Wave Mechanics: Oscillation in SpaceWaveMechanicsOscillationInSpace2026-09-11 15:56:342026-09-11 15:56:34Topicwave mechanicsspatial oscillationspatial periodicitywavelengthperiodic functionspatial snapshotdisplacement fieldperiodsinusoidal spatial pattern% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Wave Mechanics: Oscillation in Space}
WM01--WM03 described quantities that vary with time at one point. The basic
object was
\begin{equation}
u=u(t).
\end{equation}
We now make a different simplification. Freeze time and ask how a quantity
varies from place to place. The basic object becomes
\begin{equation}
\boxed{u=u(x).}
\end{equation}
This is a spatial profile or spatial snapshot. It may repeat as $x$ changes,
just as a periodic time history repeats as $t$ changes. The distance required
for one complete spatial repetition is called the \emph{wavelength}.
This lesson does not yet describe propagation. A curve that varies with $x$
is not automatically moving. Wavenumber is introduced in WM05, translating
disturbances in WM06, and the full sinusoidal traveling wave in WM07.
\section{From a time history to a spatial profile}
A graph of $u(t)$ answers the question
\begin{quote}
How does the quantity change as time passes at one fixed place?
\end{quote}
A graph of $u(x)$ answers a different question:
\begin{quote}
How does the quantity change from place to place at one fixed instant?
\end{quote}
For a stretched string, for example, imagine taking a photograph at one
instant. Each horizontal position $x$ has some transverse displacement
$u(x)$. The photograph records the string shape at that instant.
\begin{center}
\includegraphics{WM04_fig01_spatial_snapshot.png}
\vspace{0.45em}
\textbf{Figure.}
A spatial snapshot $u(x)$. The horizontal axis is position, not time. The
repeated distance between corresponding points is the wavelength $\lambda$.
Nothing in this picture alone says that the pattern is moving.
\end{center}
The independent variable has changed from time to position. That single
change creates a parallel set of ideas:
\begin{center}
\begin{tabular}{c|c}
Temporal description & Spatial description\\
\hline
$t$ & $x$\\
$u(t)$ & $u(x)$\\
period $T$ & wavelength $\lambda$\\
seconds & meters\\
\end{tabular}
\end{center}
This temporal--spatial symmetry becomes one of the organizing ideas of wave
mechanics.
\section{Spatial periodicity}
A time-periodic function satisfies
\begin{equation}
u(t+T)=u(t).
\end{equation}
A spatially periodic function obeys the analogous relation
\begin{equation}
\boxed{u(x+\lambda)=u(x).}
\end{equation}
The symbol $\lambda$ is the Greek letter lambda. It denotes a distance in
space.
If the pattern is periodic, moving one wavelength to the right brings us to an
equivalent location in the repeating pattern.
More generally,
\begin{equation}
u(x+n\lambda)=u(x),
\qquad n=0,\pm1,\pm2,\ldots
\end{equation}
because any integer number of complete spatial cycles returns to the same
pattern state.
\section{Wavelength}
The \emph{wavelength} is the smallest positive spatial distance over which the
complete pattern repeats. We write
\begin{equation}
\boxed{\lambda=\text{one complete spatial period}.}
\end{equation}
Its SI unit is the meter:
\begin{equation}
[\lambda]=\text{m}.
\end{equation}
Depending on scale, useful units can also include centimeters, millimeters,
micrometers, or kilometers.
For a sinusoidal-looking pattern, wavelength can be measured between any two
corresponding points in adjacent cycles, for example
\begin{itemize}
\item crest to next crest,
\item trough to next trough,
\item upward zero crossing to next upward zero crossing,
\item any point to the next point having the same displacement and the same local direction of change with $x$.
\end{itemize}
The last statement is the spatial counterpart of the period-measurement rule
from WM01. Merely finding the same value of $u$ is not enough because most
values occur more than once per cycle.
