Wave Mechanics: Wave SpeedWaveMechanicsWaveSpeed2026-09-11 23:54:302026-09-11 23:54:30Definitionwave mechanicswave speedpropagation speedphase velocityfrequencywavelengthperiodangular frequencywavenumberconstant phasesinusoidal traveling wave% this is the default PhysicsLibrary preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics).
% If you use \includegraphics{figure} with no file extension, PhysicsLibrary
% will prefer figure.png before falling back to figure.eps.
\usepackage{graphicx}
\DeclareGraphicsExtensions{.png,.jpg,.jpeg,.pdf,.eps}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here\section*{Wave Mechanics: Wave Speed}
WM06 introduced translating disturbances, and WM07 assembled the sinusoidal
traveling-wave form
\begin{equation}
u(x,t)=A\cos(kx-\omega t+\phi)
\end{equation}
for propagation toward increasing $x$. WM08 now asks the quantitative
question that was deliberately postponed:
\begin{quote}
How fast does a recognizable phase of the wave move through space?
\end{quote}
The answer connects the temporal and spatial descriptions developed throughout
WM01--WM07. For a sinusoidal wave, the propagation speed can be written in
three equivalent forms,
\begin{equation}
\boxed{c=\frac{\omega}{k}=\frac{\lambda}{T}=f\lambda.}
\end{equation}
These relations are standard results of elementary wave mechanics
\cite{French1971,Crawford1968,OpenStax16,Feynman47}. The purpose of this
lesson is to derive them from concepts already established in the series, not
to introduce them as formulas to memorize.
WM08 is still primarily kinematic. It tells us how wave speed is encoded in
$k$, $\omega$, $\lambda$, $T$, and $f$. Later lessons will ask a different
question: what physical properties of a medium determine that speed?
\section{Tracking a point of constant phase}
For the right-moving sinusoidal wave
\begin{equation}
u(x,t)=A\cos(kx-\omega t+\phi),
\end{equation}
define the complete phase
\begin{equation}
\theta(x,t)=kx-\omega t+\phi.
\end{equation}
A crest, a trough, or any other corresponding point on successive cycles can
be identified by holding the phase fixed. Let
\begin{equation}
\theta(x,t)=\theta_0,
\end{equation}
where $\theta_0$ is constant. Then
\begin{equation}
kx-\omega t+\phi=\theta_0.
\end{equation}
Solving for position gives
\begin{align}
kx &= \omega t+\theta_0-\phi,\\
x &= \frac{\omega}{k}t+\frac{\theta_0-\phi}{k}.
\end{align}
This is the equation of a straight line in an $x$--$t$ diagram. Its slope is
\begin{equation}
\boxed{\frac{\Delta x}{\Delta t}=\frac{\omega}{k}.}
\end{equation}
Therefore the speed of a constant-phase feature is
\begin{equation}
\boxed{c=\frac{\omega}{k}.}
\end{equation}
\begin{center}
\includegraphics{WM08_fig01_constant_phase_speed.png}
\vspace{0.45em}
\textbf{Figure.}
A fixed phase value traces a straight line in an $x$--$t$ diagram. Between two
points on that line, the propagation speed is the spatial change divided by
the elapsed time. For the right-moving sinusoid, the slope is
$\Delta x/\Delta t=\omega/k$.
\end{center}
This derivation uses only algebra and the constant-phase idea from WM07. No
wave equation is required.
\section{Direction and signed propagation velocity}
It is useful to distinguish \emph{speed}, which is nonnegative, from a signed
one-dimensional propagation velocity.
For
\begin{equation}
kx-\omega t+\phi=\theta_0,
\end{equation}
we found
\begin{equation}
x=\frac{\omega}{k}t+\text{constant},
\end{equation}
so the phase moves toward positive $x$. Its signed propagation velocity is
\begin{equation}
v_{\text{phase}}=+\frac{\omega}{k}.
\end{equation}
For the left-moving wave
\begin{equation}
u(x,t)=A\cos(kx+\omega t+\phi),
\end{equation}
constant phase gives
\begin{equation}
x=-\frac{\omega}{k}t+\text{constant},
\end{equation}
and therefore
\begin{equation}
v_{\text{phase}}=-\frac{\omega}{k}.
