---
title: Travelling Waves
module: Oscillations and Waves
moduleNumber: 7
lessonNumber: 4
order: 704
summary: >
  A wave carries a shape, not the material: each element of a rope or air column
  oscillates in place while the disturbance travels through it. Writing that shape
  as $f(x\mp vt)$ turns "the pattern moves" into a statement about the cosine's
  argument, and a local force balance on one string segment fixes the speed at
  $v=\sqrt{T/\mu}$ — restoring stiffness over inertia, with amplitude nowhere in
  it. We build the sinusoidal wave and its phase, derive the wave equation from
  Newton's second law, and follow the energy a travelling wave transports, then
  check speed and power against those predictions.
topics: [Waves]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 15 — Waves; §15-1"
---

## Disturbances, sinusoidal waves, and phase

A wave is a travelling disturbance that transfers a pattern through a medium or
field. In a mechanical wave, individual material elements usually oscillate about
local equilibrium positions while the disturbance advances. The medium does not need
to travel with the pattern. A pulse on a rope moves along the rope even though each
short segment moves mainly up and down and then returns near its original position.

Displacement specifies the departure of one material element from its equilibrium
position. In a transverse wave, displacement is perpendicular to the direction in
which the disturbance travels. A rope wave has this geometry: the disturbance
moves along the rope while displacement is vertical. In a longitudinal wave,
displacement is parallel to propagation; compressions and rarefactions move through
a spring or gas while elements shift back and forth along the same line.

Amplitude is the largest magnitude of displacement from equilibrium. It is a signed
coordinate only before its magnitude is taken: an upward crest and downward trough
may have opposite displacement but equal amplitude. Amplitude is not the distance
travelled by the disturbance, and it does not by itself specify wave speed. Medium
properties and the type of wave determine speed; source motion determines how the
disturbance is launched.

$$
% caption: A transverse pulse travels along the rope toward increasing $x$ while each
% material element moves perpendicular to that direction. The amplitude $A$ is the
% greatest displacement from the equilibrium line, not a distance the pulse has covered
% along the rope.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->,black] (0,0)--(6.20,0) node[right] {rope at rest};
  \draw[acc,very thick] plot[domain=.40:5.60,samples=180]
    (\x,{1.05*exp(-4*(\x-2.10)^2)});
  \draw[<->,black] (2.10,0.06)--(2.10,1.00);
  \node[black,right] at (2.16,0.55) {amplitude};
  \draw[fill=white,draw=black,line width=.4pt] (3.45,0) circle (1.8pt);
  \draw[->,black,thick] (3.45,0.14)--(3.45,0.78);
  \node[black,above] at (3.45,0.80) {element};
  \draw[->,acc,thick] (4.35,0.42)--(5.30,0.42);
  \node[acc,above] at (4.82,0.42) {pulse};
\end{tikzpicture}
$$

### Sinusoidal records, wavelength, and phase

A periodic travelling wave has two independent repeated intervals. A spatial
snapshot gives the wavelength $\lambda$, measured between adjacent crests, adjacent
troughs, or any two points at the same phase. A time record at one marked location
gives the period $T$, measured between repeated states of the same oscillator. The
frequency is $f=1/T$. Matching a crest in one snapshot with the corresponding crest
after a later time interval gives the wave speed directly; matching all crests shows
that the whole pattern translates when the medium is nondispersive.

For sinusoidal motion, the repeated intervals connect to the angular quantities through

$$
k=\frac{2\pi}{\lambda},
\qquad
\omega=2\pi f,
\qquad
v=\frac{\omega}{k}=\lambda f.
$$

The last relation joins a measured spatial period and a measured temporal period for
one coherent travelling wave. It does not say that an arbitrary frequency determines
speed. On a uniform, nondispersive string, the medium fixes $v$; a higher driving
frequency produces a shorter wavelength. In water waves, guided waves, and material
media with dispersion, different frequency components can have different phase
speeds, so a single $\lambda f$ value applies only to the component whose wavelength
and frequency were measured together.

$$
% caption: A spatial snapshot of a sinusoidal wave. The wavelength $\lambda$ is the
% distance between two successive crests, equivalently two successive troughs; a
% crest-to-trough distance is half a wavelength, which guards against a factor-of-two
% reading error.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->,black] (.2,0)--(6.30,0) node[right] {position};
  \draw[->,black] (.4,-1.35)--(.4,1.75) node[above] {displacement};
  \draw[acc,very thick] plot[domain=.55:5.95,samples=220]
    (\x,{1.15*sin(150*(\x-.55))});
  \draw[fill=white,draw=black,line width=.4pt] (1.15,1.15) circle (1.6pt);
  \draw[fill=white,draw=black,line width=.4pt] (3.55,1.15) circle (1.6pt);
  \draw[<->,black] (1.15,1.48)--(3.55,1.48);
  \node[black,above] at (2.35,1.44) {one wavelength};
\end{tikzpicture}
$$

A phase difference is a fraction of a full cycle. Two marked elements separated by
$\Delta x$ in a sinusoidal snapshot have phase difference
$\Delta\phi=k\Delta x$, subject to adding any integer multiple of $2\pi$. At one
location, a time delay $\Delta t$ corresponds to
$\Delta\phi=\omega\Delta t$. A half-cycle delay reverses displacement; a
quarter-cycle delay places one element at maximum speed when the other reaches
maximum displacement. These statements require a stated propagation direction and
reference phase. The phrase “out of phase” needs a numerical phase difference or a
clear geometric condition to be testable.

