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For pressure-dominated linear cosmological density perturbations, ${\ddot{\delta }}_k+2H{\dot{\delta }}_k+{\omega }^2{\delta }_k=0$. Writing ${\delta }_k=\chi /a$ gives $\ddot{\chi }+({\omega }^2-\ddot{a}/a)\chi =0$. In the WKB approximation, the oscillation amplitude is $1/(a\sqrt{\omega })$. For constant sound speed it is proportional to $a^{-1/2}$; the local frozen-frequency amplitude damping time $H^{-1}$ does not imply a universal global envelope $a^{-1}$.

In a thin constant-density disk with free surfaces $z=\pm H$, neglect slow radial gradients and let perturbations vary as $e^{ikr-i\omega t}$. With $h=p^{\prime }/\rho$, the local inviscid equations are $-i\omega u-2\Omega v=-ikh$, $-i\omega v+{\kappa }^{2}u/(2\Omega )=0$, $-i\omega w=-h_z$ and $iku+w_z=0$. Eliminating velocity gives $h_{zz}=k^2{\omega }^{2}h/({\omega }^2-{\kappa }^2)$. This local WKB approximation assumes $H/r\ll 1$ and $|kr|\gg 1$ and has both surface and inertial wave branches.

// Target: astrophysics.bigb

An oscillator with frequency $\omega (t)=e^{-\epsilon t}$ leaves the WKB approximation when $|{\omega }_t|/{\omega }^2=\epsilon e^{\epsilon t}$ becomes order one. The shifted variable $T=\epsilon t-\log ({\epsilon }^{-1})$ changes its leading undamped equation to $Y_{TT}+e^{-2T}Y=0$. With $Z=e^{-T}$ this is the order-zero Bessel differential equation. Large-$Z$ Bessel functions match the earlier oscillations, while the small-$Z$ constant and logarithmic solutions become a constant and a linear function of late time.

For a particle of fixed energy $E$, a classically allowed region has positive classical kinetic energy $E-V(x)$. For a constant potential energy, the stationary wavefunction is a combination of $e^{ikx}$ and $e^{-ikx}$, where $k=\sqrt{2m(E-V)}/\hslash$ is the wavenumber. In a varying potential energy this local value underlies the WKB approximation away from turning points.

After the logarithmic radial coordinate $r=e^x$, the Langer replacement $l(l+1)\mapsto (l+1/2{)}^2$ turns the radial Coulomb problem into a two-turning-point WKB approximation problem. Its action integral giveswhich in this Coulomb case agrees with the exact dependence on the principal quantum number.

For $S<0$ and constant positive buoyancy frequency, the critical height is $U_0/|S|$. A stationary upward ray approaches it with vertical group velocity proportional to $U^2$, requiring infinite ideal ray travel time. Large gradient Richardson number can preserve the short-wavelength WKB approximation, but parcel-displacement amplitude grows and eventually invalidates linearity.

A level where the squared vertical wavenumber of an internal gravity wave touches zero quadratically without changing sign. The ordinary WKB approximation fails there; unlike a simple sign-changing turning point, this does not itself provide an evanescent region on one side. Finite-wavelength analysis determines how solutions connect across or meet a boundary at the level.

The rapidly varying real phase of a short-wavelength field written $a(\mathbf{r},t)e^{iS(\mathbf{r},t)}$. Its derivatives define the local wavevector $\mathbf{k}=\nabla S$ and angular frequency $\omega =-{\partial }_{t}S$. The WKB approximation assumes that $a$ and background parameters vary little across one wavelength.

This nonlinear ordinary differential equation governs the amplitude of a complex oscillator solution. Independent real solutions of $y^{\prime \prime }+p(x)y=0$ can be combined into a nonvanishing complex solution; its modulus satisfies the equation with $c$ equal to the squared Wronskian. A real positive amplitude and a conserved phase flux give an exact representation, not merely a WKB approximation.

Substitution of $y=A{e}^{i{ϵ}^{-1}\int \sigma dX}$ into ${ϵ}^2{y}^{\prime \prime }+k^{2}y=0$ gives the displayed real amplitude equation and conserved imaginary-part flux. Conversely these equations produce an exact complex solution wherever $A\ne 0$. The WKB approximation is obtained by expanding this exact system where $|k|$ stays positive.

