Wave Mechanics: Wavenumber
WaveMechanicsWavenumber
2026-09-11 17:17:15
2026-09-11 17:17:15
Definition
removed double References heading
wave mechanics
wavenumber
angular wavenumber
wavelength
spatial phase
spatial frequency
radians per meter
periodic function
sinusoidal spatial pattern
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\section*{Wave Mechanics: Wavenumber}
WM04 introduced a spatially periodic profile $u(x)$ and defined wavelength
$\lambda$ as the smallest positive distance over which the pattern repeats:
\begin{equation}
u(x+\lambda)=u(x).
\end{equation}
For a sinusoidal spatial pattern we wrote
\begin{equation}
u(x)=A\cos\left(2\pi\frac{x}{\lambda}\right).
\end{equation}
The factor
\begin{equation}
\frac{2\pi}{\lambda}
\end{equation}
appears so often in wave mechanics that it is given its own symbol. We define
\begin{equation}
\boxed{k=\frac{2\pi}{\lambda}.}
\end{equation}
The quantity $k$ is called the \emph{angular wavenumber}, or simply
\emph{wavenumber} in much of physics. It measures how rapidly phase changes
with distance. In this Wave Mechanics series, the symbol $k$ will always mean
this angular spatial rate. This convention is standard in wave equations and
sinusoidal wave notation \cite{French1971,Crawford1968,OpenStax}.
WM05 is still a spatial-only lesson. No pattern is assumed to move. The
translating disturbance $F(x\mp ct)$ is introduced in WM06, and the full
sinusoidal traveling wave appears in WM07.
\section{From one wavelength to one angular cycle}
In WM02, one temporal cycle corresponded to an angular advance of $2\pi$
radians. The same idea applies in space.
If position increases by one wavelength,
\begin{equation}
\Delta x=\lambda,
\end{equation}
then the spatial pattern advances by one complete phase cycle,
\begin{equation}
\Delta\theta=2\pi.
\end{equation}
Therefore the phase advance per unit distance is
\begin{equation}
\frac{\Delta\theta}{\Delta x}
=\frac{2\pi}{\lambda}.
\end{equation}
This is precisely the quantity we call $k$:
\begin{equation}
\boxed{k=\frac{2\pi}{\lambda}.}
\end{equation}
Equivalently,
\begin{equation}
\boxed{\lambda=\frac{2\pi}{k}.}
\end{equation}
\begin{center}
\includegraphics{WM05_fig01_wavelength_to_phase.png}
\vspace{0.45em}
\textbf{Figure.}
One wavelength $\lambda$ in space corresponds to one angular phase cycle of
$2\pi$. Wavenumber $k$ measures how many radians of spatial phase are
accumulated per unit distance.
\end{center}
This is the spatial counterpart of angular frequency:
\begin{equation}
\omega=\frac{2\pi}{T}.
\end{equation}
The parallel structure is
\begin{center}
\begin{tabular}{c|c}
Temporal quantity & Spatial quantity\\
\hline
period $T$ & wavelength $\lambda$\\
angular frequency $\omega=2\pi/T$ & wavenumber $k=2\pi/\lambda$\\
radians per second & radians per meter\\
\end{tabular}
\end{center}
This temporal--spatial symmetry is one of the main reasons angular frequency
and angular wavenumber are so useful in wave mechanics.
\section{Units of wavenumber}
Because wavelength has units of length,
\begin{equation}
[k]=\frac{1}{\text{m}}.
\end{equation}
In physical discussion it is often more informative to say
\begin{equation}
\boxed{k\text{ is measured in radians per meter}.}
\end{equation}
For example,
\begin{equation}
k=8\,\text{rad/m}
\end{equation}
means that phase increases by $8$ radians for each meter of increasing $x$.
Formally, plane angle is a dimensionless quantity in SI and the radian is the
coherent unit used for plane angle \cite{NIST2001}. Therefore the dimensional
unit of $k$ can be written simply as $\text{m}^{-1}$. Keeping the word
``radian'' in the interpretation is nevertheless useful because it reminds us
that $k$ is an \emph{angular phase rate}, not merely a count of cycles per
meter.
\section{Spatial phase}
The sinusoidal spatial pattern can now be written more compactly as
\begin{equation}
\boxed{u(x)=A\cos(kx+\phi).}
\end{equation}
The quantity
\begin{equation}
\boxed{\theta(x)=kx+\phi}
\end{equation}
is the \emph{spatial phase}.
The phase constant $\phi$ has the same meaning introduced in WM02 and WM03: it
specifies where in the cycle the pattern begins at the chosen origin $x=0$.
At $x=0$,
\begin{equation}
\theta(0)=\phi.
\end{equation}
If $\phi=0$, the simple cosine profile begins at a maximum:
\begin{equation}
u(0)=A.
\end{equation}
As $x$ increases, the term $kx$ advances the phase.
\section{How distance maps into phase}
For $\phi=0$, consider positions separated by fractions of one wavelength.