\section{Period and wavelength are analogous, not identical}
Period and wavelength both measure repetition, but they measure it along
different independent variables.
\begin{center}
\includegraphics{WM04_fig02_time_vs_space_periodicity.png}
\vspace{0.45em}
\textbf{Figure.}
Temporal and spatial periodicity are mathematically parallel. The upper graph
repeats after a time $T$; the lower graph repeats after a distance $\lambda$.
The two quantities have different physical dimensions and must not be
interchanged.
\end{center}
Period satisfies
\begin{equation}
u(t+T)=u(t),
\end{equation}
while wavelength satisfies
\begin{equation}
u(x+\lambda)=u(x).
\end{equation}
Their units make the distinction unavoidable:
\begin{align}
[T]&=\text{s},\\
[\lambda]&=\text{m}.
\end{align}
A statement such as ``the wavelength is $0.5\,\text{s}$'' is dimensionally
wrong. Likewise, ``the period is $2\,\text{m}$'' is wrong.
\section{Amplitude remains a vertical scale}
The meaning of amplitude does not change when we switch from a time graph to a
spatial graph. For a profile centered on $u=0$,
\begin{equation}
A=\max |u(x)|.
\end{equation}
Amplitude measures the size of the dependent variable. Wavelength measures
the horizontal repetition distance.
These are independent features. Two spatial patterns can have the same
amplitude but different wavelengths.
\begin{center}
\includegraphics{WM04_fig03_compare_wavelengths.png}
\vspace{0.45em}
\textbf{Figure.}
Two periodic spatial profiles with the same amplitude but different
wavelengths. The shorter wavelength repeats more frequently in space, but its
vertical scale need not be larger or smaller.
\end{center}
Thus a short wavelength does not mean a large amplitude, and a long wavelength
does not mean a small amplitude.
\section{A simple sinusoidal spatial pattern}
A convenient example of a periodic spatial pattern is
\begin{equation}
u(x)=A\cos\left(2\pi\frac{x}{\lambda}\right).
\end{equation}
This equation is useful because increasing $x$ by one wavelength changes the
cosine argument by $2\pi$:
\begin{align}
u(x+\lambda)
&=A\cos\left(2\pi\frac{x+\lambda}{\lambda}\right)\\
&=A\cos\left(2\pi\frac{x}{\lambda}+2\pi\right)\\
&=A\cos\left(2\pi\frac{x}{\lambda}\right)\\
&=u(x).
\end{align}
So the function has spatial period $\lambda$ exactly as required.
This form contains the quantity
\begin{equation}
\frac{2\pi}{\lambda},
\end{equation}
which measures angular phase change per unit distance. WM05 gives that
quantity its own symbol, $k$, and develops its physical meaning as
\emph{wavenumber}. For now, it is enough to notice that one wavelength
corresponds to one angular cycle of $2\pi$.
\section{The smallest positive repeat distance}
As with time periodicity, a pattern that repeats after $\lambda$ also repeats
after $2\lambda$, $3\lambda$, and so on. Wavelength normally means the
smallest positive repeat distance.
Suppose a pattern satisfies
\begin{equation}
u(x+0.40\,\text{m})=u(x)
\end{equation}
for every $x$, and no smaller positive distance has this property. Then
\begin{equation}
\boxed{\lambda=0.40\,\text{m}.}
\end{equation}
The pattern also repeats after $0.80\,\text{m}$ and $1.20\,\text{m}$, but those
are two and three wavelengths rather than new wavelengths.
\begin{center}
\includegraphics{WM04_fig04_spatial_repeat.png}
\vspace{0.45em}
\textbf{Figure.}
Points separated by one wavelength occupy equivalent locations in a repeating
spatial pattern. The marked points have the same displacement and the same
local orientation of the curve.
\end{center}
\section{Equal displacement does not necessarily mean one wavelength}
Consider the sinusoidal spatial profile. The curve can pass through the same
value of $u$ twice within one wavelength, once while increasing with $x$ and
once while decreasing with $x$.
