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A normal mode of a uniformly bending elastic filament fixed in position and slope at both ends satisfies $W^{\prime \prime \prime \prime }=k^{4}W$. With $\beta =kL>0$, the allowed wavenumbers solve $\cos \beta \cosh \beta =1$. The first root is $4.730040745$ and ${\beta }_n=(n+\frac{1}{2})\pi +O(e^{-(n+1/2)\pi })$ for $n\ge 1$. The apparent root at zero is not an eigenfunction: the zero-eigenvalue cubic polynomial satisfying all four clamped boundary conditions is identically zero. This spectrum differs from that of a filament clamped only at one end.

The elastic energy $U=A\int (W^{\prime \prime }{)}^{2}dx/2$ minimized without forcing gives $W^{\prime \prime \prime \prime }=0$. With fixed $\int W^{2}dx$, stationarity of $U-A{k}^4\int W^{2}dx/2$ gives $W^{\prime \prime \prime \prime }=k^{4}W$: a bending eigenmode. A general admissible shape expands in these normal modes. The natural boundary conditions for a free endpoint are $W^{\prime \prime }(L)=W^{\prime \prime \prime }(L)=0$, while the clamp imposes $W(0)=W^{\prime }(0)=0$.

If the mass and stiffness matrices commute with a permutation representation, each symmetry subspace is invariant under the generalized eigenvalue problem for small oscillations. On an irreducible subspace, Schur lemma forces a commuting operator to be scalar, producing a degenerate normal mode multiplet.

Let $c_s^2=(dP/d\Sigma {)}_0$ be the barotropic closure of a razor-thin disk sound-speed squared and ${\kappa }_r$ the radial epicyclic frequency, with ${\kappa }_r^2=2(2-q_r){\Omega }^2$ for orbital shear parameter $q_r$. For a radial axisymmetric normal mode in a uniform barotropic shearing sheet, let $U_x,U_y$ be velocity amplitudes and $\sigma$ the surface-density amplitude. Linear momentum gives $-i\omega U_x=2\Omega U_y-ik{c}_s^2\sigma /{\Sigma }_0$ and $-i\omega U_y=-{\kappa }_r^2{U}_x/(2\Omega )$; mass conservation gives $\omega \sigma =k{\Sigma }_0{U}_x$. Eliminating the amplitudes for $\omega \ne 0$ proves the displayed dispersion relation. The rotational term is epicyclic restoration; the pressure term is acoustic restoration. Without a perturbed gravitational potential, no self-gravity term belongs in the formula.

Two equal stable density jumps of size $\Delta \rho /2$ at $z=\pm h/2$ lie in the global linear shear flow $U=\Delta Uz/h$. Decaying normal modes haveThe jump conditions for stratified inviscid shear flow give the determinant equationWriting $\alpha =kh/2$, $J=g\Delta \rho h/({\rho }_0\Delta U^2)$ and $\tilde{c}=2c/\Delta U$, this isThe two roots for ${\tilde{c}}^2$ are real; one is negative exactly when $2\alpha /(1+e^{-2\alpha })<J<2\alpha /(1-e^{-2\alpha })$. At large $\alpha$, this narrow band centres on $J=2\alpha$, where the isolated counterpropagating interfacial gravity waves have the same zero laboratory speed. This realizes counterpropagating wave instability.

The temporal frequency of a normal mode. In $\dot{z}=iAz$, a real eigenvalue of $A$ is a signed eigenfrequency; its positive imaginary part after damping gives exponential decay in $e^{i\lambda t}$.

For a stratified parallel shear flow, the gradient Richardson number compares the squared buoyancy frequency with squared vertical shear. The Miles–Howard theorem excludes exponentially growing inviscid normal modes when this ratio is at least $1/4$ everywhere. Where $U^{\prime }=0$, the equivalent criterion is written directly as $N^2\ge (U^{\prime }{)}^2/4$.

A stationary point in a non-monotone velocity profile can support a quarter-power critical layer whose phase-speed correction has positive imaginary part. The resulting normal mode grows at a rate proportional to the square root of the wavenumber. Such unbounded high-frequency amplification demonstrates linear instability of the frozen layer; nonlinear evolution and precise well-posedness consequences require separate analysis.

