Wave Mechanics: ResonanceWaveMechanicsResonance2026-09-12 00:36:032026-09-12 00:36:03Definitionwave mechanicsresonancestanding wavesnormal modesnatural frequencydriving frequencyharmonicsdampingforced oscillationresonance curvestring modes% 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: Resonance}
WM10 showed that a finite wave system can support standing-wave patterns with
fixed nodes and antinodes. Once boundary conditions are imposed, not every
wavelength is allowed. Only certain standing-wave patterns satisfy the
boundaries, and each allowed pattern has its own natural frequency.
WM11 asks the next question:
\begin{center}
\emph{What happens when an external periodic driver tries to force the system to
oscillate?}
\end{center}
The answer is \emph{resonance}. A periodically driven system responds most
strongly when the driving frequency is near one of its natural frequencies.
This connection between natural modes, driving, damping, and large response is
central to strings, air columns, structures, electrical resonators, optical
cavities, and many other wave systems \cite{French1971,Crawford1968,OpenStax156,OpenStax166}.
\section{Natural modes come first}
Consider a string of length $L$ fixed at both ends. The displacement must
vanish at
\begin{equation}
x=0
\qquad\text{and}\qquad
x=L.
\end{equation}
A convenient standing-wave form is
\begin{equation}
u(x,t)=B\sin(kx)\cos(\omega t).
\end{equation}
The first boundary condition is automatically satisfied because
\begin{equation}
\sin(0)=0.
\end{equation}
The second boundary condition requires
\begin{equation}
\sin(kL)=0.
\end{equation}
Therefore
\begin{equation}
kL=n\pi,
\qquad
n=1,2,3,\ldots
\end{equation}
and the allowed wavenumbers are
\begin{equation}
\boxed{k_n=\frac{n\pi}{L}.}
\end{equation}
Using
\begin{equation}
k_n=\frac{2\pi}{\lambda_n},
\end{equation}
we obtain
\begin{equation}
\boxed{\lambda_n=\frac{2L}{n}.}
\end{equation}
If the wave speed is $v$, then WM08 gives
\begin{equation}
f=\frac{v}{\lambda}.
\end{equation}
Hence the allowed natural frequencies are
\begin{equation}
\boxed{f_n=\frac{nv}{2L}=nf_1.}
\end{equation}
The lowest natural frequency is the \emph{fundamental frequency},
\begin{equation}
\boxed{f_1=\frac{v}{2L}.}
\end{equation}
For an ideal uniform string fixed at both ends, the higher natural frequencies
are integer multiples of the fundamental and are commonly called harmonics
\cite{OpenStax166}.
\begin{center}
\includegraphics{WM11_fig01_normal_modes_and_frequencies.png}
\vspace{0.45em}
\textbf{Figure.}
The first three normal modes of a string fixed at both ends. Each allowed
spatial pattern has its own natural frequency: $f_1$, $2f_1$, and $3f_1$.
\end{center}
\section{Normal mode is not the same as resonance}
These two terms are closely related but describe different ideas.
A \emph{normal mode} is an allowed pattern in which the system can oscillate
freely at one of its natural frequencies. It is a property of the system and
its boundary conditions.
\emph{Resonance} describes the response of the system to an external periodic
driver. If the driver frequency $f_d$ is close to a natural frequency $f_n$,
the corresponding mode can respond with a much larger amplitude than it does
when the system is driven far from that frequency.
Thus the logical order is
\begin{equation}
\boxed{\text{boundary conditions}
\;\longrightarrow\;
\text{normal modes}
\;\longrightarrow\;
\text{natural frequencies}
\;\longrightarrow\;
\text{resonant response}.}
\end{equation}
\section{Why repeated driving can build a large response}
A periodic driver supplies a small amount of motion or energy on every cycle.
When the timing is favorable, successive pushes reinforce the motion already
present. The familiar example is a swing: well-timed small pushes can build a
large oscillation.
The same idea applies to wave systems. Suppose a string is driven sinusoidally
at one end. A disturbance travels along the string, reflects, and returns.
At certain driving frequencies, the repeated forcing is synchronized with an
allowed standing-wave pattern. The response then builds strongly.
If the driver is badly detuned from all natural frequencies, successive cycles
are not consistently reinforcing. The response is generally much smaller.
\begin{center}
\includegraphics{WM11_fig04_resonant_vs_detuned_driving.png}
\vspace{0.45em}
\textbf{Figure.}
Schematic comparison of resonant and detuned driving. Near a natural
frequency, successive pushes reinforce the oscillation coherently; off
resonance, the forcing does not remain synchronized with the response.
\end{center}
\section{Driving frequency and natural frequency}
Let
\begin{equation}
f_d
\end{equation}
denote the driving frequency and let
\begin{equation}
f_n
\end{equation}
denote one natural frequency of the system.