Using
\begin{equation}
k=\frac{2\pi}{\lambda},
\end{equation}
we obtain
\begin{align}
x=0 &\quad\Rightarrow\quad kx=0,\\
x=\frac{\lambda}{4} &\quad\Rightarrow\quad kx=\frac{\pi}{2},\\
x=\frac{\lambda}{2} &\quad\Rightarrow\quad kx=\pi,\\
x=\frac{3\lambda}{4} &\quad\Rightarrow\quad kx=\frac{3\pi}{2},\\
x=\lambda &\quad\Rightarrow\quad kx=2\pi.
\end{align}
\begin{center}
\includegraphics{WM05_fig03_spatial_phase_map.png}
\vspace{0.45em}
\textbf{Figure.}
A distance of one wavelength maps to a phase advance of $2\pi$. Quarter-
wavelength steps correspond to phase advances of $\pi/2$.
\end{center}
This mapping is exactly analogous to the temporal relation
\begin{equation}
\omega t=2\pi\frac{t}{T}.
\end{equation}
In space,
\begin{equation}
\boxed{kx=2\pi\frac{x}{\lambda}.}
\end{equation}
\section{Short wavelength means large wavenumber}
The relation
\begin{equation}
k=\frac{2\pi}{\lambda}
\end{equation}
shows that $k$ and $\lambda$ are inversely related.
If wavelength decreases, the pattern completes more phase cycles within the
same physical distance, so $k$ increases. If wavelength increases, phase
accumulates more slowly with distance, so $k$ decreases.
\begin{center}
\includegraphics{WM05_fig02_compare_wavenumbers.png}
\vspace{0.45em}
\textbf{Figure.}
Two spatial sinusoids with equal amplitude but different wavenumbers. The
larger wavenumber has the shorter wavelength and accumulates phase more rapidly
with distance.
\end{center}
Thus
\begin{equation}
\boxed{\text{short }\lambda\ \Longleftrightarrow\ \text{large }k}
\end{equation}
and
\begin{equation}
\boxed{\text{long }\lambda\ \Longleftrightarrow\ \text{small }k.}
\end{equation}
Wavenumber is therefore a useful measure of spatial oscillation density.
\section{Worked example 1: wavelength to wavenumber}
Suppose
\begin{equation}
\lambda=0.80\,\text{m}.
\end{equation}
Then
\begin{align}
k
&=\frac{2\pi}{\lambda}\\
&=\frac{2\pi}{0.80\,\text{m}}\\
&\approx 7.85\,\text{rad/m}.
\end{align}
Hence
\begin{equation}
\boxed{k\approx7.85\,\text{rad/m}.}
\end{equation}
A useful interpretation is that the spatial phase advances by about $7.85$
radians for each meter.
\section{Worked example 2: wavenumber to wavelength}
Suppose
\begin{equation}
k=12.0\,\text{rad/m}.
\end{equation}
Using
\begin{equation}
\lambda=\frac{2\pi}{k},
\end{equation}
we obtain
\begin{align}
\lambda
&=\frac{2\pi}{12.0\,\text{m}^{-1}}\\
&\approx0.524\,\text{m}.
\end{align}
Therefore
\begin{equation}
\boxed{\lambda\approx0.524\,\text{m}.}
\end{equation}
\section{Worked example 3: reading a spatial sinusoid}
Suppose $x$ is measured in meters and
\begin{equation}
u(x)=0.025\cos\left(4\pi x+\frac{\pi}{6}\right).
\end{equation}
Comparing with
\begin{equation}
u(x)=A\cos(kx+\phi),
\end{equation}
we identify
\begin{align}
A&=0.025,\\
k&=4\pi\,\text{rad/m},\\
\phi&=\frac{\pi}{6}.
\end{align}
The wavelength is
\begin{align}
\lambda
&=\frac{2\pi}{k}\\
&=\frac{2\pi}{4\pi\,\text{m}^{-1}}\\
&=0.50\,\text{m}.
\end{align}
Thus
\begin{equation}
\boxed{\lambda=0.50\,\text{m}.}
\end{equation}
The phase at the origin is $\pi/6$, so the pattern does not begin at the same
point in its cycle as a zero-phase cosine.
\section{The phase constant produces a spatial shift}
Consider
\begin{equation}
u(x)=A\cos(kx+\phi).
\end{equation}
A cosine maximum occurs whenever the phase equals $2\pi n$. For the maximum
nearest the origin, choose a convenient integer $n$ and solve
\begin{equation}
kx+\phi=2\pi n.
\end{equation}
This gives
\begin{equation}
x=\frac{2\pi n-\phi}{k}.
\end{equation}
For the branch with $n=0$,
\begin{equation}
\boxed{x=-\frac{\phi}{k}.}
\end{equation}
Thus a positive phase constant shifts the corresponding cosine maximum toward
negative $x$.
\begin{center}
\includegraphics{WM05_fig04_phase_constant_shift.png}
\vspace{0.45em}
\textbf{Figure.}
Two spatial sinusoids with the same $A$ and $k$ but different phase constants.
