Wave Mechanics: Translating Disturbances
WaveMechanicsTranslatingDisturbances
2026-09-11 18:00:36
2026-09-11 18:00:36
Definition
wave mechanics
traveling disturbance
translating pulse
wave propagation
traveling wave
shape-preserving translation
wave speed
right-moving wave
left-moving wave
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\section*{Wave Mechanics: Translating Disturbances}
WM04 and WM05 described patterns that vary with position. In those lessons,
we could draw a spatial profile such as $u(x)$, but nothing in the mathematics
said that the profile moved. WM06 adds the missing ingredient: time.
The central question is simple:
\begin{quote}
How do we write a mathematical function whose shape moves through space
without changing form?
\end{quote}
The answer is one of the most useful constructions in all of wave mechanics:
\begin{equation}
\boxed{u(x,t)=F(x-ct)}
\end{equation}
for motion toward increasing $x$, and
\begin{equation}
\boxed{u(x,t)=F(x+ct)}
\end{equation}
for motion toward decreasing $x$.
These forms describe a translated disturbance of arbitrary shape. The
function $F$ need not be sinusoidal. It may represent a pulse, a localized
bump, or any other profile that preserves its shape while translating. This
construction is standard in introductory treatments of traveling waves
\cite{French1971,Crawford1968,OpenStax,Feynman}.
WM06 deliberately stops before introducing the sinusoidal traveling-wave form
$A\cos(kx-\omega t+\phi)$. That combination is the subject of WM07.
\section{Start with a fixed shape}
Suppose that at time $t=0$ a disturbance has the spatial profile
\begin{equation}
u(x,0)=F(x).
\end{equation}
Here $F$ is simply a rule that assigns a disturbance value to each position.
For example, $F$ could describe a single localized pulse on a string.
The important point is that $F$ describes the \emph{shape}. The argument of
$F$ tells us where we are sampling that shape.
If the entire shape later moves to the right without stretching, compressing,
or changing amplitude, then every recognizable feature of the profile must
appear at a larger value of $x$ as time increases.
\section{Translation to the right}
Assume the disturbance moves toward increasing $x$ with constant speed $c$.
After a time $t$, the shape has shifted a distance
\begin{equation}
ct.
\end{equation}
To determine the disturbance now observed at position $x$, ask which point of
the original profile has arrived there. That point began at
\begin{equation}
x-ct.
\end{equation}
Therefore
\begin{equation}
\boxed{u(x,t)=F(x-ct).}
\end{equation}
\begin{center}
\includegraphics{WM06_fig01_right_moving_pulse.png}
\vspace{0.45em}
\textbf{Figure.}
A localized disturbance translated toward increasing $x$. During each time
interval $\Delta t$, every recognizable feature moves the same distance
$c\Delta t$, so the shape is preserved.
\end{center}
This minus sign is sometimes surprising. The reason becomes clear when we
track one identifiable feature of the shape.
\section{Why the minus sign gives motion toward positive x}
Let one particular feature of the profile correspond to a fixed argument
$\xi_0$ of the function $F$. For the right-moving form,
\begin{equation}
x-ct=\xi_0.
\end{equation}
Solving for the position of that feature gives
\begin{equation}
\boxed{x=\xi_0+ct.}
\end{equation}
As $t$ increases, $x$ increases. Therefore the feature moves toward positive
$x$.
Over a time interval $\Delta t$,
\begin{equation}
\Delta x=c\,\Delta t.
\end{equation}
If ordinary derivative notation is familiar, the same statement can be written
\begin{equation}
\frac{dx}{dt}=c.
\end{equation}
No partial derivatives or wave equation are needed for this argument. We are
only tracking the motion of one recognizable feature.
\begin{center}
\includegraphics{WM06_fig03_constant_feature_xt.png}
\vspace{0.45em}
\textbf{Figure.}
A fixed feature of $F(x-ct)$ satisfies $x-ct=\xi_0$, so its position follows
$x=\xi_0+ct$. On a graph of position versus time, the slope is the propagation
speed $c$.
\end{center}
\section{Translation to the left}
Now consider
\begin{equation}
u(x,t)=F(x+ct).
\end{equation}
Again track a fixed feature by setting the argument equal to a constant:
\begin{equation}
x+ct=\xi_0.
\end{equation}
Then
\begin{equation}
\boxed{x=\xi_0-ct.}
\end{equation}
As time increases, the position decreases. Therefore
\begin{equation}
\boxed{F(x+ct)\text{ moves toward negative }x.}
\end{equation}
The two sign conventions are summarized by
\begin{equation}
\boxed{
\begin{array}{ll}
F(x-ct) & \text{moves toward }+x,\\
F(x+ct) & \text{moves toward }-x.