An oscilloscope or video trace can determine phase from zero crossings as well as
from peaks. A rising zero crossing and a falling zero crossing are separated by half
a period and therefore have opposite phase slopes. Marking the crossing direction
prevents a half-period ambiguity. With noisy data, fit a sinusoid over several cycles
or correlate a reference waveform with the measured record. The fit should report
frequency, phase convention, fitting interval, and residual scale rather than only a
rounded phase angle.

$$
% caption: A time record at one fixed material element. The period $T$ is the interval
% between successive states of the same phase, here two upward zero crossings; the
% frequency is $f=1/T$. Crossing direction, not the zero reading alone, fixes the phase.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->,black] (.2,0)--(6.30,0) node[right] {time};
  \draw[->,black] (.55,-1.55)--(.55,1.75) node[above] {displacement};
  \draw[acc,very thick] plot[domain=.70:5.95,samples=220]
    (\x,{1.15*sin(150*(\x-.70))});
  \draw[fill=white,draw=black,line width=.4pt] (.70,0) circle (1.5pt);
  \draw[fill=white,draw=black,line width=.4pt] (3.10,0) circle (1.5pt);
  \draw[<->,black] (.70,-1.20)--(3.10,-1.20);
  \node[black,below] at (1.90,-1.16) {one period};
\end{tikzpicture}
$$

### Comparing phase at separated sensors

Two sensors along a string or air column report the same travelling pattern at
different times. In a monochromatic wave, the measured delay between matching phase
points is $\tau=\Delta x/v$. A sensor separation equal to an integer number of
wavelengths produces records in phase; a separation of one half wavelength produces
opposite displacements at the same clock time. Intermediate separations provide a
phase estimate but require a known branch: a measured phase of $30^\circ$ can also
represent $390^\circ$, $750^\circ$, and further full-cycle additions over a long
baseline.

Use either a short baseline, where the branch is known from geometry, or several
separations fitted to a straight phase-versus-distance trend. The slope gives
$k$, and a separate time record gives $\omega$. Dividing then gives phase speed.
Phase fitting remains applicable when a sharp pulse is unavailable, but it is vulnerable to
reflections. A reflected component changes the local phase and can make a single
sensor pair report a speed that varies with position. Amplitude minima and abrupt
phase changes in a spatial scan signal that the pure travelling-wave model has broken
down.

Wave speed is measured from the motion of a reproducible feature, such as a pulse
maximum, a sharp leading edge, or a specified phase point. If that feature moves a
distance over a measured time interval, its average speed is distance divided by
elapsed time. The observation must identify the same feature at both times. A
broadening pulse can give different speeds for its peak, its leading edge, and its
energy envelope in a dispersive medium; this first treatment assumes the disturbance
retains a recognizable shape over the measurement interval.

### Wave functions, phase, and propagation direction

A sinusoidal travelling wave can be written as

$$
y(x,t)=A\cos(kx-\omega t+\phi).
$$

The argument of the cosine is the phase. Points with equal phase have equal
displacement and sit at the same stage of their local oscillation. Holding the
phase constant, $kx-\omega t=\text{const}$, gives $x=(\omega/k)t+\text{const}$:
the feature moves toward increasing $x$ for the minus sign before $\omega t$.
Replacing that sign by a plus sends the feature toward decreasing $x$. The sign
test beats reading direction from a single snapshot, which carries no time
information.

The wave number $k$ measures phase change per unit distance, in radians per metre;
the angular frequency $\omega$ measures phase change per unit time, in radians per
second. The wavelength $\lambda$ is the spatial separation of equal-phase points,
and the period $T$ is the time between repeated equal-phase states at one position,
so $k\lambda=2\pi$ and $\omega T=2\pi$. These separate the local motion of material
elements from the translation of a constant-phase feature.

$$
% caption: World lines of constant phase in the position-time plane. Each line holds
% $kx-\omega t$ fixed; the common positive slope $\d x/\d t=\omega/k$ marks rightward
% propagation, which a single displacement snapshot cannot supply.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->,black] (0,0)--(5.90,0) node[right] {position};
  \draw[->,black] (0,0)--(0,3.30) node[above] {time};
  \draw[thick] (0.50,0.30)--(2.90,2.90);
  \draw[thick] (1.70,0.30)--(4.10,2.90);
  \draw[thick] (2.90,0.30)--(5.30,2.90);
  \node[right] at (4.45,0.70) {constant phase};
  \draw[->,acc,thick] (1.55,1.05)--(2.15,1.70);
  \node[acc,left] at (1.55,1.45) {speed};
\end{tikzpicture}
$$

The phase constant $\phi$ sets the chosen origin of the wave cycle. Changing $\phi$
shifts the graph in space or time without touching amplitude, wavelength, frequency,
or speed. Comparing two measurements needs a consistent phase convention, but an
isolated phase value has no preferred physical zero. The speed of a constant-phase
feature is $\omega/k$ when the medium is nondispersive over the frequency range
considered.