The phase constant shifts the spatial pattern horizontally; it does not change
its wavelength.
\end{center}
This is the spatial counterpart of the time shift discussed in WM03.
\section{Angular wavenumber versus reciprocal wavelength}
The word \emph{wavenumber} is not used identically in every field. In this
series,
\begin{equation}
\boxed{k=\frac{2\pi}{\lambda}}
\end{equation}
is the angular wavenumber used in wave mechanics.
A different quantity is the reciprocal wavelength
\begin{equation}
\frac{1}{\lambda}.
\end{equation}
It counts spatial cycles per unit length rather than radians of phase per unit
length. In spectroscopy, the unqualified term ``wavenumber'' commonly refers
to this reciprocal-wavelength quantity and is often expressed in
$\text{cm}^{-1}$ \cite{IUPAC2025}.
The two conventions differ by a factor of $2\pi$:
\begin{equation}
\boxed{k=2\pi\left(\frac{1}{\lambda}\right).}
\end{equation}
This is analogous to the distinction between ordinary frequency $f$ and
angular frequency $\omega$:
\begin{equation}
\omega=2\pi f.
\end{equation}
For clarity, this Wave Mechanics series will consistently use $k$ for angular
wavenumber.
\section{Dimensional check of the cosine argument}
The argument of a trigonometric function must represent a pure phase. In
\begin{equation}
\cos(kx+\phi),
\end{equation}
we have
\begin{equation}
[k]=\text{m}^{-1},
\qquad
[x]=\text{m}.
\end{equation}
Therefore
\begin{equation}
[kx]=1,
\end{equation}
with the phase interpreted in radians. The phase constant $\phi$ is also an
angle, so the sum $kx+\phi$ is physically meaningful.
This dimensional check is valuable. An expression such as
\begin{equation}
\cos(k+x)
\end{equation}
would generally be meaningless because $k$ and $x$ have different dimensions.
\section{Wavenumber is not wave speed}
The symbol $k$ tells us how rapidly a phase pattern changes in space. It does
not tell us how rapidly that pattern moves through space.
At this point in the series we still have only
\begin{equation}
u=u(x).
\end{equation}
No time variable appears, so no propagation speed can yet be inferred.
Later, spatial phase $kx$ will be combined with temporal phase $\omega t$ in
an expression such as
\begin{equation}
kx-\omega t+\phi.
\end{equation}
Only then will the relation between phase evolution in space and time lead to
wave speed.
\section{Common mistakes}
\begin{itemize}
\item \textbf{Mistake:} writing $k=1/\lambda$ in a wave-mechanics equation that uses angular phase. In this series, $k=2\pi/\lambda$.
\item \textbf{Mistake:} treating a larger $k$ as a larger amplitude. Wavenumber controls horizontal spatial repetition; amplitude controls vertical scale.
\item \textbf{Mistake:} forgetting that $kx$ must be a phase. If $x$ is in meters, then $k$ must carry inverse-length units.
\item \textbf{Mistake:} confusing wavenumber $k$ with angular frequency $\omega$. The former measures phase change per distance; the latter measures phase change per time.
\item \textbf{Mistake:} assuming $k$ alone gives a wave speed. A spatial profile without time dependence does not specify propagation.
\item \textbf{Mistake:} confusing the symbol $k$ with a spring constant. The same letter is used for different quantities in different contexts; units and equations identify which meaning is intended.
\end{itemize}
\section{Summary}
WM05 converts wavelength into an angular spatial rate. The defining relation is
\begin{equation}
\boxed{k=\frac{2\pi}{\lambda},}
\end{equation}
with inverse relation
\begin{equation}
\boxed{\lambda=\frac{2\pi}{k}.}
\end{equation}
The spatial sinusoid can therefore be written as
\begin{equation}
\boxed{u(x)=A\cos(kx+\phi),}
\end{equation}
and its spatial phase is
\begin{equation}
\boxed{\theta(x)=kx+\phi.}
\end{equation}
The central temporal--spatial analogy is
\begin{center}
\begin{tabular}{c|c}
Temporal & Spatial\\
\hline
$T$ & $\lambda$\\
$f=1/T$ & reciprocal wavelength $1/\lambda$\\
$\omega=2\pi/T$ & $k=2\pi/\lambda$\\
$\omega t$ & $kx$\\
\end{tabular}
\end{center}
WM06 next replaces a fixed spatial profile with a translating disturbance and
shows why functions of the form $F(x-ct)$ and $F(x+ct)$ represent motion in
opposite directions.
\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{NIST2001}
I. M. Mills, B. N. Taylor, and A. J. Thor,
``Definitions of the Units Radian, Neper, Bel, and Decibel,''
National Institute of Standards and Technology, 2001.
\bibitem{IUPAC2025}
International Union of Pure and Applied Chemistry,
``wavenumber,''
\emph{Compendium of Chemical Terminology (the Gold Book)},
5th ed., online version 5.0.0, 2025,
doi:10.1351/goldbook.W06664.
\end{thebibliography}