Normal mode
= Normal mode
{wiki}
= Eigenmode
{synonym}
A normal mode, also called an <eigenmode>, is a pattern of free <oscillation> in a linear system whose relative displacements stay fixed while a single temporal factor changes their size and sign. For an undamped stable vibrational mode,
$$
u(x,t)=\phi(x)q(t),\qquad q(t)=A\cos(\omega t+\varphi).
$$
Here $\phi$ is the <mode shape>, $A$ sets the <amplitude>, $\omega$ is the <eigenfrequency> expressed as an <angular frequency>, and $\varphi$ is the <oscillation phase>. Every snapshot is a <scalar multiple> of the same <mode shape>; neighbouring regions may move in opposite directions. <Normal modes of a double pendulum> show that <normal modes> also describe coupled discrete systems, beyond strings. Their shapes are <eigenvectors>, rather than spatial <eigenfunctions>. https://www.feynmanlectures.caltech.edu/I_49.html[Feynman, Leighton and Sands, The Feynman Lectures on Physics, Volume I, Chapter 49]{ref}
For the ideal <wave equation on a string>, take a uniform string of length $L$ with constant <tension> $\tau$ and <linear mass density> $\mu$. Assume small transverse slope, negligible bending stiffness, no <damping>, and fixed endpoints. Its <wave speed> is $c=\sqrt{\tau/\mu}$, and the nonzero <normal modes> are
$$
u_n(x,t)=A\sin\left(\frac{n\pi x}{L}\right)\cos(\omega_nt+\varphi),\qquad \omega_n=\frac{n\pi c}{L},\qquad f_n=\frac{\omega_n}{2\pi}=\frac{nc}{2L},\qquad n=1,2,3,\ldots.
$$
The <angular frequency> $\omega_n$ is in radians per second; the <frequency> $f_n$ is in hertz. The <wavelength> is $2L/n$. The <mode shape> has $n$ loops separated by <nodes of a standing wave>, including both fixed endpoints. https://openstax.org/books/university-physics-volume-1/pages/16-6-standing-waves-and-resonance[OpenStax, University Physics Volume 1, Section 16.6]{ref}
As a worked example, choose $L=1\,\mathrm m$, $\tau=1\,\mathrm N$, and $\mu=1\,\mathrm{kg\,m^{-1}}$. Then $c=1\,\mathrm{m\,s^{-1}}$ and $f_1=0.5\,\mathrm{Hz}$, $f_2=1\,\mathrm{Hz}$, $f_3=1.5\,\mathrm{Hz}$. Their <oscillation periods> are $T_1=2\,\mathrm s$, $T_2=1\,\mathrm s$, and $T_3=2/3\,\mathrm s$. These are deliberately simple model parameters, rather than measurements of a musical instrument. The plot divides <displacement> by $A$ so that the three <mode shapes> can be compared, and uses each mode's own <oscillation period> for its snapshots.
\Image[/standing-string-normal-modes.png]
{title=First three ideal fixed-string normal modes at fractions of their own periods; black dots mark permanent nodes}
For $\varphi=0$, at $t=T_n/4$ every point has zero <displacement>, but most points have nonzero <velocity>. Only zeros of the <mode shape> are permanent <nodes of a standing wave>. At $t=T_n/2$ the original <mode shape> has reversed sign, and at $t=T_n$ it returns. The synchronized conversion between <kinetic energy> and <potential energy> is shown in <normal-mode energy exchange>.
The short derivation uses <separation of variables>: substituting $u=\phi q$ into the <wave equation> gives $\phi''+k^2\phi=0$ and $q''+c^2k^2q=0$. The fixed-end <boundary conditions> remove the cosine spatial term and require $\sin(kL)=0$, hence $k=n\pi/L$ and $\omega_n=ck$. Different geometries and <boundary conditions> give different <eigenfunctions>; a sine <mode shape> is a property of this uniform fixed string, not a universal form of an <eigenmode>.
The usefulness is that a complicated vibration becomes a collection of <harmonic oscillators>. For sufficiently regular fixed-end initial data, the <Fourier sine series> gives
$$
u(x,t)=\sum_{n=1}^{\infty}\sin\left(\frac{n\pi x}{L}\right)\left[a_n\cos(\omega_nt)+b_n\sin(\omega_nt)\right].
$$
The initial <displacement> and <velocity> determine $a_n$ and $b_n$. This is the <superposition principle> in a <Fourier sine basis>, and <modal energy distribution> measures how strongly each <normal mode> is excited. The striking simplification is that each component keeps its identity while their sum can continually change shape. https://www.feynmanlectures.caltech.edu/I_50.html[Feynman, Leighton and Sands, The Feynman Lectures on Physics, Volume I, Section 50-2]{ref}
These independent <normal modes> belong to the linear model. At larger slopes, changing <tension> introduces nonlinear effects and can couple the modes. A <linearly damped string> still admits a <Fourier sine basis>, but its temporal factors decay and need not oscillate; its <energy dissipation identity for a linearly damped string> replaces <conservation of energy>. More general <damping> need not leave undamped <mode shapes> independent.