## Wave speed and the string equation

For small transverse disturbances on a taut uniform string, wave speed is set by
tension and linear mass density:

$$
v=\sqrt{\frac{T}{\mu}}.
$$

Greater tension increases the restoring force on a displaced segment and therefore
increases speed. Greater mass per unit length increases inertia and decreases speed.
Amplitude does not appear in this ideal small-displacement result. A large-amplitude
wave can alter local tension and require a nonlinear model.

The corresponding one-dimensional wave equation has the form

$$
\frac{\partial^2 y}{\partial t^2}
=
v^2\frac{\partial^2 y}{\partial x^2}.
$$

Its speed parameter is fixed by the medium. A source may change frequency, but in a
nondispersive string that change alters wavelength, not speed. The relation
$v=\omega/k$ holds for a single sinusoidal solution and within a narrow frequency
range where speed is independent of frequency. It is not a universal speed rule in a
dispersive medium.

A time-of-flight measurement checks the speed directly.

> **Worked example.** A pulse maximum travels $\Delta x=3.60\ \mathrm m$ between two
> marked positions in $\Delta t=0.180\ \mathrm s$. The measured speed is
>
> $$
> v_{\rm meas}=\frac{\Delta x}{\Delta t}
> =\frac{3.60\ \mathrm m}{0.180\ \mathrm s}
> =20.0\ \mathrm{m\,s^{-1}}.
> $$
>
> The two marks must refer to the same pulse feature, and the interval must be long
> enough to reduce trigger uncertainty without allowing substantial pulse distortion.
> Comparing this speed with $\sqrt{T/\mu}$ from measured tension and linear density
> tests the uniform-string and small-amplitude assumptions.


A travelling string wave also transports mechanical energy. Each segment alternates
between kinetic energy from transverse velocity and elastic energy from its local
slope, while the average energy flow is along the string. A sinusoidal wave on a
uniform string has average power that increases with the square of amplitude and the square
of angular frequency when tension and linear density are fixed. Doubling amplitude
therefore requires four times the average power in the ideal model.

This scaling gives a practical measurement check. If a driver transfers 0.80 watts
to a string at one amplitude, an otherwise unchanged run at twice that amplitude
should require about 3.2 watts. A different ratio can indicate source calibration
error, damping, nonlinear tension change, or a departure from the small-amplitude
string model. Power is a rate of energy transport; it is not the transverse velocity
of one material segment.


In a longitudinal pulse, particle displacement is parallel to the direction of
propagation. Gas or spring elements move back and forth about equilibrium while
compressions and rarefactions advance through the medium. The wave speed describes
the motion of that compression pattern, not the speed of one material element.

A microphone records local pressure variation, which follows compression and
rarefaction rather than particle position itself. Two microphones at known separation
can provide a time-of-flight speed measurement when the same pressure feature is
identified in both records. Sensor phase delay, tube reflections, temperature, and
gas composition contribute to the uncertainty. The record therefore measures a local
field disturbance, not a trajectory of individual molecules.

### Local string element and the wave equation

The string-speed relation follows from Newton's second law applied to a short curved
piece of string. Let the undisturbed string lie along the $x$-axis, with tension
$T$ nearly constant and linear mass density $\mu$. A segment from $x$ to
$x+\Delta x$ has mass $\mu\Delta x$. The tension at each end acts along the local
tangent. Its horizontal components nearly cancel for a gently sloped wave, whereas
the difference of the vertical components is the transverse force.

Write the local string angle as $\theta$. At small slope,
$\sin\theta\simeq\tan\theta\simeq \partial y/\partial x$. The vertical force on
the segment is consequently

$$
F_y
=T\left[
\left.\frac{\partial y}{\partial x}\right|_{x+\Delta x}
-\left.\frac{\partial y}{\partial x}\right|_{x}
\right]
\simeq T\frac{\partial^2y}{\partial x^2}\Delta x.
$$

Newton's second law gives
$\mu\Delta x\,\partial^2y/\partial t^2=F_y$. Cancelling the chosen segment length
produces the wave equation

$$
\frac{\partial^2y}{\partial t^2}
=\frac{T}{\mu}\frac{\partial^2y}{\partial x^2}.
$$

Comparison with the standard one-dimensional form identifies
$v=\sqrt{T/\mu}$. The derivation uses a local force balance; it does not assume a
sinusoidal shape. A narrow pulse, a broad crest, and a superposition of small pulses
share the same speed on a uniform string when their slopes remain small and the
tension stays approximately uniform.

