Wave Mechanics: Partial Derivatives for WavesWaveMechanicsPartialDerivativesForWaves2026-09-12 02:28:572026-09-12 02:28:57Topicwave mechanicspartial derivativefieldspatial slopetemporal rate of changecurvatureaccelerationtraveling wavechain rulewave equationpartial differential equation% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Wave Mechanics: Partial Derivatives for Waves}
A wave is a field that depends on more than one independent variable. In one
spatial dimension we write
\begin{equation}
u=u(x,t),
\end{equation}
where $x$ specifies position and $t$ specifies time. Earlier articles used
this notation geometrically and physically: a spatial snapshot is obtained by
holding time fixed, while a time history is obtained by holding position
fixed. WM12 also introduced the free-end condition
\begin{equation}
\frac{\partial u}{\partial x}=0,
\end{equation}
but deliberately postponed a systematic treatment of the derivative notation.
WM13 develops that calculus. The central idea is simple:
\begin{equation}
\boxed{\text{a partial derivative changes one independent variable while the others are held fixed}.}
\end{equation}
This is the mathematical language needed to describe local wave slope, local
velocity, curvature, acceleration, and eventually the wave equation itself.
Standard multivariable-calculus treatments emphasize that partial derivatives
are ordinary rates of change taken one variable at a time, while standard wave
texts interpret the resulting derivatives physically as slope, velocity,
curvature, and acceleration \cite{MIT1802Partial,OpenStax162,French1971,Crawford1968}.
\section{From one-variable derivatives to a field}
For an ordinary function
\begin{equation}
y=f(x),
\end{equation}
there is only one independent variable. The derivative
\begin{equation}
\frac{dy}{dx}
\end{equation}
asks how $y$ changes when $x$ changes.
For a wave field
\begin{equation}
u=u(x,t),
\end{equation}
there are two independent variables. We can ask two different local questions:
\begin{itemize}
\item How does $u$ change from one nearby position to another at the same time?
\item How does $u$ change from one nearby time to another at the same position?
\end{itemize}
These questions lead to different derivatives.
\begin{center}
\includegraphics{WM13_fig01_partial_derivative_slices.png}
\vspace{0.45em}
\textbf{Figure.}
The same field $u(x,t)$ can be sliced in two ways. Holding $t$ fixed produces a
spatial profile whose local slope is $\partial u/\partial x$. Holding $x$
fixed produces a time history whose local slope is $\partial u/\partial t$.
\end{center}
\section{The spatial partial derivative}
At an event $(x_0,t_0)$, hold the time fixed at $t=t_0$ and compare the field at
two nearby positions:
\begin{equation}
\frac{u(x_0+\Delta x,t_0)-u(x_0,t_0)}{\Delta x}.
\end{equation}
As $\Delta x$ approaches zero, this difference quotient approaches the spatial
partial derivative:
\begin{equation}
\boxed{
\frac{\partial u}{\partial x}(x_0,t_0)
=\lim_{\Delta x\rightarrow 0}
\frac{u(x_0+\Delta x,t_0)-u(x_0,t_0)}{\Delta x}.
}
\end{equation}
The time $t_0$ does not change during this limiting process.
Geometrically,
\begin{equation}
\boxed{\frac{\partial u}{\partial x}=\text{local spatial slope of the wave profile}.}
\end{equation}
If $u$ is transverse displacement measured in meters and $x$ is measured in
meters, then
\begin{equation}
\left[\frac{\partial u}{\partial x}\right]
=\frac{\text{m}}{\text{m}}=1.
\end{equation}
The slope is dimensionless in that particular application. For other wave
variables, the units depend on the units of the field itself.
\section{The temporal partial derivative}
Now hold position fixed at $x=x_0$ and compare the field at two nearby times:
\begin{equation}
\frac{u(x_0,t_0+\Delta t)-u(x_0,t_0)}{\Delta t}.
\end{equation}
Taking the limit gives
\begin{equation}
\boxed{
\frac{\partial u}{\partial t}(x_0,t_0)
=\lim_{\Delta t\rightarrow 0}
\frac{u(x_0,t_0+\Delta t)-u(x_0,t_0)}{\Delta t}.
}
\end{equation}
The position $x_0$ does not change during this limiting process.
For a transverse string displacement,
\begin{equation}
\boxed{\frac{\partial u}{\partial t}=\text{transverse velocity of the string element at fixed }x.}
\end{equation}
If $u$ is measured in meters,
\begin{equation}
\left[\frac{\partial u}{\partial t}\right]
=\frac{\text{m}}{\text{s}}.
