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A self-similar energy-conserving strong spherical explosion into a uniform ambient density has $R_s=\xi (E{t}^2/{\rho }_0{)}^{1/5}$ and $v_s=2R_s/(5t)$. The dimensionless coefficient depends on the equation of state. Radiative losses, nonuniform media or continuous energy injection alter the similarity solution.

At a material interface, incident P or SV energy can scatter into both P and SV modes, subject to conserved horizontal slowness and traction/displacement conditions. SH is decoupled in a planar isotropic model; some conversions vanish at symmetry directions such as normal incidence.

For uniform slip on a shear fault, scalar seismic moment is $M_0=\mu Ad$, with shear modulus $\mu$, fault area $A$ and slip magnitude $d$. More generally it integrates rigidity-weighted slip. It measures source strength rather than radiated energy alone.

A seismic ray is a trajectory describing high-frequency seismic energy propagation in a medium whose properties vary slowly compared with the wavelength. In an isotropic medium it is normal to local travel-time wavefronts, and its travel time is $T=\int ds/v$ along the path.

Changes in wave speed bend rays, while interfaces can reflect or refract them. The approximation is useful for interpreting arrival times, but diffraction, interference and caustics require more than a single-ray description. In anisotropic media, energy propagation and the wavefront-normal direction need not coincide.

A seismic shadow range is an interval of receiver distances missing a specified family of geometrical rays. A velocity decrease can separate shallow-turning and deep-turning arrivals; diffraction or other wave families can still deliver energy there.

In a unit-spacing periodic spinless chain, self-paired momenta are $k=0$ and, when available, $k=\pi$. An anomalous pair of the same creation operator vanishes by the canonical anticommutation relations. Such a mode is a single normal number level, filled or empty according to its energy sign, rather than a two-distinct-mode Bogoliubov transformation. This excludes the paired-angle construction at those special momenta.

For magnetic concentration by an incompressible stagnation flow, one possible normalization gives $(g^2{)}^{\prime }=2\eta -2\omega g^2$, $f^{\prime }/f=2\omega -\lambda \eta /g^2$, and $b^{\prime \prime }+(q^{-1}+q)b^{\prime }+(\lambda -m^2/q^2)b=0$. The mode parameter $\lambda$ is not fixed by decay at infinity alone. The regular axisymmetric Gaussian has $\lambda =2$, $b=e^{-q^2/2}$ and conserved axial magnetic flux, with energy proportional to $g^{-2}$. The radial equation also has regular algebraic-tail solutions.

A weak-coupling expansion of a vacuum-subtracted soliton mass begins with the classical energy and adds the Gaussian fluctuation approximation. For Sine-Gordon theory, $M=8m/{\beta }^2+\Delta M_{1\mathrm{loop}}+O(m{\beta }^2)$. The term of order $m$ includes both the vacuum-subtracted fluctuation frequencies and the appropriate mass counterterm.

A semiconductor has an electronic band gap and a conductivity strongly affected by temperature, impurities and carrier injection. Optical absorption can create mobile electrons and holes, enabling a photodiode.

The useful feature of a semiconductor is that its charge carrier population can be controlled. Semiconductor doping changes the supply of mobile electrons or electron holes; carrier injection changes their population away from thermal equilibrium. A transistor turns that control into an electronic amplifier: a small input signal controls a larger flow of supplied energy.

For a uniform semiconductor in a weak electric field, a simple drift model writes its electrical conductivity asHere $q>0$ is the elementary charge, $n,p$ are the number densities of mobile electrons and electron holes, and ${\mu }_n,{\mu }_p$ their positive charge carrier mobilities. The two contributions add because opposite electric charges drift in opposite directions and therefore carry electric current in the same direction. This model assumes fixed temperature and mobilities and omits diffusion from density gradients.

An original dimensionless example isolates the effect of carrier injection. Suppose initially $p\ll n=n_0$ and ${\mu }_n=2{\mu }_p$. Adding equal excess populations $\Delta n=\Delta p=0.1n_0$ gives $\Delta \sigma /{\sigma }_0=0.15$, even though the added electron holes remain a minority carrier population. A semiconductor can respond strongly to a small injected population without changing its underlying band gap.

At sufficiently low temperature, thermal ionization of shallow dopants becomes inefficient, reducing the mobile carrier density. An acceptor-doped sample may then have many neutral acceptors and few mobile electron holes. Extracting an activation energy requires a model of dopant ionization and compensation; the mobile density need not have the same simple activation exponent in every compensation regime.

A search in which isotope-enriched detector material is also the decay source. germanium detectors combine a contained energy deposit with energy resolution, pulse information and material-background controls. Background peaks and resolution tails must be tested before assigning a signal.