The resonance condition is approximately
\begin{equation}
\boxed{f_d\approx f_n.}
\end{equation}
In angular-frequency language,
\begin{equation}
\boxed{\omega_d\approx\omega_n.}
\end{equation}
For a lightly damped system, the strongest response occurs very near the
natural frequency. The exact peak can be shifted slightly by damping,
depending on which response quantity is being measured. At the level of WM11,
the essential physics is that the response becomes large when the drive is
near a natural frequency \cite{OpenStax156,MIT803L3}.
\section{A single mode behaves like a driven oscillator}
A useful way to understand resonance is to focus on one normal mode at a time.
Its time-dependent amplitude can be modeled like a driven oscillator. A
standard one-coordinate model is
\begin{equation}
m\ddot q+b\dot q+\kappa q=F_0\cos(\omega_d t),
\end{equation}
where
\begin{itemize}
\item $q(t)$ is the mode amplitude,
\item $m$ is an effective modal mass,
\item $\kappa$ is an effective restoring coefficient,
\item $b$ represents damping,
\item $F_0$ is the driving-force amplitude,
\item $\omega_d$ is the driving angular frequency.
\end{itemize}
The undamped natural angular frequency is
\begin{equation}
\boxed{\omega_0=\sqrt{\frac{\kappa}{m}}.}
\end{equation}
For steady sinusoidal forcing, the response amplitude has the standard form
\begin{equation}
\boxed{
Q(\omega_d)=
\frac{F_0}
{\sqrt{(\kappa-m\omega_d^2)^2+(b\omega_d)^2}}
}.
\end{equation}
This formula is not derived in WM11; it is included to show mathematically why
a response peak appears near the natural frequency. OpenStax and MIT 8.03
develop the driven damped oscillator in detail \cite{OpenStax156,MIT803L3}.
\section{The resonance curve}
If the steady-state response amplitude is plotted against driving frequency,
a resonance curve appears.
\begin{center}
\includegraphics{WM11_fig02_resonance_curve.png}
\vspace{0.45em}
\textbf{Figure.}
Schematic resonance curves for the same natural frequency with different
amounts of damping. Less damping produces a taller, narrower peak; more
damping produces a lower, broader response.
\end{center}
The plot contains three important ideas.
\begin{enumerate}
\item Far below resonance, the system responds but the amplitude is usually modest.
\item Near resonance, the response amplitude can become large.
\item Far above resonance, the system cannot follow the driver efficiently and the response decreases.
\end{enumerate}
Damping changes the resonance dramatically. With weak damping, the peak is
high and narrow. With stronger damping, the peak is lower and broader
\cite{OpenStax156}.
\section{Real resonance peaks are finite}
An ideal undamped oscillator driven exactly at its natural frequency is a
special mathematical limit. In that limit, continuous coherent driving can
make the amplitude grow without bound in the idealized model.
Real systems always contain some combination of damping, energy leakage,
material loss, radiation, friction, nonlinear effects, or finite driving time.
These effects limit the amplitude.
Therefore, when a real resonance curve is measured, one usually sees a finite
peak with a finite width rather than an infinite spike.
\section{A wave system can have many resonances}
A finite string has many allowed modes, not just one. Each natural mode can
therefore produce its own resonant response.
For a fixed-fixed ideal string,
\begin{equation}
f_n=nf_1.
\end{equation}
Sweeping the driving frequency upward can therefore produce a sequence of
response peaks near
\begin{equation}
f_1,\quad 2f_1,\quad 3f_1,\quad\ldots
\end{equation}
\begin{center}
\includegraphics{WM11_fig03_multiple_mode_resonances.png}
\vspace{0.45em}
\textbf{Figure.}
Schematic frequency sweep of a finite string. Large responses occur near the
natural frequencies $f_1$, $f_2$, and $f_3$, with a different standing-wave
mode associated with each peak.
\end{center}
This is one of the most important experimental signatures of a bounded wave
system: a frequency sweep reveals a spectrum of resonances.
\section{Fundamental, harmonics, and overtones}
For an ideal string fixed at both ends,
\begin{equation}
f_n=nf_1.
\end{equation}
The terminology is:
\begin{itemize}
\item $f_1$: fundamental frequency or first harmonic,
\item $f_2=2f_1$: second harmonic and first overtone,
\item $f_3=3f_1$: third harmonic and second overtone,
\item and so on.
\end{itemize}
The word \emph{overtone} counts frequencies above the fundamental, whereas the
word \emph{harmonic} counts the fundamental itself as the first harmonic.
This distinction prevents a common indexing error.
Not every physical system has natural frequencies that are exact integer
multiples of the fundamental. The harmonic relation is a special consequence
of the ideal string model and its boundary conditions.
\section{Resonance does not create energy}
A resonant system can have a large amplitude even when the applied periodic
force is comparatively small. This does not violate energy conservation.
The driver performs work on the system repeatedly. Near resonance, the
forcing is timed so that energy is transferred efficiently into the oscillation.
Damping and other losses remove energy at the same time. In steady state, the
average input from the driver balances the average losses.
A quantitative treatment of wave energy, power, and energy flux is deferred to
the later energy section of the Wave Mechanics series.
