denominator cancel and $f_o=f_s$. For speeds small compared with $c$, the first-order shift is $(f_o-f_s)/f_s\simeq (u_s-u_o)/c$; the difference between moving-source and moving-receiver experiments appears beyond this first order.

A one-dimensional collapsible tube model couples area $\alpha$, mean axial velocity $u$ and internal pressure $p$: ${\alpha }_t+(\alpha u{)}_x=0$ and $u_t+u{u}_x=-p_x-R(\alpha )u$. A wall relation specifies internal minus external pressure as a function of area. Decreasing area generally increases the friction coefficient $R$, while positive wall stiffness gives real small-amplitude wave speed.

For constant wave speed $c>0$, the whole-line Cauchy problemhas the unique classical solution, for $f\in C^2(\mathbb{R})$ and $g\in C^1(\mathbb{R})$,The first term evolves displacement released from rest; the second evolves initial velocity. The factor $1/(2c)$ converts an integral of velocity over distance into displacement. On a finite string the formula applies before the backward characteristic curves reach an endpoint, or globally after an extension that enforces the boundary conditions. ^[ref]

For a completely smooth example, choose a bump function with compact support:where $a>0$ sets the half-width and $A$ sets the initial peak. ThenAt $t=0$ the two copies coincide, recovering peak $A$. At $t=a/(2c)$, $u(0,t)=A{e}^{-1/3}\simeq 0.716531A$. At $t=3a/(2c)$ the pulses are separated, centred at $x=\pm 3a/2$, each of width $2a$ and peak $A/2$. Zero initial velocity forces equal left and right contributions; a purely right-moving initial profile instead requires $g=-c{f}^{\prime }$. Thus “released from rest” and “launched to the right” describe different Cauchy data.

The domain of dependence of $(x_{\ast },t_{\ast })$, for $t_{\ast }\ge 0$, is contained in $[x_{\ast }-c{t}_{\ast },x_{\ast }+c{t}_{\ast }]$ on the initial line. More precisely, displacement contributes only through the two endpoints, while initial velocity contributes throughout the interval. With $c=1$, $x_{\ast }=0.5$ and $t_{\ast }=1.5$, that interval is $[-1,2]$. Changing either initial datum outside it cannot change $u(0.5,1.5)$. If both initial data vanish outside $[-a,a]$, the solution vanishes outside $[-a-ct,a+ct]$: this is finite propagation speed. The expanding interval bounds possible influence; it need not be filled by nonzero displacement, as the separated bumps demonstrate.

Historically, d'Alembert studied the vibrating-string problem in “Recherches sur la courbe que forme une corde tendue mise en vibration”, in the Berlin Academy's volume for the year 1747, pages 214–219. The memoir introduces the differential equation and integrates it through two arbitrary one-variable functions. The original geometric problem makes the formula's enduring appeal clear: arbitrary initial shapes can be handled without first decomposing them into normal modes. ^[ref]

For a brief derivation, use characteristic coordinates $\xi =x-ct$ and $\eta =x+ct$. The chain rule gives $u_{tt}-c^2{u}_{xx}=-4c^2{u}_{\xi \eta }$, so $u=F(\xi )+G(\eta )$. The initial conditions yield $F+G=f$ and $-c{F}^{\prime }+c{G}^{\prime }=g$. Integrating these relations gives the displayed formula; the arbitrary integration constants cancel in their sum.

A wave equation with a linear damping term and positive wave speed $c$ separates into spatial normal modes and damped harmonic oscillators. For a spatial eigenvalue $k^2$, the time factor obeys $T^{\prime \prime }+\alpha T^{\prime }+c^2{k}^{2}T=0$. Its characteristic roots are $-\alpha /2\pm \sqrt{{\alpha }^2/4-c^2{k}^2}$. Thus underdamping requires $|\alpha |<2c|k|$, critical damping gives a repeated root and the second solution $t{e}^{-\alpha t/2}$, and overdamping gives two real exponential rates. Negative $\alpha$ amplifies rather than damps the motion.