A noninteracting perfect fluid in general relativity has ${\nabla }_{\mu }{T}_i^{\mu \nu }=0$, so ${\dot{\rho }}_i+3H({\rho }_i+P_i)=0$. If energy is transferred between species, source terms $Q_i$ appear and only their sum vanishes. The Friedmann equation still involves the sum of all component stress tensors because all components share one spacetime metric.

An energy and orbital angular momentum eigenstate for a real two-dimensional central potential can be chosen with integer $k$ and real radial function $f$. Its radial equation isQuantum degeneracy permits other stationary states, including $f(r)\cos (k\varphi )/\sqrt{\pi }$, which are not single angular exponentials. Separation of variables supplies a basis rather than describing every superposition in a degenerate eigenspace.

Equivalent chemical bonds in an intact molecule need not have equal successive bond-dissociation enthalpies. Once one chemical bond is removed, the next reactant has a different electronic quantum state and may relax to a different molecular geometry. The successive steps compare different species. This distinction also explains why a tabulated average chemical bond strength is not necessarily the cost of the first cleavage.^[ref]

For an invented species $A{B}_2$, stipulate the following gas-phase steps, with all enthalpies at the same temperature and the same standard states:The symbols stand for hypothetical species with consistently chosen electronic states. They are not values for water or another measured molecule. By Hess's law, the total enthalpy of atomization for $A{B}_2(g)\to A(g)+2B(g)$ is $700\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$; the arithmetic mean per broken chemical bond is $350\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Neither individual step equals that mean. Nor must the second step be cheaper: here it is deliberately more expensive. An exact sequence sums exactly; substituting averages taken from other molecules generally gives only an estimate.

Serge Haroche developed experiments that probe photons stored in a cavity without immediately absorbing them. Passing chemical atoms interact with the cavity field and carry information about its photon number to an external detector.

Repeated nondestructive probes can progressively narrow the possible photon number, allowing the evolution of a measurement result to be followed. The approach separates the system carrying the stored energy from the successive particles carrying information about it, enabling experimental studies of quantum measurement and the quantum-to-classical transition.^[ref]

In a series RLC circuit, the same electric current $i=q^{\prime }$ flows through the resistor with electrical resistance $R$, the inductor, and the capacitor. Choose $q$ as the signed electric charge on the capacitor plate into which positive $i$ flows. With terminal drops measured in that direction, Kirchhoff's voltage law givesHere $V(t)$ is the applied electric potential difference; assume constant $L>0$, $C>0$, and $R>0$. With the applied source set to zero and the loop kept closed, the free response satisfiesThe angular frequency ${\omega }_0$ is the value for lossless free oscillation; $\alpha$ is the decay parameter. The three possibilities are underdamped RLC response for $\alpha <{\omega }_0$, critically damped RLC response for $\alpha ={\omega }_0$, and overdamped RLC response for $\alpha >{\omega }_0$. ^[ref].

For a fair comparison, start each circuit with the same electric charge $q(0)=q_0$ and zero electric current $q^{\prime }(0)=0$. Set $\tau ={\omega }_{0}t$ and $a=\alpha /{\omega }_0$; then $x=q/q_0$ obeys $d^{2}x/d{\tau }^2+2adx/d\tau +x=0$. The plot compares $a=0.5,1,2$. These dimensionless curves apply to any $L,C$ after rescaling time and electric charge.

The stored energy is the sum of magnetic energy and capacitor energy:Differentiating and using the circuit equation gives the complete power balance,In the free response, $E^{\prime }=-R{i}^2\le 0$: Joule heating accounts exactly for the lost energy. For the plotted initial conditions, $E(0)=q_0^2/(2C)$ and $E/E(0)=x^2+(dx/d\tau {)}^2$. Since every free solution tends to zero for $R>0$,In the lossless limit $R=0$, energy stays constant and shuttles between the capacitor and inductor. Thus an oscillation of electric charge and a decrease of total energy are compatible statements.

The analogy with a damped harmonic oscillator is exact within these ideal models: electric charge corresponds to displacement, electric current to velocity, inductance to mass, $1/C$ to spring constant, and $R$ to the coefficient of a resisting force proportional to velocity. The two stored energy terms play the roles of kinetic energy and elastic potential energy. ^[ref].

Differentiating the driven equation also gives $L{i}^{\prime \prime }+R{i}^{\prime }+i/C=V^{\prime }$. When using this equation to solve for electric current, retain the original constraint $L{i}^{\prime }(0)+Ri(0)+q(0)/C=V(0)$: the initial electric charge fixes the initial rate of change of electric current.