$$
% caption: A short curved string element. The two end tensions are equal in magnitude
% and directed along the local tangents; their horizontal parts nearly cancel while the
% vertical difference, set by the change of slope across $\d x$, drives the transverse
% acceleration $a_y$.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[acc,very thick] (.45,.15) .. controls (2.10,.90) and (3.75,.90) .. (5.75,.10);
  \draw[fill=white,draw=black,line width=.4pt] (2.20,.56) circle (1.5pt);
  \draw[fill=white,draw=black,line width=.4pt] (3.85,.58) circle (1.5pt);
  \draw[->,black,thick] (2.20,.56)--(1.15,.16);
  \node[black,below] at (1.05,.06) {$T$};
  \draw[->,black,thick] (3.85,.58)--(4.95,.18);
  \node[black,below] at (5.05,.08) {$T$};
  \draw[<->,black] (2.20,-.55)--(3.85,-.55);
  \node[black,below] at (3.02,-.51) {element};
  \draw[->,acc,thick] (3.02,.78)--(3.02,.04);
  \node[acc,right] at (3.12,.42) {$a_y$};
\end{tikzpicture}
$$

Curvature rather than displacement controls the immediate transverse acceleration.
At the top of a smooth crest, displacement is large while the curvature points
downward, so the segment accelerates toward equilibrium. At an inflection point,
displacement can be nonzero while curvature vanishes and the segment has zero
instantaneous transverse acceleration. This distinction explains why a drawn wave
profile cannot be read as a force diagram by comparing height alone.

The small-slope condition limits the formula. If a string segment turns through a
large angle, the horizontal tension components change enough to alter the local
tension, and the approximation $\sin\theta\simeq\tan\theta$ fails. The pulse can
steepen, its speed can depend on amplitude, and different portions of the profile can
travel at different rates. A laboratory string also loses energy through air drag,
internal friction, and a driver that does not hold tension perfectly constant. Those
effects alter amplitude and may broaden a pulse without changing the local definition
of speed.

$$
% caption: Curvature, not displacement, sets the instantaneous transverse acceleration.
% At the crest the profile curves downward, so $a_y$ points toward equilibrium; at the
% inflection point the curvature vanishes and $a_y=0$ even though the displacement there
% is nonzero.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->,black] (.2,0)--(6.05,0) node[right] {$x$};
  \draw[->,black] (.45,-1.30)--(.45,1.95) node[above] {$y$};
  \draw[acc,very thick] plot[domain=.55:5.50,samples=200]
    (\x,{1.45*cos(60*(\x-2.45))});
  \draw[fill=white,draw=black,line width=.4pt] (2.45,1.45) circle (1.6pt);
  \draw[->,acc,thick] (2.45,1.33)--(2.45,.70);
  \node[acc,right] at (2.52,1.02) {$a_y$};
  \node[black,above] at (2.45,1.60) {crest};
  \draw[fill=white,draw=black,line width=.4pt] (3.95,0) circle (1.6pt);
  \draw[black,thin] (3.98,.06)--(4.55,.50);
  \node[black,right] at (4.50,.52) {zero curvature};
\end{tikzpicture}
$$

## Pulse-speed measurement

A speed experiment needs a distance reference, a feature definition, and a timing
rule. Place two optical gates, motion sensors, or video markers a measured distance
$\Delta x$ apart. Record the times at which the same crest or sharp pulse maximum
crosses the two positions. The estimate is

$$
v_{\rm meas}=\frac{\Delta x}{\Delta t}.
$$

The feature choice becomes important when a pulse deforms. A maximum can shift
relative to a leading edge after dispersion or reflection. State whether the analysis
uses peak time, threshold crossing, cross-correlation of the full records, or a
phase marker. Cross-correlation can use more of a noisy waveform, but it assumes that
the record at the second sensor is a delayed version of the first. A reflected pulse
or amplitude-dependent distortion violates that assumption.

For independent distance and timing uncertainties, a first relative estimate is

$$
\frac{\delta v}{v}
\simeq
\sqrt{\left(\frac{\delta(\Delta x)}{\Delta x}\right)^2+
\left(\frac{\delta(\Delta t)}{\Delta t}\right)^2}.
$$

Increasing sensor separation reduces the fractional timing contribution, provided the
pulse remains identifiable. Repeating runs at several tensions tests the predicted
relation $v^2=T/\mu$. A graph of measured $v^2$ against tension should be linear
through the origin within uncertainty for a uniform string. A nonzero intercept can
indicate tension calibration error, a mass-density error, or a systematic delay in the
timing electronics.

## Energy and power on a string

A travelling disturbance transfers energy because each small string element has
transverse kinetic energy and elastic energy associated with its slope. On a string
with linear density $\mu$ and tension $T$, the energy in a short element $\d x$ is

$$
\d E=
\left[
\frac12\mu\left(\frac{\partial y}{\partial t}\right)^2+
\frac12T\left(\frac{\partial y}{\partial x}\right)^2
\right]\d x.
$$

The first term is kinetic energy per unit length. The second follows from the small
increase in string length caused by a transverse slope. It is a local elastic-energy
term, valid when slopes are small and tension is approximately constant. A point at a
crest can have zero instantaneous transverse speed and therefore zero kinetic-energy
density, while its slope can also vanish and give zero elastic-energy density. Energy
need not be concentrated where the plotted displacement is largest.

A general profile need not have equal kinetic and elastic energy densities at one
position and time. Their sum is the conserved quantity for an ideal uniform
string. In a real string, internal friction and air drag convert some of this
mechanical energy to heat. A driver can replace that loss, maintaining a nearly steady
amplitude only when its average input balances the average loss and outgoing power.