\end{equation}
If $c$ denotes the positive speed magnitude, then in either direction
\begin{equation}
\boxed{c=\left|v_{\text{phase}}\right|=\frac{\omega}{k}}
\end{equation}
for positive $k$ and $\omega$.
Later in the series, dispersive waves will require a more careful distinction
between phase velocity and group velocity. For the present one-dimensional
sinusoidal wave, the constant-phase speed is the propagation speed we are
studying.
\section{One wavelength in one period}
There is a second, highly physical route to the same result.
Consider a particular crest of a right-moving periodic wave. After one full
period $T$, that crest has advanced by one wavelength $\lambda$. Therefore
\begin{equation}
\boxed{c=\frac{\lambda}{T}.}
\end{equation}
\begin{center}
\includegraphics{WM08_fig02_wavelength_per_period.png}
\vspace{0.45em}
\textbf{Figure.}
The same phase feature observed one period later has advanced one wavelength.
The corresponding change in the $x$--$t$ plane is $\Delta x=\lambda$ during
$\Delta t=T$, so $c=\lambda/T$.
\end{center}
The same statement can be verified directly from phase. Suppose two events
are separated by
\begin{equation}
\Delta x=\lambda,
\qquad
\Delta t=T.
\end{equation}
The change in phase is
\begin{align}
\Delta\theta
&=k\lambda-\omega T\\
&=\left(\frac{2\pi}{\lambda}\right)\lambda
-\left(\frac{2\pi}{T}\right)T\\
&=2\pi-2\pi\\
&=0.
\end{align}
Thus those two events lie on the same constant-phase track.
\section{From period to frequency: deriving $c=f\lambda$}
WM01 introduced
\begin{equation}
f=\frac{1}{T}.
\end{equation}
Substituting this into
\begin{equation}
c=\frac{\lambda}{T}
\end{equation}
gives
\begin{equation}
\boxed{c=f\lambda.}
\end{equation}
This relation can be read in words:
\begin{quote}
wave speed = cycles per second $\times$ distance per cycle.
\end{quote}
The dimensional check is immediate:
\begin{equation}
\left(\frac{1}{\text{s}}\right)(\text{m})
=\frac{\text{m}}{\text{s}}.
\end{equation}
OpenStax states the same fundamental relationship as
$v=\lambda/T=\lambda f$ for traveling waves \cite{OpenStax16}.
\section{Showing that $\omega/k$ and $f\lambda$ are the same quantity}
From the definitions developed earlier,
\begin{equation}
\omega=2\pi f
\end{equation}
and
\begin{equation}
k=\frac{2\pi}{\lambda}.
\end{equation}
Therefore
\begin{align}
\frac{\omega}{k}
&=\frac{2\pi f}{2\pi/\lambda}\\
&=f\lambda.
\end{align}
Equivalently, using $\omega=2\pi/T$,
\begin{align}
\frac{\omega}{k}
&=\frac{2\pi/T}{2\pi/\lambda}\\
&=\frac{\lambda}{T}.
\end{align}
Hence
\begin{equation}
\boxed{c=\frac{\omega}{k}=\frac{\lambda}{T}=f\lambda.}
\end{equation}
\begin{center}
\includegraphics{WM08_fig03_speed_relation_map.png}
\vspace{0.45em}
\textbf{Figure.}
The same propagation speed can be calculated from temporal frequency and
wavelength, from period and wavelength, or from angular frequency and angular
wavenumber. The three forms are algebraically equivalent.
\end{center}
\section{Units of $\omega/k$}
WM02 gave angular frequency units of radians per second, while WM05 gave
angular wavenumber units of radians per meter. Therefore
\begin{equation}
\frac{\omega}{k}
=\frac{\text{rad}/\text{s}}{\text{rad}/\text{m}}
=\frac{\text{m}}{\text{s}}.
\end{equation}
The angular measure cancels, leaving the dimensions of speed.
This check is especially useful because $\omega$ and $k$ can look abstract.
Their ratio has an immediately familiar mechanical unit.