For an attractive Coulomb potential $V(r)=-\kappa /r$ with $\kappa >0$, let $m$ be the relative-motion mass, using the reduced mass for a two-body system. A bound energy $E<0$ definesIn the orbital angular momentum sector $\ell$, the Radial Schrodinger equation isExtract the regular origin power and decaying exponential by $R=r^{\ell }{e}^{-\gamma r}S(r)$, with $S={\sum }_{k\ge 0}{a}_k{r}^k$. Equating powers gives the recurrence relationThe regular confluent hypergeometric function otherwise contains an exponentially growing large-radius part. A physical bound state therefore requires termination: if $S$ has degree $q\ge 0$, then $a_{q+1}=0$, soHere $N$ is the principal quantum number and $q$ counts positive radial nodes. For $\ell =0$, the recurrence reduces to $a_k=(2\gamma k-\beta )a_{k-1}/[k(k+1)]$; that denominator is an s-wave specialization, not the general angular-momentum recurrence. Regularity at the origin and normalizability at infinity are both essential boundary conditions.

For a well of depth $V_0$ and full width $2a$, measured from zero exterior potential energy, even-state matching gives $q=k\tan ka$. If $a\sqrt{m{V}_0}/\hslash \ll 1$, then $ka\ll 1$ and $q\simeq 2m{V}_{0}a/{\hslash }^2$. Using $E_0=-{\hslash }^2{q}^2/(2m)$ gives the displayed leading binding energy. It agrees with the attractive delta-potential result for integrated strength $2a{V}_0$.

The sheet elastocapillary energy ratio compares capillary and bending energy scales for an elastic sheet:Here $L$ is a lateral size, $\gamma$ is surface tension, $B$ is plate bending stiffness and ${\ell }_{\mathrm{ec}}$ is the elastocapillary length. The ratio is a dimensionless quantity: $\gamma L^2$ and $B$ both have units of energy.

The estimate assumes bending through an angle of order one over size $L$. Then curvature scales as $1/L$ and integrating the bending energy density over area $L^2$ gives a scale $B$. A comparable change in interfacial area gives a surface energy scale $\gamma L^2$. The same balance underlies the capillary origami study's definition of elastocapillary length.^[ref]

For an original calculation, take $B=2\times 10^{-8}\mathrm{N}\mathrm{m}$, $\gamma =0.05N/m$ and $L=1\mathrm{mm}$. Then $\eta =2.5$. Doubling $L$ alone raises it to 10. Halving thickness at fixed $L$, Young's modulus and Poisson's ratio divides $B$ by eight and raises $\eta$ to 20.

Changing all dimensions gives another trend. Write $s=L/L_0$ and keep material properties and surface tension fixed. At fixed thickness, $\eta /{\eta }_0=s^2$; at fixed aspect ratio $h/L$, $B/B_0=s^3$ and $\eta /{\eta }_0=s^{-1}$. Thus halving $L$ alone gives one quarter of the initial ratio, while halving both $L$ and $h$ doubles it.

A larger ratio makes capillary energy more competitive with bending, but does not establish complete closure. Actual wetting, contact angle, droplet volume, contact-line pinning and accessible deformation paths also matter. The ratio compares scales rather than computing the full change in energy along a folding path.

A shell model of turbulence replaces a spatially resolved fluid by a sequence of amplitudes associated with progressively smaller length scales. A typical choice assigns a complex velocity amplitude $u_n$ to the wave-number shell $k_n=k_0{\lambda }^n$, with $\lambda >1$. Neighboring shells exchange energy through nonlinear couplings, while forcing supplies energy and viscosity damps high wave numbers. The model retains a scale cascade at far lower cost than direct numerical simulation, while discarding most spatial geometry.^[ref]

The basic energy bookkeeping is transparent. WriteIf the nonlinear coupling and end conditions satisfy ${\sum }_n\mathrm{Re}(u_n^{\ast }{N}_n)=0$, differentiating givesThe nonlinearity transfers energy between shells without creating its total. Viscous damping becomes faster at larger $k_n$: an isolated unforced linear mode decays as $e^{-\nu k_n^{2}t}$.

For an illustrative constant-flux estimate, a transfer time ${\tau }_n\sim (k_n|u_n|{)}^{-1}$ gives $\epsilon \sim |u_n{|}^2/{\tau }_n\sim k_n|u_n{|}^3$, hence $|u_n|\sim (\epsilon /k_n{)}^{1/3}$. Deviations of higher moments from a single scaling rule can then diagnose intermittency. The estimate is dimensional reasoning, not an exact solution. The published improved coupling was designed to reduce spurious shell correlations while preserving useful conservation and scaling properties.^[ref]