\end{equation}
This is not the same as the propagation speed of the wave. The wave may move
through the medium with speed $c$ while individual material points move up and
down with velocity $\partial u/\partial t$.
\begin{center}
\includegraphics{WM13_fig02_difference_quotients.png}
\vspace{0.45em}
\textbf{Figure.}
The spatial and temporal difference quotients approach the same event from two
different directions in the $(x,t)$ domain. A partial derivative varies one
independent variable at a time.
\end{center}
\section{Notation}
Several notations are common:
\begin{equation}
\frac{\partial u}{\partial x},
\qquad
u_x,
\end{equation}
and
\begin{equation}
\frac{\partial u}{\partial t},
\qquad
u_t.
\end{equation}
The compact subscript notation is especially useful when expressions become
long. For example,
\begin{equation}
u_{xx}=\frac{\partial^2u}{\partial x^2},
\qquad
u_{tt}=\frac{\partial^2u}{\partial t^2}.
\end{equation}
In this series, both notations will be used. The fraction-like notation often
makes the physical variable being held fixed easier to see, while the compact
notation keeps wave equations readable.
\section{Second partial derivatives}
The first spatial derivative gives slope. Differentiate again with respect to
position:
\begin{equation}
\boxed{u_{xx}=\frac{\partial^2u}{\partial x^2}.}
\end{equation}
This measures how rapidly the slope changes with position. In one-dimensional
wave problems it is a measure of local curvature. OpenStax uses this
interpretation directly when developing the linear wave equation
\cite{OpenStax162}.
For a string displacement,
\begin{equation}
[u_{xx}]=\frac{\text{m}}{\text{m}^2}=\frac{1}{\text{m}}.
\end{equation}
Likewise, differentiate the temporal derivative again:
\begin{equation}
\boxed{u_{tt}=\frac{\partial^2u}{\partial t^2}.}
\end{equation}
For transverse displacement,
\begin{equation}
\boxed{u_{tt}=\text{transverse acceleration of the material point at fixed }x}
\end{equation}
with units
\begin{equation}
[u_{tt}]=\frac{\text{m}}{\text{s}^2}.
\end{equation}
This pair,
\begin{equation}
\boxed{u_{xx}\leftrightarrow\text{spatial curvature}},
\qquad
\boxed{u_{tt}\leftrightarrow\text{local acceleration}},
\end{equation}
is the essential mathematical structure behind the one-dimensional wave
equation.
\section{Example: derivatives of a sinusoidal traveling wave}
Consider the right-moving wave
\begin{equation}
u(x,t)=A\cos(kx-\omega t+\phi).
\end{equation}
Define the phase
\begin{equation}
\theta=kx-\omega t+\phi.
\end{equation}
Then
\begin{equation}
u=A\cos\theta.
\end{equation}
When differentiating with respect to $x$, time is held fixed. The chain rule
gives
\begin{align}
u_x
&=\frac{\partial}{\partial x}\left(A\cos\theta\right)\\
&=-A\sin\theta\frac{\partial\theta}{\partial x}\\
&=-Ak\sin\theta.
\end{align}
Differentiate once more:
\begin{align}
u_{xx}
&=-Ak\cos\theta\frac{\partial\theta}{\partial x}\\
&=-Ak^2\cos\theta.
\end{align}
Since
\begin{equation}
u=A\cos\theta,
\end{equation}
we obtain
\begin{equation}
\boxed{u_{xx}=-k^2u.}
\end{equation}
Now differentiate with respect to time while holding $x$ fixed:
\begin{align}
u_t
&=\frac{\partial}{\partial t}\left(A\cos\theta\right)\\
&=-A\sin\theta\frac{\partial\theta}{\partial t}\\
&=A\omega\sin\theta.
\end{align}
A second time derivative gives
\begin{align}
u_{tt}
&=A\omega\cos\theta\frac{\partial\theta}{\partial t}\\
&=-A\omega^2\cos\theta.
\end{align}
Thus
\begin{equation}
\boxed{u_{tt}=-\omega^2u.}
\end{equation}
The two second-derivative identities are therefore
\begin{equation}
\boxed{u_{xx}=-k^2u,}
\qquad
\boxed{u_{tt}=-\omega^2u.}
\end{equation}
\begin{center}
\includegraphics{WM13_fig03_spatial_derivative_family.png}
\vspace{0.45em}
\textbf{Figure.}
For a sinusoidal spatial profile, the first spatial derivative is shifted by a
quarter cycle, while the second spatial derivative is the negative of the
original shape multiplied by $k^2$.
\end{center}
\section{A first glimpse of the wave equation}
For the sinusoidal traveling wave,
\begin{equation}
u_{tt}=-\omega^2u
\end{equation}
and
\begin{equation}
u_{xx}=-k^2u.