\section{Worked example 1: find the resonant frequencies of a string}
A string of length
\begin{equation}
L=1.20\,\text{m}
\end{equation}
supports waves with speed
\begin{equation}
v=180\,\text{m/s}.
\end{equation}
The fundamental frequency is
\begin{align}
f_1&=\frac{v}{2L}\\
&=\frac{180}{2(1.20)}\,\text{Hz}\\
&=\boxed{75\,\text{Hz}}.
\end{align}
Therefore
\begin{align}
f_2&=\boxed{150\,\text{Hz}},\\
f_3&=\boxed{225\,\text{Hz}},\\
f_4&=\boxed{300\,\text{Hz}}.
\end{align}
If the string is driven near $225\,\text{Hz}$, the third mode is expected to
respond strongly.
\section{Worked example 2: identify the mode from a driving frequency}
A fixed-fixed string has fundamental frequency
\begin{equation}
f_1=40\,\text{Hz}.
\end{equation}
A driver operates at
\begin{equation}
f_d=160\,\text{Hz}.
\end{equation}
For an ideal string,
\begin{equation}
f_n=nf_1.
\end{equation}
Hence
\begin{align}
n&=\frac{f_d}{f_1}\\
&=\frac{160}{40}\\
&=\boxed{4}.
\end{align}
The driver is tuned to the fourth harmonic, so the fourth standing-wave mode
is expected to be strongly excited.
\section{Worked example 3: detuning}
Suppose one resonance occurs at
\begin{equation}
f_n=100\,\text{Hz}.
\end{equation}
Compare two driving frequencies:
\begin{equation}
f_{d1}=99\,\text{Hz},
\qquad
f_{d2}=70\,\text{Hz}.
\end{equation}
Their detunings are
\begin{align}
|f_{d1}-f_n|&=1\,\text{Hz},\\
|f_{d2}-f_n|&=30\,\text{Hz}.
\end{align}
All else being equal, the $99\,\text{Hz}$ drive lies much closer to the natural
frequency and is therefore expected to produce the larger steady response.
The exact amplitudes cannot be determined without additional information about
damping and coupling strength.
\section{Common mistakes}
\begin{itemize}
\item \textbf{Mistake:} using \emph{resonance} and \emph{normal mode} as synonyms. A normal mode is an allowed free pattern; resonance is a forced response near a natural frequency.
\item \textbf{Mistake:} assuming the driving frequency changes the natural frequency. The driver selects how strongly existing modes are excited; the natural frequencies are properties of the system.
\item \textbf{Mistake:} assuming every frequency produces a standing wave with large amplitude. Large standing-wave response occurs near allowed resonances.
\item \textbf{Mistake:} assuming the resonance amplitude is infinite in a real system. Damping and other losses limit the response.
\item \textbf{Mistake:} confusing harmonic number with overtone number. The second harmonic is the first overtone.
\item \textbf{Mistake:} assuming all systems have $f_n=nf_1$. That relation holds for the ideal fixed-fixed string but is not universal.
\end{itemize}
\section{What WM11 adds to the wave-mechanics language}
WM10 established standing waves. WM11 adds the distinction among three related
ideas:
\begin{equation}
\boxed{\text{mode shape}\quad\leftrightarrow\quad\text{natural frequency}\quad\leftrightarrow\quad\text{resonant response}.}
\end{equation}
For a fixed-fixed ideal string,
\begin{equation}
\boxed{k_n=\frac{n\pi}{L}},
\qquad
\boxed{\lambda_n=\frac{2L}{n}},
\qquad
\boxed{f_n=\frac{nv}{2L}}.
\end{equation}
An external driver produces a strong response when
\begin{equation}
\boxed{f_d\approx f_n.}
\end{equation}
Damping controls how tall and how broad that resonant response becomes. These
ideas prepare the way for modal expansion, Fourier methods, and the study of
energy flow in wave systems.
\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{OpenStax156}
William Moebs, Samuel J. Ling, and Jeff Sanny,
\emph{University Physics, Volume 1},
OpenStax, 2016,
Section 15.6, ``Forced Oscillations.''
\bibitem{OpenStax166}
William Moebs, Samuel J. Ling, and Jeff Sanny,
\emph{University Physics, Volume 1},
OpenStax, 2016,
Section 16.6, ``Standing Waves and Resonance.''
\bibitem{Feynman49}
Richard P. Feynman, Robert B. Leighton, and Matthew Sands,
\emph{The Feynman Lectures on Physics, Volume I},
Chapter 49, ``Modes.''
\bibitem{MIT803L3}
Massachusetts Institute of Technology,
\emph{8.03SC Physics III: Vibrations and Waves},
Lecture 3, ``Driven Oscillators, Transient Phenomena, Resonance,''
Fall 2016, MIT OpenCourseWare.
\bibitem{MIT803L9}
Massachusetts Institute of Technology,
\emph{8.03SC Physics III: Vibrations and Waves},
Lecture 9, ``Wave Equation, Standing Waves, Fourier Series,''
Fall 2016, MIT OpenCourseWare.
\end{thebibliography}