For $y_{tt}+e^{-2\epsilon t}(\epsilon y_t+y)=0$, $y(0)=0$, $y_t(0)=1$, set $r=e^{-2\epsilon t}/2$. The exact equation becomes $r{y}_{rr}+(1-r)y_r+y/(2{\epsilon }^2)=0$. Let $m(t)=M(-1/(2{\epsilon }^2),1,r)$, using the Kummer function. It is regular at $r=0$ and tends to one. The Wronskian $W=m{y}_t-m_{t}y$ obeys $W_t=-\epsilon e^{-2\epsilon t}W$, with $W(0)=M(-1/(2{\epsilon }^2),1,1/2)$. Integration gives the displayed exact late slope. The other local solution grows at most logarithmically in $r$, so $m_{t}y\to 0$ and $W\to y_t$. This provides a check on a WKB approximation even near phases where its leading predicted slope vanishes.

For a real scalar wave equation $E^{\prime \prime }+k^{2}E=0$, the next order of the WKB approximation gives $2k{a}^{\prime }+k^{\prime }a=0$. Thus $a=A/\sqrt{k}$, and the leading Wronskian flux of a right-travelling component is $|A{|}^2$ independently of the local wavenumber.

The Geiger-Nuttall law expresses the extreme sensitivity of an alpha decay half-life to its Q value. In a restricted range of heavy nuclei, a useful local form is ${\log }_{10}[T_{1/2}/T_{\ast }]\simeq a{Z}_d/\sqrt{Q_{\alpha }}+b$, where $Z_d$ is the daughter atomic number, and the constants depend on the chosen units and nuclear family. It is named for Hans Geiger and John Mitchell Nuttall.

The Coulomb barrier outside the daughter has $V(r)=2Z_d\alpha /r$ in natural units. Its outer turning point is $r_t=2Z_d\alpha /Q_{\alpha }$. The WKB approximation gives a penetration probabilitywhere $\mu$ is the daughter-alpha reduced mass. The leading large-barrier exponent is $2\pi (2Z_d\alpha )\sqrt{\mu /(2Q_{\alpha })}$, producing the approximate $Z_d/\sqrt{Q_{\alpha }}$ dependence. Finite nuclear size modifies the exponent and the slowly varying intercept. The nuclear decay constant also contains an assault frequency and an alpha-cluster formation probability. Extra orbital angular momentum raises the barrier further, and odd-nucleon structure can introduce substantial hindrance. Thus extrapolation from even-even decays gives an order-of-magnitude reference rather than a precision prediction for an odd-odd nucleus.

A periodic spatial internal-wave ray toward which neighbouring ray paths converge after reflection in a confined stratified basin. A fixed point of an internal-wave ray return map is attracting when its derivative has magnitude below one. Viscous attenuation of an internal-wave beam and internal-wave breaking regularize the ideal concentration of energy. A spatial ray construction does not remove turning-level limitations of the WKB approximation.

A smooth invertible change of variables in $u_{xx}-f(x)u=0$ introduces a first derivative in time. Rescaling by ${\dot{x}}^{-1/2}$ removes it. Choosing $\dot{x}=f^{-1/2}$ locally for $f>0$ produces the constant leading coefficient used in the WKB approximation; negative $f$ requires a complex branch or the corresponding oscillatory real formulation.

For a slowly varying oscillator with rapidly decaying forcing, first construct the localized particular solution $u_p$. Apply the initial conditions to the residual $u_h$ to determine its WKB approximation amplitude and phase constants. Ignoring the early forcing because it later becomes small loses the permanent free oscillations it excites. In $u^{\prime \prime }+4e^{-2\epsilon t}u=8e^{-2t}$, the particular expansion is $u_p=e^{-2t}[1+\epsilon (t+1/2)]+O({\epsilon }^2)$, whose initial value shifts the first homogeneous cosine coefficient by $-\epsilon /2$.