\section{Frequency is not wave speed}
A common misconception is that a higher-frequency wave must travel faster.
The relation
\begin{equation}
c=f\lambda
\end{equation}
shows why that conclusion does not follow from frequency alone. If two waves
travel at the same speed, a higher frequency must be accompanied by a shorter
wavelength:
\begin{equation}
\lambda=\frac{c}{f}.
\end{equation}
For example, suppose the propagation speed is
\begin{equation}
c=4\,\text{m}/\text{s}.
\end{equation}
Then a $1\,\text{Hz}$ wave has
\begin{equation}
\lambda=4\,\text{m},
\end{equation}
while a $2\,\text{Hz}$ wave has
\begin{equation}
\lambda=2\,\text{m}.
\end{equation}
Both propagate at the same speed.
\begin{center}
\includegraphics{WM08_fig04_same_speed_different_waves.png}
\vspace{0.45em}
\textbf{Figure.}
Two waves can have different frequencies and wavelengths while sharing the
same propagation speed. At fixed $c$, increasing $f$ requires decreasing
$\lambda$ so that the product $f\lambda$ remains unchanged.
\end{center}
In many familiar approximately nondispersive situations, waves of different
frequencies travel at nearly the same speed in a fixed medium. Feynman uses
sound and light as examples when introducing the distinction between
nondispersive and dispersive propagation \cite{Feynman47}. The later Wave
Mechanics lessons on dispersion will revisit this point in detail.
\section{Wave speed is not the local speed of the medium}
WM06 emphasized that propagation of a disturbance does not require each
material element to travel with the disturbance. The same distinction remains
important here.
For a transverse wave on a string, a marked piece of string may move mostly up
and down while a crest travels horizontally along the string. The quantity
$c$ describes the speed of the crest or other constant-phase feature, not the
instantaneous transverse speed of that marked piece of string.
Thus two different velocities can appear in the same physical problem:
\begin{itemize}
\item the \emph{wave propagation speed}, describing movement of the pattern;
\item the \emph{local material velocity}, describing motion of the medium at one location.
\end{itemize}
They should not be confused.
\section{Kinematics versus dynamics: what determines $c$?}
The relations derived in WM08 are kinematic. They tell us how the observed
wave quantities must fit together:
\begin{equation}
c=f\lambda=\frac{\omega}{k}.
\end{equation}
They do not yet tell us what sets the numerical value of $c$ for a physical
system.
That is a dynamical question. For example, the speed of a mechanical wave can
depend on properties such as tension, inertia, stiffness, density, or
compressibility. Standard wave treatments distinguish this medium-dependent
physics from the kinematic relation among speed, wavelength, and frequency
\cite{French1971,Crawford1968,OpenStax16,Feynman47}.
Later in this series, the one-dimensional string wave equation will be derived
from Newton's law. At that point the speed will emerge from the properties of
the string itself. WM08 therefore completes the kinematic foundation without
prematurely assuming the governing PDE.
\section{Worked example 1: frequency and wavelength}
A periodic wave has
\begin{equation}
f=3.0\,\text{Hz},
\qquad
\lambda=0.80\,\text{m}.
\end{equation}
Its speed is
\begin{align}
c&=f\lambda\\
&=(3.0\,\text{s}^{-1})(0.80\,\text{m})\\
&=\boxed{2.4\,\text{m}/\text{s}}.
\end{align}
The result says that a constant-phase feature advances $2.4\,\text{m}$ each
second.
\section{Worked example 2: angular frequency and wavenumber}
Suppose a right-moving sinusoidal wave has
\begin{equation}
k=4.0\,\text{rad}/\text{m},
\qquad
\omega=12.0\,\text{rad}/\text{s}.
\end{equation}
Then
\begin{align}
c&=\frac{\omega}{k}\\
&=\frac{12.0\,\text{rad}/\text{s}}
{4.0\,\text{rad}/\text{m}}\\
&=\boxed{3.0\,\text{m}/\text{s}}.
\end{align}
The radians cancel, leaving the expected units of speed.