### Instantaneous power flow

Energy crosses a fixed position because the tension force on the string element does
work as that boundary moves transversely. With positive power defined as transfer in
the positive $x$ direction, the instantaneous power is

$$
P(x,t)=-T
\frac{\partial y}{\partial x}
\frac{\partial y}{\partial t}.
$$

The sign contains physical information. A positive value means energy is crossing the
chosen position toward increasing $x$; a negative value means the transfer is toward
decreasing $x$. The result does not require a sinusoidal wave. It follows from the
tension force at the boundary and the boundary's transverse velocity.

Combining this power expression with the wave equation gives the one-dimensional
energy-balance relation

$$
\frac{\partial u}{\partial t}+\frac{\partial P}{\partial x}=0,
\qquad
u=
\frac12\mu y_t^2+\frac12T y_x^2.
$$

Energy density inside a short interval rises when more power enters than leaves and
falls when the outgoing power is larger. The local balance tracks both the stored
energy and its direction of transport, which a single total-energy value cannot.

### Average power of a sinusoidal wave

A right-moving sinusoidal wave has

$$
y(x,t)=A\cos(kx-\omega t),
\qquad
v=\frac{\omega}{k}=\sqrt{\frac{T}{\mu}},
$$

the instantaneous power becomes

$$
P(x,t)=T k\omega A^2\sin^2(kx-\omega t).
$$

It is nonnegative for this propagation direction. Using $Tk^2=\mu\omega^2$ gives the
cycle average

$$
\langle P\rangle
=\frac12\mu v\omega^2A^2
=\frac12T k\omega A^2.
$$

Average power grows with the square of amplitude and frequency. Doubling amplitude
quadruples the required mean power. At fixed frequency and amplitude, increasing
tension changes both wave speed and the relation between wavelength and frequency;
the form using $\mu v\omega^2$ keeps the measured speed explicit.

The instantaneous power oscillates because energy density is transported
nonuniformly through the profile. Its average over one period is the quantity measured
by a calorimetric absorber or by a driver operating for many cycles. Reporting only a
peak displacement does not determine average power unless $\omega$, $\mu$, and the
propagation speed are also known.

$$
% caption: Mean power of a sinusoidal string wave against amplitude squared, at fixed
% tension, linear density, and frequency. The relation is linear through the origin with
% slope $\tfrac12\mu v\omega^2$; scatter off the line signals amplitude calibration
% error or loss.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->,black] (0,0)--(5.50,0) node[right] {amplitude squared};
  \draw[->,black] (0,0)--(0,3.10) node[above] {mean power};
  \draw[acc,very thick] (.35,.20)--(4.95,2.70);
  \foreach \x/\y in {1.05/.58,2.05/1.12,3.05/1.66,4.05/2.20} {
    \draw[fill=acc!12,draw=acc,line width=.4pt] (\x,\y) circle (1.9pt);
  }
  \node[acc,right] at (2.95,2.55) {linear f\/it};
\end{tikzpicture}
$$

### Experimental checks on power transfer

Measure string tension, linear density, frequency, and amplitude independently before
using the sinusoidal power formula. A load cell gives tension, a measured length and
mass give $\mu$, and a motion sensor or calibrated video trace gives frequency and
amplitude. The amplitude used in the formula is the travelling-wave amplitude at the
measurement location; a changing envelope or driver-region motion can give a
different value.

A direct mechanical check uses the local relation $P=-Ty_xy_t$. Estimate slope from
two nearby displacement measurements and transverse velocity from a time derivative,
then average their product over many cycles. The spacing must be short enough to
resolve the local slope but large enough that displacement noise does not dominate
the difference quotient. Comparing this result with
$\mu v\omega^2A^2/2$ tests the small-slope model and the amplitude calibration.

The driver electrical input is usually larger than the power carried by the string.
Losses in the driver, support, and surrounding air account for part of the difference.
A power estimate should therefore state the measurement location, averaging time,
tension, density, and whether the reported quantity is source input or mechanical
power in the string.

The energy-density expression also gives an internal check for a sinusoidal
wave. The kinetic part is largest where the string crosses equilibrium, because
transverse speed is largest there. The elastic part is largest at the same positions,
because the local slope is largest there. At a crest or trough both terms vanish in
the ideal sinusoidal model. Energy is carried through those positions by the changing
power flow from neighboring elements; zero instantaneous density at one point does
not imply zero average power along the string.

A single right-moving sinusoid has kinetic and elastic energy densities that are
equal at every position and time under the small-slope string model. This equality
uses the wave-speed relation between tension and linear density. It is not a general
property of every mechanical disturbance. A pulse with an arbitrary shape can have
locally unequal contributions while conserving their sum and transferring energy
according to the same power-balance equation. Measuring both slope and transverse
velocity is therefore a more general test than assuming a sinusoidal waveform.