For a locally oscillatory wave written $u(\mathbf{x},t)=A(\mathbf{x},t)\cos \varphi (\mathbf{x},t)$, the local wavevector is ${\mathbf{k}}_{\mathrm{loc}}=\nabla \varphi$, where $\nabla$ is the spatial gradient. The local wavelength $2\pi /|{\mathbf{k}}_{\mathrm{loc}}|$ measures local wave phase spacing normal to the wavefront. Interpreting it as a crest separation requires the amplitude and local wavevector to vary little over one wavelength, as in the WKB approximation. Along a unit direction $\mathbf{e}$ the corresponding local repeat distance is $2\pi /|{\mathbf{k}}_{\mathrm{loc}}\cdot \mathbf{e}|$, if that dot product is nonzero; oblique sampling measures this projected spacing.

A real snapshot alone does not uniquely specify $A$ and $\varphi$: their meaning requires a chosen oscillatory component or physical approximation. Strong interference, zeros of amplitude, and rapidly changing media can defeat a single-carrier description. If $\nabla \varphi =0$, the local reciprocal formula gives no finite wavelength; it does not establish an exact global spatial period.

For $k(t)=k_0\sqrt{1+(t/T{)}^2}$, factor resistive damping out of the shearing-mode equations by $R=e^{-D}G$, $S=e^{-D}H$, $D=\eta \int k^{2}dt$. They imply $G^{\prime \prime }={\alpha }^2{k}^{2}G$, so a growing WKB approximation has $G\propto k^{-1/2}{e}^{|\alpha |\int kdt}$ and $H\simeq i\mathrm{sgn}(\alpha )kG$. Its magnetic energy is proportional to $k\exp [2|\alpha |\int kdt-2\eta \int k^{2}dt]$. At long times the positive quadratic exponent is overtaken by negative cubic resistive damping. A well-separated growing interval exists for sufficiently small positive magnetic diffusivity, with $\eta k_0/|\alpha |\ll 1$ at fixed nonzero $|\alpha |k_{0}T$, and the displayed leading peak time. This is transient amplification, not an asymptotically self-sustaining dynamo.

For $ϵy^{\prime \prime }+s^2{y}^{\prime }+y=0$ on $s\ge 0$, removing the drift by $y=e^{-s^3/(6ϵ)}v$ givesThe transformed potential changes sign at $s_t\sim \sqrt{2}{ϵ}^{1/4}$. WKB approximation is oscillatory inside and exponential outside this turning region; an Airy turning-point connection formula applies over width ${ϵ}^{5/12}$. At the endpoint the wavelength is $\sqrt{ϵ}$, while the physical exponential prefactor changes on the envelope scale ${ϵ}^{1/3}$. Balancing only diffusion and drift and declaring a single ${ϵ}^{1/3}$ layer misses the zeroth-order term and the turning region. Matching constants may be singular at homogeneous boundary-value resonances.

The WKB approximation, also called JWKB, describes a short local de Broglie wavelength in a slowly varying potential energy. For $p(x)=\sqrt{2m[E-V(x)]}$ in a classically allowed region, its two local waves areThe phase follows the classical momentum; its amplitude varies to conserve probability current. In a forbidden region, putting $\kappa (x)=\sqrt{2m[V(x)-E]}$ gives $\psi \simeq {\kappa }^{-1/2}[C{e}^{{\int }^x\kappa /\hslash }+D{e}^{-{\int }^x\kappa /\hslash }]$, with boundary conditions selecting the physical combination.

The WKB validity condition is $\hslash |p^{\prime }|/p^2\ll 1$, with the analogous condition on $\kappa$. It fails at a turning point where $E=V$. For a simple turning point, locally linearizing V yields an Airy function; matching fixes the relative amplitudes and a $\pi /4$ phase. Two such turning points give the two-turning-point WKB quantization conditionFor an opaque barrier, the WKB tunnelling exponent instead givesThe factor two distinguishes probability from amplitude. Matching determines the prefactor; the exponential alone is not accurate near the barrier top. These formulas connect bound state energy eigenvalues and quantum tunnelling to classical actions. ^[ref]