We can verify the result using wavelength and frequency:
\begin{align}
\lambda&=\frac{2\pi}{k}=\frac{2\pi}{4.0}
=\frac{\pi}{2}\,\text{m},\\
f&=\frac{\omega}{2\pi}=\frac{12.0}{2\pi}
\approx1.91\,\text{Hz}.
\end{align}
Then
\begin{equation}
f\lambda\approx(1.91)\left(\frac{\pi}{2}\right)
\approx3.0\,\text{m}/\text{s}.
\end{equation}
\section{Worked example 3: infer wavelength at fixed speed}
A wave travels through a system at
\begin{equation}
c=150\,\text{m}/\text{s}
\end{equation}
and has frequency
\begin{equation}
f=50\,\text{Hz}.
\end{equation}
From
\begin{equation}
c=f\lambda,
\end{equation}
we obtain
\begin{align}
\lambda
&=\frac{c}{f}\\
&=\frac{150\,\text{m}/\text{s}}{50\,\text{s}^{-1}}\\
&=\boxed{3.0\,\text{m}}.
\end{align}
If the frequency doubled while the propagation speed remained unchanged, the
wavelength would be cut in half.
\section{Common mistakes}
\begin{itemize}
\item \textbf{Mistake:} assuming high frequency automatically means high wave speed. Speed depends on the product $f\lambda$, not on $f$ alone.
\item \textbf{Mistake:} using $c=f/\lambda$. Dimensional analysis immediately rejects this because $f/\lambda$ does not have units of speed.
\item \textbf{Mistake:} writing $c=k/\omega$. The correct angular form is $c=\omega/k$.
\item \textbf{Mistake:} forgetting propagation direction. The speed magnitude is positive, but the signed phase velocity is positive for $kx-\omega t$ and negative for $kx+\omega t$ when $k,\omega>0$.
\item \textbf{Mistake:} confusing the wave's propagation speed with the local velocity of a material element.
\item \textbf{Mistake:} assuming WM08 explains what physical property sets $c$. The present derivation is kinematic; the medium-dependent dynamics are developed later.
\end{itemize}
\section{What WM08 completes}
WM00--WM08 now provide a complete introductory language for one-dimensional
traveling waves without beginning from the wave equation.
The central temporal quantities are
\begin{equation}
T,\qquad f=\frac{1}{T},\qquad \omega=2\pi f.
\end{equation}
The central spatial quantities are
\begin{equation}
\lambda,\qquad k=\frac{2\pi}{\lambda}.
\end{equation}
The right-moving sinusoidal traveling wave is
\begin{equation}
u(x,t)=A\cos(kx-\omega t+\phi),
\end{equation}
and the spatial and temporal descriptions are connected by
\begin{equation}
\boxed{c=\frac{\omega}{k}=\frac{\lambda}{T}=f\lambda.}
\end{equation}
Every quantity in this equation has now been introduced separately and given a
physical interpretation.
\section{Connection to the next block}
WM08 closes the first production block of the Wave Mechanics series. The next
block begins with WM09, where the two-variable field $u(x,t)$ is studied more
formally.
The sequence will then introduce partial derivatives, coupled oscillators, the
continuum limit, and finally derive the one-dimensional wave equation from the
physics of a continuous medium.
The important order is
\begin{equation}
\boxed{\text{kinematics of waves}\;\longrightarrow\;
\text{dynamics of the medium}\;\longrightarrow\;
\text{wave equation}.}
\end{equation}
\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{OpenStax16}
Samuel J. Ling, Jeff Sanny, and William Moebs,
\emph{University Physics, Volume 1},
OpenStax, 2016,
Chapter 16, especially Sections 16.1--16.2 and the Chapter 16 key equations.
\bibitem{Feynman47}
Richard P. Feynman, Robert B. Leighton, and Matthew Sands,
\emph{The Feynman Lectures on Physics, Volume I},
Chapter 47, ``Sound. The wave equation,'' especially Sections 47--1 and 47--4.
\bibitem{MIT803}
Massachusetts Institute of Technology,
\emph{8.03SC Physics III: Vibrations and Waves},
MIT OpenCourseWare, introductory traveling-wave materials.
\end{thebibliography}