Wave Mechanics: SuperpositionWaveMechanicsSuperposition2026-09-12 00:03:232026-09-12 00:05:04Topicwave mechanicssuperpositioninterferencelinear wavesconstructive interferencedestructive interferencephase differencepulse overlapresultant amplitudelinearity% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Wave Mechanics: Superposition}
WM00--WM08 developed the language of one-dimensional traveling waves. We can
now describe a disturbance such as
\begin{equation}
u_1(x,t)=A_1\cos(kx-\omega t+\phi_1)
\end{equation}
or a second disturbance
\begin{equation}
u_2(x,t)=A_2\cos(kx-\omega t+\phi_2).
\end{equation}
WM09 asks what happens when two disturbances occupy the same region of space
at the same time.
For a \emph{linear} wave system, the answer is the principle of superposition:
\begin{equation}
\boxed{u(x,t)=u_1(x,t)+u_2(x,t)+\cdots.}
\end{equation}
The total disturbance is the algebraic sum of the individual disturbances at
each position and time. This principle is central to wave mechanics and is
the foundation of interference, beats, standing waves, Fourier methods, and
normal-mode analysis \cite{French1971,Crawford1968,OpenStax165}.
Superposition is not a statement that all physical waves always add linearly.
It applies when the governing response is linear, or when a physical system is
being modeled within a regime where nonlinear effects are negligible. Later
in this series the one-dimensional linear wave equation will provide a direct
mathematical reason why sums of solutions are again solutions
\cite{Feynman47}.
\section{Point-by-point addition}
The word \emph{superposition} means that two disturbances can occupy the same
place simultaneously and their instantaneous effects add.
Suppose, at one particular event $(x_0,t_0)$,
\begin{equation}
u_1(x_0,t_0)=3\,\text{mm}
\end{equation}
and
\begin{equation}
u_2(x_0,t_0)=-1\,\text{mm}.
\end{equation}
Then the total displacement there is
\begin{equation}
u(x_0,t_0)=3\,\text{mm}-1\,\text{mm}=2\,\text{mm}.
\end{equation}
The same addition is performed independently at every other point.
\begin{center}
\includegraphics{WM09_fig01_pointwise_superposition.png}
\vspace{0.45em}
\textbf{Figure.}
Two arbitrary spatial disturbances and their point-by-point sum. At the
marked position $x_0$, the value in the bottom panel is obtained by adding the
values from the upper two panels. Superposition is local in this sense: the
sum is formed at each position and time.
\end{center}
The disturbances do not need to have the same shape. They do not even need to
be sinusoidal. If the system is linear, the instantaneous total is still the
algebraic sum.
\section{Interference is the visible consequence of superposition}
When two or more waves overlap, their superposition produces a new pattern.
This overlap phenomenon is called \emph{interference}.
The word \emph{interference} does not mean that the waves permanently damage
or obstruct one another. In a linear medium, two pulses may overlap strongly,
produce a temporary resultant disturbance, and then continue propagating with
their original shapes.
\begin{center}
\includegraphics{WM09_fig03_pulses_pass_through.png}
\vspace{0.45em}
\textbf{Figure.}
Two equal pulses approach, overlap, and then separate. During complete
overlap, the algebraic sum reaches twice the amplitude of either pulse. In the
ideal linear model, the component pulses emerge unchanged after the encounter.
\end{center}
This behavior distinguishes waves from colliding rigid objects. The material
of a medium may move locally, but the wave patterns can pass through one
another because the disturbance variables add rather than exclude one another.
\section{Constructive interference}
Consider two identical sinusoidal waves with the same amplitude, wavenumber,
angular frequency, and phase:
\begin{align}
u_1(x,t)&=A\cos(kx-\omega t),\\
u_2(x,t)&=A\cos(kx-\omega t).
\end{align}
Their sum is
\begin{align}
u(x,t)
&=u_1+u_2\\
&=2A\cos(kx-\omega t).
\end{align}
Thus the resultant amplitude is
\begin{equation}
\boxed{A_R=2A.}
\end{equation}
The waves are said to interfere \emph{constructively}. Crest aligns with
crest and trough aligns with trough.
\section{Destructive interference}
Now let the second wave be shifted in phase by $\pi$:
\begin{align}
u_1(x,t)&=A\cos(kx-\omega t),\\
u_2(x,t)&=A\cos(kx-\omega t+\pi).
\end{align}
Because
\begin{equation}
\cos(\theta+\pi)=-\cos\theta,
\end{equation}
we obtain
\begin{align}
u(x,t)
&=A\cos\theta-A\cos\theta\\
&=0.
\end{align}
For two equal-amplitude waves exactly $\pi$ out of phase,
\begin{equation}
\boxed{A_R=0.}
\end{equation}
This is complete destructive interference.
\begin{center}
\includegraphics{WM09_fig02_constructive_destructive.png}
\vspace{0.45em}
\textbf{Figure.}
At zero phase difference, equal waves add constructively and the resultant
amplitude is $2A$. At a phase difference of $\pi$, equal waves cancel point by
point and the resultant amplitude is zero. These are the two limiting cases
of interference.