Arrival-time location estimates a hypocenter and origin time by comparing picked phase arrivals with predicted travel times. Linearization gives residual changes $\delta t_i=\delta t_0+\nabla T_i\cdot \delta \mathbf{x}$. Poorly distinct derivative columns cause tradeoffs, while incorrect wave speeds or phase labels cause systematic bias even with many stations.

In a smooth isotropic medium, an elastic ray is the path followed by a local wavefront normal, with the appropriate P or S wave speed $c$. Arclength $s$ gives the displayed trajectory and travel-time equations. Its neighboring ray paths determine geometrical spreading, while the eikonal equation determines the normals. A caustic occurs where the neighboring paths focus and the elementary ray wave amplitude description ceases to be uniform.

For the ideal uniform wave equation on a string, the wave speed is $c=\sqrt{\tau /\mu }$, where $\tau$ is constant tension and $\mu$ is linear mass density. Assume small transverse slopes, negligible bending stiffness and no damping. Fix both ends at zero displacement, giving the homogeneous Dirichlet boundary conditions $y(0,t)=y(L,t)=0$. Our standing wave already satisfies the first. For a nonzero amplitude, the second requires $\sin (kL)=0$, henceThe value $n=0$ gives zero displacement in this sine family, not a vibrating normal mode. Each positive $n$ fits $n$ half-wavelengths into the string: it has $n$ antinodes, $n-1$ interior nodes of a standing wave, and two endpoint nodes of a standing wave. For $L=1\mathrm{m}$ and $c=20\mathrm{m}{\mathrm{s}}^{-1}$, $f_n=10n\mathrm{Hz}$; the third normal mode has $f_3=30\mathrm{Hz}$ and wavelength $2/3\mathrm{m}$.

The equally spaced eigenfrequencies follow from this string's uniform wave speed and fixed-end boundary conditions. They are not a universal spectrum for every vibrating object. Changing an endpoint constraint changes the allowed mode shapes; changing tension changes wave speed. At fixed $L$ and $\mu$, quadrupling $\tau$ doubles every $f_n$. The more general construction of arbitrary initial motion belongs to normal mode and wave equation on a string.

A spatially constant blowup profile for a wave equation solves an ordinary differential equation. Choose smooth compactly supported initial data which agree with the profile on a ball larger than its blowup time times the wave speed. Finite propagation speed preserves agreement in an interior light cone, forcing finite-time breakdown of the localized solution. For $u_{tt}-\Delta u=u^3$, the profile $u(t)=\sqrt{2}/(1-t)$ is one example.

For a slow uniform layer over a faster half-space, the nth Love wave branch reaches the half-space shear wave speed at the displayed threshold. Its half-space decay exponent then vanishes, so the threshold is a limiting nonlocalized state, not a trapped finite-energy mode. The fundamental has zero threshold and exists at every positive frequency; overtones require frequency strictly above their positive threshold. At high frequency each fixed branch approaches the layer shear wave speed from above.

A strip of membrane with fixed edges $y=0,b$ supports transverse normal modes proportional to $\sin (n\pi y/b)$, $n\ge 1$. Separation of the two-dimensional wave equation gives the displayed dispersion relation. Confinement produces longitudinal wave dispersion even when the membrane's underlying wave speed $v$ is independent of frequency.

A normal mode, also called an eigenmode, is a pattern of free oscillation in a linear system whose relative displacements stay fixed while a single temporal factor changes their size and sign. For an undamped stable vibrational mode,Here $\varphi$ is the mode shape, $A$ sets the amplitude, $\omega$ is the eigenfrequency expressed as an angular frequency, and $\phi$ is the oscillation phase. Every snapshot is a scalar multiple of the same mode shape; neighbouring regions may move in opposite directions. Normal modes of a double pendulum show that normal modes also describe coupled discrete systems, beyond strings. Their shapes are eigenvectors, rather than spatial eigenfunctions. ^[ref]