\end{array}}
\end{equation}
\begin{center}
\includegraphics{WM06_fig02_sign_convention.png}
\vspace{0.45em}
\textbf{Figure.}
The sign inside the argument is opposite the direction in which the shape
translates. A minus sign gives motion toward $+x$; a plus sign gives motion
toward $-x$.
\end{center}
\section{A quick sign test}
A reliable way to avoid memorizing the sign rule is to track a feature.
Suppose the feature is initially located where the argument is zero.
For $F(x-ct)$, the zero-argument feature satisfies
\begin{equation}
x-ct=0,
\end{equation}
so
\begin{equation}
x=ct.
\end{equation}
It moves right. For $F(x+ct)$, the zero-argument feature satisfies
\begin{equation}
x+ct=0,
\end{equation}
so
\begin{equation}
x=-ct.
\end{equation}
It moves left. This feature-tracking argument is more robust than trying to
remember a verbal rule.
\section{Dimensional check}
The quantities inside the argument of $F$ must be compatible. Since $x$ is a
position,
\begin{equation}
[x]=\text{length}.
\end{equation}
If $c$ is a speed and $t$ is a time, then
\begin{equation}
[ct]=\frac{\text{length}}{\text{time}}\,\text{time}
=\text{length}.
\end{equation}
Therefore the combinations $x-ct$ and $x+ct$ are dimensionally meaningful.
An expression such as $F(x-c)$ would generally be invalid because position and
speed do not have the same dimensions.
\section{The shape can be arbitrary}
Nothing in the translation argument required a sinusoid. For example, a
localized pulse could be described by
\begin{equation}
F(\xi)=A\exp\left[-\left(\frac{\xi}{a}\right)^2\right],
\end{equation}
where $a$ sets the width of the pulse. A right-moving version is
\begin{equation}
u(x,t)=A\exp\left[-\left(\frac{x-ct}{a}\right)^2\right].
\end{equation}
Every feature of the pulse translates by the same distance $ct$. The profile
is not required to be periodic.
This is conceptually important. Traveling-wave mathematics applies to pulses
and other disturbances, not only to endlessly repeating sinusoids
\cite{OpenStax,Feynman}.
\section{Snapshot versus time history}
Once a disturbance depends on both position and time,
\begin{equation}
u=u(x,t),
\end{equation}
there are two different one-dimensional views.
At a fixed time $t=t_0$,
\begin{equation}
u(x,t_0)
\end{equation}
is a \emph{spatial snapshot}. It shows the shape across space at one instant.
At a fixed position $x=x_0$,
\begin{equation}
u(x_0,t)
\end{equation}
is a \emph{time history}. It shows what one observer at one location measures
as the disturbance passes.
These two views were previewed in WM00. WM06 is the first lesson in which both
views belong to the same moving object.
\section{Propagation is not necessarily material transport}
A moving disturbance should not automatically be interpreted as the bulk
motion of the material through which it travels. In a transverse wave on a
string, for example, the disturbance may propagate horizontally while
individual pieces of the string move mainly up and down around their local
equilibrium positions.
\begin{center}
\includegraphics{WM06_fig04_wave_vs_medium_motion.png}
\vspace{0.45em}
\textbf{Figure.}
For a transverse string example, the disturbance can propagate along the
string while individual material elements move locally. Propagation speed and
material-element velocity are different physical quantities.
\end{center}
This distinction is emphasized in standard wave treatments and is especially
important once energy transport is introduced later in the series
\cite{OpenStax,French1971}.
\section{Worked example 1: identify the direction}
Consider
\begin{equation}
u(x,t)=F(x-3t),
\end{equation}
with $x$ in meters and $t$ in seconds. The form is $F(x-ct)$, so
\begin{equation}
c=3\,\text{m/s}.
\end{equation}
Therefore the disturbance moves toward positive $x$ at
\begin{equation}
\boxed{3\,\text{m/s}.}
\end{equation}
A feature initially at $x=2\,\text{m}$ will be located after $4\,\text{s}$ at
\begin{align}
x
&=2\,\text{m}+(3\,\text{m/s})(4\,\text{s})\\
&=14\,\text{m}.
\end{align}
\section{Worked example 2: left-moving feature}
Suppose
\begin{equation}
u(x,t)=G(x+5t).
\end{equation}
This is the form $G(x+ct)$, so the disturbance moves toward negative $x$ with
speed
\begin{equation}
c=5\,\text{m/s}.