For a vertical magnetic field in a homogeneous shearing sheet, a horizontal normal mode with growth rate $s$ and vertical Alfvén frequency ${\omega }_a=k{v}_{Az}$ satisfiesThe two roots for $s^2$ are real. For $q>0$, a growing magnetic mode exists precisely when $0<{\omega }_a^2<2q{\Omega }^2$. Magnetic tension is strong enough to couple radially displaced elements, yet weak enough to allow their separation to increase. Ogilvie's lecture on magnetorotational instability derives this relation and its finite-thickness boundary conditions.

A normal mode $e^{i(kx-\omega t)}$ has dispersion relation $\omega =Uk+(c_d-i)k^2+i\mu$. The absolute wavenumber is the saddle where $d\omega /dk=0$. The Green function of the linear complex Ginzburg-Landau equation distinguishes convective hydrodynamic instability from absolute hydrodynamic instability.

Linear instability is growth of small disturbances in the linearized dynamics. An eigenmode with positive temporal growth rate is sufficient to establish it. In a non-normal system, finite transient amplification is a different issue from the existence of an exponentially growing eigenmode. For normal modes written as $e^{i\alpha (x-ct)}$, the temporal rate is $\alpha \mathrm{Im}c$.

For a static smooth ideal magnetohydrodynamics equilibrium with conservative external forces and surface terms eliminated by the boundary conditions, the linear displacement equation is $\rho \ddot{\mathbf{\xi }}=\mathcal{F}\mathbf{\xi }$, where the force operator is a self-adjoint operator in the appropriate integral pairing. Its restoring quadratic form $\delta W=-\frac{1}{2}\int {\mathbf{\xi }}^{\ast }\cdot \mathcal{F}\mathbf{\xi }dV$ is real. Multiplying the normal mode equation by the complex conjugate of the displacement and integrating gives ${\omega }^2\int \rho |\mathbf{\xi }{|}^{2}dV=2\delta W$. Hence squared frequencies are real, and a negative restoring form gives an unstable mode through the variational Rayleigh quotient, subject to the operator domain and spectral hypotheses. Background fluid flow adds terms to this argument; the static hypothesis is essential.

A smooth inviscid stratified parallel flow with gradient Richardson number at least $1/4$ everywhere has no exponentially growing two-dimensional normal modes. Put $a=1/2$ in the power-transformed Taylor–Goldstein energy identity and take its imaginary part:The integral is positive for a nonzero mode when the numerator is nonnegative, forcing $c_i=0$. This is a modal stability theorem, not a prohibition on transient growth. Maslowe's review discusses the theorem and the role of critical layers.

The Orr-Sommerfeld equation governs the wall-normal velocity of normal mode disturbances to an incompressible viscous parallel shear flow. Here $D=d/dy$, ${\kappa }^2={\alpha }^2+{\beta }^2$, and $Re$ is the Reynolds number. It follows by eliminating pressure from the linearized Navier-Stokes equations. At a rigid no-slip wall, $\hat{v}=D\hat{v}=0$. In the inviscid limit it reduces to the Rayleigh equation for inviscid shear flow.

For ${\zeta }_{\perp }{h}_t=-B{h}_{xxxx}$, separated normal modes decay at rates $B{\alpha }_n^4/({\zeta }_{\perp }{L}^4)$. Clamping at zero and vanishing moment and shear at $L$ give $\cos {\alpha }_n\cosh {\alpha }_n=-1$. A mode isThe fourth-order operator is self-adjoint: two integration by parts operations give $\int f{g}^{\prime \prime \prime \prime }=\int f^{\prime \prime }{g}^{\prime \prime }$ under these boundary conditions. Its zero eigenspace is trivial and its eigenfunctions form an orthogonal complete basis. Thus a nonzero projection on the first mode gives long-time decay proportional to ${\varphi }_1(x/L)\exp [-B{\alpha }_1^{4}t/({\zeta }_{\perp }{L}^4)]$, where ${\alpha }_1=1.8751040687\dots$. If this projection vanishes, the lowest excited mode replaces it.

An oscillatory normal mode whose amplitude grows, corresponding to a complex growth rate with positive real part.