\end{center}
OpenStax uses the same point-by-point addition picture to define superposition
and interference, including the constructive and $\pi$-shifted destructive
cases \cite{OpenStax165}.
\section{Partial interference and phase difference}
Most overlapping sinusoidal waves are neither exactly in phase nor exactly
opposite in phase.
Let
\begin{align}
u_1&=A\cos\theta,\\
u_2&=A\cos(\theta+\Delta\phi),
\end{align}
where
\begin{equation}
\theta=kx-\omega t+\phi_1
\end{equation}
and
\begin{equation}
\Delta\phi=\phi_2-\phi_1.
\end{equation}
Using the trigonometric identity
\begin{equation}
\cos\alpha+\cos\beta
=2\cos\left(\frac{\alpha-\beta}{2}\right)
\cos\left(\frac{\alpha+\beta}{2}\right),
\end{equation}
we obtain
\begin{align}
u
&=A\cos\theta+A\cos(\theta+\Delta\phi)\\
&=2A\cos\left(\frac{\Delta\phi}{2}\right)
\cos\left(\theta+\frac{\Delta\phi}{2}\right).
\end{align}
The magnitude of the resultant amplitude is therefore
\begin{equation}
\boxed{A_R=2A\left|\cos\left(\frac{\Delta\phi}{2}\right)\right|.}
\end{equation}
This single expression contains the constructive and destructive cases:
\begin{align}
\Delta\phi=0 &\quad\Rightarrow\quad A_R=2A,\\
\Delta\phi=\pi &\quad\Rightarrow\quad A_R=0.
\end{align}
At intermediate phase differences, the interference is partial.
\begin{center}
\includegraphics{WM09_fig04_resultant_amplitude_phase.png}
\vspace{0.45em}
\textbf{Figure.}
For two equal-amplitude, equal-frequency sinusoidal waves, the resultant
amplitude varies continuously with phase difference. Constructive interference
occurs at equivalent phase differences of $2\pi n$, while complete cancellation
occurs at odd multiples of $\pi$.
\end{center}
The phase-difference dependence of the resultant amplitude is a standard
result of interference theory \cite{French1971,Crawford1968,OpenStax165}.
\section{Different amplitudes}
Complete cancellation requires more than a phase difference of $\pi$; the two
component amplitudes must also be equal.
Consider
\begin{align}
u_1&=A_1\cos\theta,\\
u_2&=A_2\cos(\theta+\Delta\phi).
\end{align}
Expanding the second cosine gives
\begin{align}
u
&=(A_1+A_2\cos\Delta\phi)\cos\theta
-A_2\sin\Delta\phi\sin\theta.
\end{align}
This combination is itself a sinusoid of the same $k$ and $\omega$. Its
amplitude is
\begin{equation}
\boxed{A_R=
\sqrt{A_1^2+A_2^2+2A_1A_2\cos\Delta\phi}.}
\end{equation}
Two useful checks are
\begin{align}
\Delta\phi=0
&\quad\Rightarrow\quad A_R=A_1+A_2,\\
\Delta\phi=\pi
&\quad\Rightarrow\quad A_R=|A_1-A_2|.
\end{align}
Thus unequal waves shifted by $\pi$ interfere destructively but do not, in
general, cancel completely.
\section{Superposition does not create or destroy the component waves}
The resultant disturbance can be larger or smaller than either component at a
given event. This does not mean that the individual waves have ceased to
exist as useful components of the description.
In a linear system we may write
\begin{equation}
u=u_1+u_2
\end{equation}
throughout the overlap. After localized pulses separate, the original pulse
shapes can reappear. For continuous sinusoidal waves, the component waves can
likewise be regarded as continuing through one another while the observable
disturbance is their sum.
Energy accounting requires more care than simply adding instantaneous
amplitudes, so energy and power are treated later in the series.
\section{Why linearity matters}
The principle of superposition is fundamentally a statement about a \emph{linear}
model.
Suppose a later governing equation has a linear operator $L$ such that
\begin{equation}
L[u]=0.
\end{equation}
If
\begin{equation}
L[u_1]=0
\qquad\text{and}\qquad
L[u_2]=0,
\end{equation}
then linearity means
\begin{equation}
L[u_1+u_2]=L[u_1]+L[u_2]=0.
\end{equation}
Therefore the sum is also a solution. Feynman explicitly demonstrates this
for the linear wave equation \cite{Feynman47}.
WM09 does not yet derive the wave equation; that comes in a later block. The
argument above is included only to show where the superposition principle will
ultimately come from mathematically.