\end{equation}
Eliminate $u$ between them:
\begin{equation}
u_{tt}=\frac{\omega^2}{k^2}u_{xx}.
\end{equation}
WM08 established
\begin{equation}
c=\frac{\omega}{k}.
\end{equation}
Therefore
\begin{equation}
\boxed{u_{tt}=c^2u_{xx}.}
\end{equation}
This is the one-dimensional linear wave equation.
At this point, however, we have only shown that a sinusoidal traveling wave has
this derivative relationship. We have \emph{not yet derived the wave equation
from the mechanics of a physical medium}. That distinction matters. The
later string-dynamics article will use Newton's second law and tension to show
why a stretched string obeys this equation and why
\begin{equation}
c=\sqrt{\frac{T}{\mu}}.
\end{equation}
The present article supplies the calculus needed for that derivation.
\section{The result is not restricted to sinusoids}
Return to the general right-moving disturbance from WM06:
\begin{equation}
u(x,t)=F(x-ct).
\end{equation}
Let
\begin{equation}
\xi=x-ct.
\end{equation}
Then
\begin{equation}
u=F(\xi).
\end{equation}
The chain rule gives
\begin{align}
u_x
&=F'(\xi)\frac{\partial\xi}{\partial x}\\
&=F'(\xi),
\end{align}
while
\begin{align}
u_t
&=F'(\xi)\frac{\partial\xi}{\partial t}\\
&=-cF'(\xi).
\end{align}
Thus
\begin{equation}
\boxed{u_t=-c\,u_x}
\end{equation}
for any sufficiently smooth right-moving disturbance of the form $F(x-ct)$.
Differentiate again:
\begin{equation}
u_{xx}=F''(\xi)
\end{equation}
and
\begin{equation}
u_{tt}=c^2F''(\xi).
\end{equation}
Therefore
\begin{equation}
\boxed{u_{tt}=c^2u_{xx}.}
\end{equation}
The same second-order relationship also holds for a left-moving disturbance
$G(x+ct)$. This explains why the wave equation naturally accommodates both
propagation directions. OpenStax makes the same connection between translating
wave functions and the linear wave equation \cite{OpenStax162}.
\begin{center}
\includegraphics{WM13_fig04_traveling_wave_slope_rate.png}
\vspace{0.45em}
\textbf{Figure.}
For a right-moving translating profile $F(x-ct)$, the local temporal change and
local spatial slope are related by $u_t=-c\,u_x$. Differentiating again gives
$u_{tt}=c^2u_{xx}$.
\end{center}
\section{Partial derivative versus total derivative}
The notation
\begin{equation}
\frac{\partial u}{\partial t}
\end{equation}
means that $x$ is held fixed while time changes. This is appropriate for a
sensor mounted at one location or for a particular material element on an ideal
string labeled by its equilibrium coordinate $x$.
But suppose instead that we follow a moving observation point $x=x(t)$. Then
the measured quantity is
\begin{equation}
u(x(t),t).
\end{equation}
Its ordinary time derivative requires the multivariable chain rule:
\begin{equation}
\boxed{
\frac{du}{dt}
=\frac{\partial u}{\partial t}
+\frac{dx}{dt}\frac{\partial u}{\partial x}.
}
\end{equation}
The first term is the local time change at fixed position. The second appears
because the observer is moving through a spatially varying field.
This distinction becomes important in fluid mechanics, electromagnetism,
continuum mechanics, and transport theory. For the present Wave Mechanics
series, the key lesson is simply that $\partial/\partial t$ means \emph{fixed
spatial coordinate}, whereas $d/dt$ may describe a path through the $(x,t)$
domain.
\section{Mixed partial derivatives}
A field can also be differentiated once with respect to each independent
variable:
\begin{equation}
u_{xt}
=\frac{\partial}{\partial t}\left(\frac{\partial u}{\partial x}\right),
\end{equation}
or
\begin{equation}
u_{tx}
=\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial t}\right).
\end{equation}
For sufficiently smooth functions, these are equal:
\begin{equation}
\boxed{u_{xt}=u_{tx}.}
\end{equation}
This result is often called equality of mixed partial derivatives or Clairaut's
theorem. Mixed derivatives are not needed for the basic one-dimensional wave
equation, but they appear frequently in more advanced field theories and
coordinate transformations. Standard multivariable-calculus treatments cover
these derivative rules in detail \cite{MIT1802Partial}.
\section{Worked example 1: spatial and temporal derivatives}
Consider
\begin{equation}
u(x,t)=0.020\,\text{m}\cos(4x-10t),
\end{equation}
with $x$ in meters and $t$ in seconds.
The spatial derivative is
\begin{equation}
u_x=-0.080\sin(4x-10t).