Amplitude uncertainty enters the average-power result twice in fractional form
because power is proportional to amplitude squared. A five-percent amplitude
calibration uncertainty alone produces roughly a ten-percent power uncertainty before
tension, density, frequency, and speed uncertainties are included. A camera estimate
can also underestimate amplitude when exposure time averages rapid transverse motion.
Use a shutter interval short compared with the period, or fit a time-resolved position
record. Tension should be measured while the driver is operating, since heating or a
moving support can change it during a run.

Power balance is a further experimental check. Measure the mechanical power
entering a dissipative termination or compare the decrease of a calibrated driver's
energy store with the string power over a stated interval. The comparison must use an
interval containing many periods; an instantaneous force or displacement measurement
has phase dependence and does not equal the cycle average. Agreement within
uncertainty supports the assumed uniform tension and small-slope model. A systematic
shortfall in downstream power indicates distributed losses or an amplitude that
changes along the measured length.

| reported quantity | independent measurement | reduction used | result that calls for a check |
|---|---|---|---|
| tension $T$ | in-line load cell while the driver runs | $v=\sqrt{T/\mu}$ | Speed inferred from arrival times changes while the load-cell value is stable. |
| linear density $\mu$ | mass and active length of the same string segment | $\mu=m/L$ | A fitted value differs systematically between segments or after a change of string. |
| local amplitude $A$ | time-resolved displacement fit at the stated position | Travelling-wave sinusoid, with peak or rms convention recorded | A standing-wave envelope or driver displacement is substituted for the local travelling amplitude. |
| mean string power | synchronized local slope and transverse-velocity records | $\langle P\rangle=\langle-Ty_xy_t\rangle$ | The local result and $\mu v\omega^2A^2/2$ disagree beyond the propagated calibration uncertainty. |
| downstream energy loss | absorber heating or a calibrated energy-store change over many periods | Compare energy change divided by the common interval with mean power | The difference grows with distance, indicating distributed loss or an amplitude gradient. |

## Longitudinal pulses and material limits

In a longitudinal mechanical wave, material displacement is parallel to the direction
of travel. A local compression raises pressure or axial stress; a local expansion lowers
it. The restoring response of the medium passes that stress change to neighbouring
material, while inertia limits how rapidly each element accelerates. Wave speed is set
by the ratio of restoring stiffness to inertia, not by the amplitude of a sufficiently
small pulse. A larger stiffness raises the speed; greater mass density lowers it.

In a fluid, the relevant stiffness is the bulk modulus,

$$
B=-V\frac{\d p}{\d V}.
$$

The negative sign records that an increase in pressure decreases volume. Small
longitudinal pressure disturbances travel at

$$
c=\sqrt{\frac{B}{\rho}},
$$

where $\rho$ is equilibrium mass density. The modulus in this relation must match the
thermodynamic process of the disturbance. Sound in a gas commonly changes pressure too
quickly for substantial heat transfer, so an adiabatic bulk modulus is appropriate;
using an isothermal compressibility then predicts the wrong speed. An ideal gas has
the adiabatic result $c=\sqrt{\gamma p/\rho}$.

$$
% caption: A longitudinal pulse in a compressible column. Elements oscillate along the
% travel direction, bunching into a compression and thinning into a rarefaction; the
% pulse speed is the speed of that pattern, fixed by stiffness over density.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
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\useasboundingbox (-0.30,-0.25) rectangle (6.65,3.05);
\draw[black,thick] (.50,.70) rectangle (6.10,1.85);
\foreach \x in {0.95,1.12,1.29,1.46,1.63,1.80,1.97,2.14,3.25,3.75,4.25,4.75,5.25,5.72} {
  \draw[fill=acc!14,draw=acc,line width=.35pt] (\x,1.27) circle (.055);
}
\draw[->,black,very thick] (.85,2.35)--(2.15,2.35);
\node[black,above] at (1.50,2.35) {pulse};
\node[black,below] at (1.55,.70) {compression};
\node[black,below] at (4.30,.70) {rarefaction};
\end{tikzpicture}
$$

Several elastic responses are possible in solids. In a large,
isotropic solid, a compressional wave changes both volume and shape, giving the
longitudinal speed

$$
c_L=\sqrt{\frac{K+4G/3}{\rho}},
$$

where $K$ is bulk modulus and $G$ is shear modulus. A long slender bar with sides
free to move instead has the one-dimensional approximation
$c=\sqrt{E/\rho}$, with $E$ the Young modulus. The formulas do not conflict: they
refer to different deformation constraints. Substituting a bar’s Young modulus into a
three-dimensional bulk-wave calculation, or using a fluid bulk modulus for a solid
bar, silently changes the physical model.

Pulse measurements determine speed from distance and arrival time rather than from a
sketch of a compressed region. Two receivers at known positions $x_1$ and $x_2$ give
$c=(x_2-x_1)/(t_2-t_1)$ when they track the same recognizable pulse feature. A sharp
leading edge, a correlation peak, or a fitted pulse centre can supply that feature,
but the choice must be used consistently at both receivers. Trigger delay cancels in a
receiver-to-receiver difference only if the timing paths are stable. A single source
and one receiver requires an independently calibrated launch time and is more exposed
to electronic delay.