Real physical systems can become nonlinear at sufficiently large amplitude or
under other conditions where the restoring response is not proportional to the
disturbance. In such regimes, simple superposition may fail. OpenStax makes
the same distinction between linear and nonlinear waves \cite{OpenStax165}.
\section{A preview of counter-propagating-wave superposition}
A particularly important future application occurs when equal sinusoidal waves
move in opposite directions:
\begin{align}
u_1&=A\cos(kx-\omega t),\\
u_2&=A\cos(kx+\omega t).
\end{align}
Their sum is
\begin{equation}
\boxed{u=2A\cos(kx)\cos(\omega t).}
\end{equation}
This expression no longer has the form of a single wave translating rigidly to
one side. It is the mathematical seed of a standing wave. Nodes,
antinodes, resonance, and normal modes are intentionally deferred to later
lessons.
\section{Worked example 1: point-by-point addition}
At one position and time, two disturbances have values
\begin{equation}
u_1=4.0\,\text{mm},
\qquad
u_2=-1.5\,\text{mm}.
\end{equation}
The total displacement is
\begin{align}
u&=u_1+u_2\\
&=4.0\,\text{mm}-1.5\,\text{mm}\\
&=\boxed{2.5\,\text{mm}}.
\end{align}
The signs matter because superposition is an algebraic, not arithmetic,
addition.
\section{Worked example 2: equal waves with a phase difference}
Two equal sinusoidal waves have amplitude
\begin{equation}
A=4.0\,\text{mm}
\end{equation}
and phase difference
\begin{equation}
\Delta\phi=\frac{\pi}{3}.
\end{equation}
The resultant amplitude is
\begin{align}
A_R
&=2A\left|\cos\left(\frac{\Delta\phi}{2}\right)\right|\\
&=2(4.0\,\text{mm})\cos\left(\frac{\pi}{6}\right)\\
&=8.0\,\text{mm}\left(\frac{\sqrt{3}}{2}\right)\\
&=\boxed{4\sqrt{3}\,\text{mm}}\\
&\approx\boxed{6.93\,\text{mm}}.
\end{align}
The result lies between zero and the fully constructive value $2A=8.0\,\text{mm}$.
\section{Worked example 3: destructive interference with unequal amplitudes}
Let
\begin{equation}
A_1=5.0\,\text{mm},
\qquad
A_2=3.0\,\text{mm},
\qquad
\Delta\phi=\pi.
\end{equation}
Then
\begin{align}
A_R
&=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\pi}\\
&=\sqrt{(5.0)^2+(3.0)^2-2(5.0)(3.0)}\,\text{mm}\\
&=\sqrt{4}\,\text{mm}\\
&=\boxed{2.0\,\text{mm}}.
\end{align}
The waves interfere destructively, but the cancellation is incomplete because
their amplitudes are unequal.
\section{Common mistakes}
\begin{itemize}
\item \textbf{Mistake:} adding amplitudes without considering sign or phase. Superposition adds the instantaneous disturbances, not merely their positive amplitude magnitudes.
\item \textbf{Mistake:} assuming destructive interference always gives zero. Complete cancellation requires matching amplitudes and the appropriate phase difference.
\item \textbf{Mistake:} thinking two pulses bounce off one another like rigid objects. In an ideal linear system, the component disturbances pass through and the observable overlap is their sum.
\item \textbf{Mistake:} assuming superposition is universal. It is a property of linear wave models and can fail in nonlinear regimes.
\item \textbf{Mistake:} treating a large resultant amplitude as evidence that the component waves have merged permanently. The decomposition into linear components remains valid while the system remains linear.
\end{itemize}
\section{What WM09 adds to the wave-mechanics language}
The first block established how one wave is described. WM09 adds the rule for
combining multiple linear waves:
\begin{equation}
\boxed{u=\sum_i u_i.}
\end{equation}
From this single principle follow the basic ideas of constructive interference,
destructive interference, partial interference, and the temporary overlap of
traveling pulses.
For equal sinusoidal components with phase difference $\Delta\phi$,
\begin{equation}
\boxed{A_R=2A\left|\cos\left(\frac{\Delta\phi}{2}\right)\right|,}
\end{equation}
and for unequal amplitudes,
\begin{equation}
\boxed{A_R=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\Delta\phi}.}
\end{equation}
These results will reappear throughout later wave mechanics, especially in
standing waves, Fourier expansions, modal analysis, acoustics, optics, and
quantum mechanics.
\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{OpenStax165}
Samuel J. Ling, Jeff Sanny, and William Moebs,
\emph{University Physics, Volume 1},
OpenStax, 2016,
Section 16.5, ``Interference of Waves.''
\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 the discussion of the
linearity of the wave equation and superposition of solutions.
\bibitem{MIT803}
Massachusetts Institute of Technology,
\emph{8.03SC Physics III: Vibrations and Waves},
MIT OpenCourseWare, materials on traveling waves, interference, and
superposition.
\end{thebibliography}