\end{equation}
If a particular feature is at $x=7\,\text{m}$ when $t=0$, then after
$1.5\,\text{s}$ its position is
\begin{align}
x
&=7\,\text{m}-(5\,\text{m/s})(1.5\,\text{s})\\
&=-0.5\,\text{m}.
\end{align}
Thus
\begin{equation}
\boxed{x=-0.5\,\text{m}.}
\end{equation}
\section{Worked example 3: recover the original shape}
Suppose a right-moving disturbance is
\begin{equation}
u(x,t)=F(x-2t).
\end{equation}
At $t=0$,
\begin{equation}
u(x,0)=F(x).
\end{equation}
At $t=3\,\text{s}$,
\begin{equation}
u(x,3)=F(x-6).
\end{equation}
The profile at $t=3\,\text{s}$ is therefore the original profile shifted
$6\,\text{m}$ toward positive $x$. The shape has not been redefined. Only its
position has changed.
\section{What is and is not assumed}
The formulas $F(x\mp ct)$ make a specific assumption: the disturbance
translates at constant speed while preserving its shape.
This is a useful idealization, but not every physical disturbance behaves this
way forever. Real waves may attenuate, disperse, reflect, or change shape.
Those effects require additional physics.
For WM06, however, shape-preserving translation is exactly the right starting
point because it isolates the kinematics of propagation before introducing the
wave equation or more complicated media.
\section{Common mistakes}
\begin{itemize}
\item \textbf{Mistake:} assuming the minus sign means motion toward negative $x$. For $F(x-ct)$, tracking a fixed feature gives $x=\xi_0+ct$, so the motion is toward positive $x$.
\item \textbf{Mistake:} thinking $F$ must be a sine or cosine. $F$ can describe an arbitrary translated shape.
\item \textbf{Mistake:} confusing the propagation of the disturbance with the motion of the material itself.
\item \textbf{Mistake:} reading $c$ as an amplitude. The quantity $c$ has units of speed and controls horizontal translation with time.
\item \textbf{Mistake:} writing $F(x\mp c)$ when time dependence is intended. The distance translated after time $t$ is $ct$.
\item \textbf{Mistake:} assuming that every physical wave must preserve its shape exactly. The forms in this lesson describe nondispersive shape-preserving translation.
\end{itemize}
\section{Connection to the next lesson}
WM05 gave the spatial sinusoid
\begin{equation}
u(x)=A\cos(kx+\phi).
\end{equation}
WM06 has now shown how an arbitrary shape moves through expressions such as
\begin{equation}
F(x-ct).
\end{equation}
In WM07 these two ideas will be combined. A sinusoidal profile will be made to
translate, producing the familiar traveling-wave form
\begin{equation}
A\cos(kx-\omega t+\phi).
\end{equation}
At that point every symbol in the expression will have been introduced
separately before being assembled into a complete wave.
\section*{Summary}
The central results of WM06 are
\begin{equation}
\boxed{u(x,t)=F(x-ct)}
\end{equation}
for shape-preserving motion toward positive $x$, and
\begin{equation}
\boxed{u(x,t)=F(x+ct)}
\end{equation}
for shape-preserving motion toward negative $x$.
Tracking a fixed feature gives
\begin{equation}
x=\xi_0+ct
\end{equation}
for the right-moving case and
\begin{equation}
x=\xi_0-ct
\end{equation}
for the left-moving case.
The disturbance may be periodic or nonperiodic. The function $F$ describes
its shape; the combination $x\mp ct$ translates that shape through space.
\section*{References}
\begin{thebibliography}{9}
\bibitem{French1971}
A.~P. French,
\emph{Vibrations and Waves},
M.I.T. Introductory Physics Series,
W. W. Norton \& Company, 1971.
\bibitem{Crawford1968}
Frank S. Crawford, Jr.,
\emph{Waves},
Berkeley Physics Course, Volume 3,
McGraw-Hill, 1968.
\bibitem{OpenStax}
Samuel J. Ling, Jeff Sanny, and William Moebs,
\emph{University Physics, Volume 1},
OpenStax, 2016,
Chapter 16, especially Section 16.2, ``Mathematics of Waves.''
\bibitem{Feynman}
Richard P. Feynman, Robert B. Leighton, and Matthew Sands,
\emph{The Feynman Lectures on Physics, Volume I},
Chapter 47, ``Sound. The Wave Equation,'' especially the discussion of
traveling disturbances of the form $f(x-vt)$.
\bibitem{MITOCW}
Massachusetts Institute of Technology OpenCourseWare,
\emph{18.03 Differential Equations: Waves interactive demonstration},
illustrating left- and right-moving functions of translated arguments.
\end{thebibliography}