The main uncertainty is often not the time digitizer. Receiver spacing must be measured
along the actual propagation path, including a curved tube or bar, and the pulse must
remain recognizable over that path. A broad pulse gives an arrival-time ambiguity that
should be estimated from its rise time and signal-to-noise ratio. Repeating the
measurement over several spacings strengthens the test: a graph of arrival time
against distance should be linear if one speed describes the medium and the selected
feature is stable. A nonzero fitted intercept indicates a fixed electronic delay
without changing the slope-based speed.

Acoustic impedance, defined for small plane longitudinal waves as $Z=\rho c$, limits
how readily a pulse can carry stress and particle motion across a material join. Speed
alone is insufficient: two media can have similar speeds but very different densities
and therefore different impedance. A large impedance mismatch restricts the energy
that reaches the downstream medium and can leave a weak transmitted pulse even with a
well-calibrated source. The boundary condition is local stress and particle-motion
continuity, not an arbitrary matching of pulse heights. Coupling layers, surface gaps,
and imperfect contact introduce additional losses beyond the material impedance.

Transmission measurements need a stated boundary and detector convention. A probe may
respond to pressure, displacement, particle speed, or an electrical signal proportional
to one of them; ratios of those signals are not automatically energy ratios. Use a
reference run in a uniform sample to establish source repeatability and probe gain,
then compare the joined sample under the same drive, geometry, and bandwidth. Keep the
excitation small enough that the modulus and speed remain in the linear regime. Large
pulses can change temperature, contact conditions, or local material response and make
a small-amplitude impedance model inapplicable.

> **Worked example.** Two probes in a rod are separated by $\Delta x=0.800\ \mathrm m$,
> and their common pulse centres differ by $\Delta t=0.160\ \mathrm{ms}$. The speed is
>
> $$
> c=\frac{\Delta x}{\Delta t}
> =\frac{0.800\ \mathrm m}{0.160\times10^{-3}\ \mathrm s}
> =5.00\times10^3\ \mathrm{m\,s^{-1}}.
> $$
>
> With a spacing uncertainty of $1\ \mathrm{mm}$ ($0.13\%$) and a timing uncertainty
> of $2\ \mathrm{\mu s}$ ($1.3\%$), the timing term dominates the fractional
> uncertainty. Reporting the pulse feature, receiver separation, fitted time offset,
> and material temperature makes this value testable against $\sqrt{E/\rho}$ or the
> appropriate bulk-wave model.

The density in a speed formula is the density of the medium carrying the pulse under
the conditions of the measurement. Temperature, composition, and static pressure can
change density and stiffness together. In gases, treating pressure as a separate
control while silently using a density measured at another temperature is inconsistent.
In liquids, dissolved gas or small bubbles can greatly reduce the effective bulk
stiffness and produce a speed far below that of the pure liquid. A measured speed that
disagrees with a tabulated value should therefore prompt checks of temperature,
composition, pressure, and entrained gas before invoking an unfamiliar wave model.

The adiabatic assumption for a gas has a time-scale condition. During a fast acoustic
compression, neighbouring regions exchange too little heat to maintain one common
temperature, so the pressure response follows an adiabatic relation. At very low
frequency or in a geometry with strong thermal contact, heat exchange can move the
response toward an isothermal value. The measured speed then lies closer to the
stiffness appropriate to that process. The distinction is not a correction appended to
the formula after measurement; it determines which modulus belongs in the formula in
the first place.

In a solid bar, end loading and sensor mounting can alter the apparent pulse shape
without changing the bulk material speed. A thick adhesive layer, a loosely clamped
probe, or a finite source contact introduces a local delay and can spread the leading
edge. Differential timing with two probes reduces source delay, but it does not remove
a different coupling delay at each probe. Interchanging probe locations or measuring
several separations helps isolate a fixed hardware offset from a propagation-time
slope. The same procedure also exposes a weakly dispersive situation in which the
chosen feature changes as the pulse travels.

Receiver bandwidth sets another limit. A narrow-band probe rounds a sharp pulse and
can shift the apparent time of its peak relative to the leading edge. Cross-correlation
of two full time records can be less sensitive to this choice than threshold crossing,
provided both channels have matched filtering and the correlation peak is well defined.
The arrival-time uncertainty should include sampling interval, trigger jitter, noise,
and any systematic timing offset. Quoting only the instrument’s nominal clock
resolution overstates precision when the pulse rise time is much longer.

Impedance combines the two material properties that govern a
plane longitudinal wave: stress amplitude is proportional to impedance times particle
speed. A source coupled to a high-impedance sample can need more stress to produce the
same particle motion than it would in a low-impedance sample. At a material join, the
downstream probe signal depends on the impedance pair, contact layer, beam area, and
probe sensitivity. A small transmitted electrical signal may represent a true small
mechanical pulse, a sensor with low response in that material, or both. Calibration on
each side of the join is required before interpreting signal amplitude as transmitted
mechanical energy.

Transmission limits should be separated from distributed attenuation. A join can
reduce a pulse immediately at the boundary, whereas attenuation gradually reduces it
with propagation distance through material loss, scattering from inhomogeneity, or
geometrical spreading. Receiver measurements at several positions in a uniform region
distinguish a gradual distance trend from a change associated with the join. The model
must also state whether the pulse is approximately plane over the probe area. A small
point source in a three-dimensional body spreads geometrically, so amplitude changes
cannot be assigned to impedance alone.