For the ideal wave equation on a string, take a uniform string of length $L$ with constant tension $\tau$ and linear mass density $\mu$. Assume small transverse slope, negligible bending stiffness, no damping, and fixed endpoints. Its wave speed is $c=\sqrt{\tau /\mu }$, and the nonzero normal modes areThe angular frequency ${\omega }_n$ is in radians per second; the frequency $f_n$ is in hertz. The wavelength is $2L/n$. The mode shape has $n$ loops separated by nodes of a standing wave, including both fixed endpoints. ^[ref]

As a worked example, choose $L=1\mathrm{m}$, $\tau =1\mathrm{N}$, and $\mu =1\mathrm{kg}{\mathrm{m}}^{-1}$. Then $c=1\mathrm{m}{\mathrm{s}}^{-1}$ and $f_1=0.5\mathrm{Hz}$, $f_2=1\mathrm{Hz}$, $f_3=1.5\mathrm{Hz}$. Their oscillation periods are $T_1=2\mathrm{s}$, $T_2=1\mathrm{s}$, and $T_3=2/3\mathrm{s}$. These are deliberately simple model parameters, rather than measurements of a musical instrument. The plot divides displacement by $A$ so that the three mode shapes can be compared, and uses each mode's own oscillation period for its snapshots.

For $\phi =0$, at $t=T_n/4$ every point has zero displacement, but most points have nonzero velocity. Only zeros of the mode shape are permanent nodes of a standing wave. At $t=T_n/2$ the original mode shape has reversed sign, and at $t=T_n$ it returns. The synchronized conversion between kinetic energy and potential energy is shown in normal-mode energy exchange.

The short derivation uses separation of variables: substituting $u=\varphi q$ into the wave equation gives ${\varphi }^{\prime \prime }+k^2\varphi =0$ and $q^{\prime \prime }+c^2{k}^{2}q=0$. The fixed-end boundary conditions remove the cosine spatial term and require $\sin (kL)=0$, hence $k=n\pi /L$ and ${\omega }_n=ck$. Different geometries and boundary conditions give different eigenfunctions; a sine mode shape is a property of this uniform fixed string, not a universal form of an eigenmode.

The usefulness is that a complicated vibration becomes a collection of harmonic oscillators. For sufficiently regular fixed-end initial data, the Fourier sine series givesThe initial displacement and velocity determine $a_n$ and $b_n$. This is the superposition principle in a Fourier sine basis, and modal energy distribution measures how strongly each normal mode is excited. The striking simplification is that each component keeps its identity while their sum can continually change shape. ^[ref]

These independent normal modes belong to the linear model. At larger slopes, changing tension introduces nonlinear effects and can couple the modes. A linearly damped string still admits a Fourier sine basis, but its temporal factors decay and need not oscillate; its energy dissipation identity for a linearly damped string replaces conservation of energy. More general damping need not leave undamped mode shapes independent.

On the whole line, the homogeneous one-dimensional wave equation with constant wave speed $c>0$ isHere $u$ is displacement in the string interpretation; $f$ is initial displacement and $g$ is initial velocity. Its two directions of travel giveThis is D'Alembert's formula, including both Cauchy data. The countertravelling decomposition of the one-dimensional wave equation identifies $F(x-ct)$ as the right-moving travelling wave and $G(x+ct)$ as the left-moving travelling wave. Their sum is permitted by the superposition principle. For $f\in C^2$ and $g\in C^1$, it is a classical solution. More generally, travelling waves with profiles $F$ and $G$ that are locally integrable functions solve the equation as distributions, with derivatives interpreted distributionally.^[ref]

The synthetic diagram uses the smooth bump function $b$ defined in D'Alembert's formula, with $f(x)=b(x)$ and $c=1$. Releasing this displacement from rest, $g=0$, gives $F=G=f/2$. Each emerging travelling wave has half the original peak. Launching the identical initial displacement solely to the right instead requires $g=-c{f}^{\prime }$ and gives $F=f$, $G=0$. At $t=1.5$, the released profiles have peaks $1/2$ at $x=\pm 1.5$, while the launched profile has peak $1$ at $x=1.5$. These are calculated model values. A purely left-moving launch requires $g=c{f}^{\prime }$. The initial velocity therefore selects the balance of directions; the initial shape alone does not.