Therefore a distance between two equal-displacement points may be only a
fraction of a wavelength.
For a reliable measurement of $\lambda$, compare corresponding pattern states:
crest to crest, trough to trough, or a zero crossing with the same crossing
direction.
This is directly analogous to the temporal result from WM01 and WM03: equal
displacement does not necessarily identify the same phase state.
\section{Worked example 1: reading wavelength from repeated crests}
Suppose adjacent crests of a spatial profile occur at
\begin{equation}
x_1=0.35\,\text{m},
\qquad
x_2=1.10\,\text{m}.
\end{equation}
Because adjacent crests are corresponding points in neighboring cycles,
\begin{align}
\lambda
&=x_2-x_1\\
&=1.10\,\text{m}-0.35\,\text{m}\\
&=\boxed{0.75\,\text{m}}.
\end{align}
If another crest occurs at $1.85\,\text{m}$, the same spacing is confirmed:
\begin{equation}
1.85\,\text{m}-1.10\,\text{m}=0.75\,\text{m}.
\end{equation}
\section{Worked example 2: counting spatial cycles}
A periodic pattern has wavelength
\begin{equation}
\lambda=0.25\,\text{m}.
\end{equation}
How many complete wavelengths fit in a distance of $2.0\,\text{m}$?
The number of spatial cycles is
\begin{equation}
N=\frac{L}{\lambda}.
\end{equation}
Thus
\begin{align}
N
&=\frac{2.0\,\text{m}}{0.25\,\text{m}}\\
&=\boxed{8}.
\end{align}
The meter units cancel, leaving a dimensionless cycle count.
\section{What a spatial snapshot does not tell us}
A single function $u(x)$ describes shape in space. It does not tell us how
that shape changes with time.
From one frozen profile alone we cannot determine whether the pattern
\begin{itemize}
\item is stationary,
\item moves to the right,
\item moves to the left,
\item changes shape,
\item or belongs to a standing-wave pattern.
\end{itemize}
Time dependence requires a second independent variable. Eventually we will
write
\begin{equation}
u=u(x,t).
\end{equation}
But introducing both variables too early hides the separate meanings of period
and wavelength. WM04 keeps the spatial idea isolated so that the later
combination is easier to interpret.
\section{Common mistakes}
\begin{itemize}
\item \textbf{Mistake:} calling a graph of $u(x)$ a time history. Its horizontal axis is position.
\item \textbf{Mistake:} treating wavelength and period as interchangeable. Wavelength has dimensions of length; period has dimensions of time.
\item \textbf{Mistake:} measuring between arbitrary equal-displacement points. Corresponding points must represent the same location within the spatial cycle.
\item \textbf{Mistake:} assuming a shorter wavelength means a larger amplitude. Wavelength and amplitude describe independent geometric features.
\item \textbf{Mistake:} assuming that a sinusoidal spatial profile is automatically a traveling wave. Propagation requires time dependence.
\end{itemize}
\section{Summary}
WM04 replaces the one-point temporal description $u(t)$ with a spatial profile
\begin{equation}
\boxed{u=u(x).}
\end{equation}
A spatially periodic function satisfies
\begin{equation}
\boxed{u(x+\lambda)=u(x),}
\end{equation}
where $\lambda$ is the smallest positive repeat distance, or wavelength.
The central temporal--spatial analogy is
\begin{center}
\begin{tabular}{c|c}
Temporal & Spatial\\
\hline
$t$ & $x$\\
$T$ & $\lambda$\\
$u(t+T)=u(t)$ & $u(x+\lambda)=u(x)$\\
seconds & meters\\
\end{tabular}
\end{center}
A useful sinusoidal example is
\begin{equation}
u(x)=A\cos\left(2\pi\frac{x}{\lambda}\right).
\end{equation}
WM05 will turn the spatial angular rate $2\pi/\lambda$ into a named quantity,
\emph{wavenumber}, and show why spatial phase is naturally measured in radians
per meter.