For independent distance and timing uncertainties, the fractional uncertainties in a
two-probe speed result combine approximately as

$$
\left(\frac{\sigma_c}{c}\right)^2\simeq
\left(\frac{\sigma_{\Delta x}}{\Delta x}\right)^2+
\left(\frac{\sigma_{\Delta t}}{\Delta t}\right)^2.
$$

Increasing probe separation reduces the fractional effect of a fixed timing uncertainty, but too great a
separation can broaden or attenuate the pulse. An appropriate spacing balances those two
limits. Repeated measurements at each spacing estimate random scatter, while an
independent distance calibration and a delay-intercept fit address systematic error.

## Model checks and reproducible measurement

Every wave-speed formula contains a model choice. The fluid expression
$c=\sqrt{B/\rho}$ assumes a small longitudinal disturbance in a medium that can be
represented by one equilibrium density and one bulk modulus for the relevant pressure
process. The bar expression $c=\sqrt{E/\rho}$ assumes primarily axial strain in a
slender member with sides free enough to follow that deformation. Neither formula
licenses a speed calculation from density alone. State the geometry, static loading,
temperature, and process condition that justify the selected stiffness before comparing
a measured speed with a prediction.

Dimensional analysis catches many setup errors. A modulus has units of pressure,
which are $\mathrm{kg\,m^{-1}\,s^{-2}}$. Dividing by density gives
$\mathrm{m^2\,s^{-2}}$, and the square root gives speed. If a calculation returns
metres per second squared, a density has been omitted or an extra time factor has
entered. The same check applies to the timing estimate: distance divided by a time
difference has speed units, whereas distance divided by a clock reading with an
unremoved trigger delay does not necessarily represent propagation speed.

Limiting checks test the physical direction of a result. At fixed density, a larger
bulk or Young modulus must give a larger predicted longitudinal speed. At fixed
stiffness, adding mass density must lower it. If $B$ or $E$ is allowed to approach
zero, the model predicts a vanishing speed because there is no restoring response. If
stiffness is allowed to grow without bound, the predicted speed grows without bound;
real materials cease to satisfy the simple elastic model before that mathematical limit
becomes a practical design option. These checks do not prove a formula, but a result
that fails them cannot be correct.

Source frequency and propagation speed are separate measurements. A driver can repeat
similar pulses at a chosen repetition rate, or it can provide a periodic mechanical
drive at a chosen frequency. That source setting controls how often the disturbance is
launched; it does not set the speed selected by medium stiffness and density. Once a
speed has been measured for a nearly single-frequency drive, the associated wavelength
can be found from $\lambda=c/f$. Reversing the relation and treating a chosen source
frequency as evidence for a material speed is circular unless wavelength has been
measured independently in the same medium.

Frequency changes can still alter a practical measurement without changing the
intended material speed. A higher-frequency drive may give a shorter pulse feature,
improve timing discrimination, or exceed the bandwidth of a source, probe, adhesive
layer, or digitizer. Pulse shape may then change enough that a peak at one receiver is
not the same physical marker as a peak at the other. The remedy is to define the
arrival feature from the recorded waveform, verify that it remains stable over the
probe separation, and report the drive bandwidth along with the timing method.

A complete procedure begins by conditioning the sample and recording its temperature,
dimensions, composition, static stress, and mounting. Measure the probe separation
along the propagation path with an uncertainty traceable to the geometry. Establish a
source level low enough to keep the response linear, then acquire a baseline record
with no launched pulse to quantify electrical noise and timing jitter. A reference
measurement over a short known path checks the sign convention and verifies that the
two timing channels have stable relative delay.

Next, launch repeated pulses and store the full waveform at both probes rather than
only a threshold-crossing time. Select an arrival marker before examining the speed
result: a rising-edge fraction, a fitted pulse centre, or a correlation maximum are
common defensible choices. Apply the same marker rule to every record. Compute
$\Delta t=t_2-t_1$ for every accepted pair, reject records only by written quality
criteria, and summarize both the mean and scatter. A record with saturated electronics,
an ambiguous marker, or a changed source amplitude should be flagged rather than
silently adjusted to agree with the mean.

Repeat at three or more probe spacings when the apparatus permits it. Fit arrival time
against distance; the slope gives inverse speed and the intercept estimates a common
time offset. Inspect residuals rather than relying only on the fitted slope. Curvature
or a spacing-dependent change in pulse shape indicates that one constant-speed model
does not describe the chosen marker over the full path. In that case, report the range
over which the local estimate is valid instead of averaging incompatible records.

The final report should contain the material model, stiffness and density values with
their conditions, source waveform and bandwidth, probe locations, arrival-marker rule,
timing calibration, fitted speed, and separate random and systematic uncertainties.
Joined samples require the coupling description and a statement of whether the downstream
signal is used only for timing or also for an amplitude comparison. This documentation
permits independent reproduction and separates travel-time evidence from the
elastic-material model.