\end{equation}
The units are dimensionless because displacement has units of meters and $x$
has units of meters.
The temporal derivative is
\begin{equation}
u_t=0.200\,\text{m/s}\sin(4x-10t).
\end{equation}
At a given event, $u_x$ tells us the local slope of the string while $u_t$ tells
us the local transverse velocity of that material point.
\section{Worked example 2: second derivatives}
For the same wave,
\begin{equation}
u_{xx}=-16u
\end{equation}
and
\begin{equation}
u_{tt}=-100u.
\end{equation}
Therefore
\begin{equation}
u_{tt}=\frac{100}{16}u_{xx}.
\end{equation}
The propagation speed is
\begin{equation}
c=\frac{\omega}{k}=\frac{10}{4}=2.5\,\text{m/s},
\end{equation}
so
\begin{equation}
c^2=6.25\,\text{m}^2/\text{s}^2.
\end{equation}
Thus
\begin{equation}
\boxed{u_{tt}=c^2u_{xx}.}
\end{equation}
\section{Worked example 3: a non-sinusoidal pulse}
Let
\begin{equation}
u(x,t)=\exp\left[-(x-3t)^2\right].
\end{equation}
This has the translating form
\begin{equation}
F(x-ct)
\end{equation}
with
\begin{equation}
c=3.
\end{equation}
Without carrying out the full algebra, the chain-rule result immediately gives
\begin{equation}
\boxed{u_t=-3u_x}
\end{equation}
and
\begin{equation}
\boxed{u_{tt}=9u_{xx}.}
\end{equation}
This is an important conceptual point: the derivative relationship is a
property of translating waveforms, not only of sine and cosine functions.
\section{Common mistakes}
\begin{itemize}
\item \textbf{Mistake:} differentiating both $x$ and $t$ at once. A partial derivative changes one independent variable while holding the others fixed.
\item \textbf{Mistake:} interpreting $u_t$ as the propagation speed of the wave. For a string, $u_t$ is the transverse velocity of a material point; the propagation speed is $c$.
\item \textbf{Mistake:} treating $u_x$ as a time derivative because the wave is moving. $u_x$ is the slope of a spatial snapshot at fixed time.
\item \textbf{Mistake:} forgetting the chain rule when differentiating $\cos(kx-\omega t+\phi)$.
\item \textbf{Mistake:} missing the minus sign in $u_{xx}=-k^2u$ or $u_{tt}=-\omega^2u$.
\item \textbf{Mistake:} assuming the appearance of $u_{tt}=c^2u_{xx}$ here is already a complete physical derivation of the string wave equation. WM13 verifies the differential relationship for translating waves; the later mechanics derivation explains why a real stretched string obeys it.
\end{itemize}
\section{What WM13 adds to the wave-mechanics language}
WM13 turns the graphical ideas of snapshot and time history into calculus.
At fixed time,
\begin{equation}
\boxed{u_x=\frac{\partial u}{\partial x}=\text{spatial slope}}
\end{equation}
and
\begin{equation}
\boxed{u_{xx}=\frac{\partial^2u}{\partial x^2}=\text{spatial curvature measure}.}
\end{equation}
At fixed position,
\begin{equation}
\boxed{u_t=\frac{\partial u}{\partial t}=\text{local rate of field change}}
\end{equation}
and, for string displacement,
\begin{equation}
\boxed{u_{tt}=\frac{\partial^2u}{\partial t^2}=\text{local transverse acceleration}.}
\end{equation}
For a translating wave $F(x-ct)$,
\begin{equation}
\boxed{u_t=-c\,u_x}
\end{equation}
and
\begin{equation}
\boxed{u_{tt}=c^2u_{xx}.}
\end{equation}
These tools prepare the way for the physical derivation of the one-dimensional
string wave equation from force balance and Newton's second law.
\section{References}
\begin{thebibliography}{9}
\bibitem{MIT1802Partial}
Massachusetts Institute of Technology,
\emph{18.02SC Multivariable Calculus},
Unit 2, ``Partial Derivatives,''
MIT OpenCourseWare.
\bibitem{OpenStax162}
William Moebs, Samuel J. Ling, and Jeff Sanny,
\emph{University Physics, Volume 1},
OpenStax, 2016,
Section 16.2, ``Mathematics of Waves,'' especially the treatment of partial
derivatives, slope, acceleration, curvature, and the linear wave equation.
\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{MIT803Textbook}
Massachusetts Institute of Technology,
\emph{8.03SC Physics III: Vibrations and Waves - The Physics of Waves},
Fall 2016, MIT OpenCourseWare,
sections introducing the one-dimensional wave equation and traveling-wave
solutions.
\end{thebibliography}