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A uniformly illuminated rectangle of width $w$ and height $h$ has field proportional to $\mathrm{sinc}(q_{x}w/2)\mathrm{sinc}(q_{y}h/2)$, with angular spatial wavenumbers $q_x,q_y$ measured in inverse length. The first envelope zeros give $w=2\pi /|q_{x,0}|$ and $h=2\pi /|q_{y,0}|$. Two identical rectangles with horizontal centre separation $s$ introduce an additional $\cos (q_{x}s/2)$ factor, so the interference period is $\Delta q_x=2\pi /s$.

The factors $2\pi$ depend on using angular spatial frequency; axes in cycles per length would use the reciprocal widths directly. Fourier light intensity alone cannot determine an overall aperture translation or distinguish a mask from its inverted copy. Extra assumptions about shape and coherent illumination are necessary for a unique geometrical reconstruction.

A closed rectangular perfect conductor cavity has discrete normal modes selected by zero tangential electric field at the walls. Each frequency has the displayed form. With all three indices positive there are two independent polarizations of an electromagnetic wave; modes with exactly one zero index have one, and those with two zero indices are absent. For example the $n=0$ family can have $E_z\propto \sin (\pi lx/L_x)\sin (\pi my/L_y)$, independent of $z$. It satisfies the wall conditions because $E_z$ is normal to the $z$ faces and tangential to the side walls. Omitting such zero-index families is harmless for leading bulk density of states, but changes the low-temperature physics of a thin cavity.

Take a straight hollow rectangular electromagnetic waveguide with vacuum interior $0<x<a$, $0<y<b$ and axis $z$, where $a>b>0$. Its four walls are ideal perfect conductors. The time-dependent Maxwell equations require zero tangential electric field and zero normal oscillating magnetic field at these walls. Real metal has finite electrical conductivity, so real guides lose electromagnetic energy by Joule heating; the perfect-wall model isolates confinement and wave dispersion from that loss.

The simplest propagating transverse electric mode of a rectangular waveguide is ${\mathrm{TE}}_{10}$. “Transverse electric” means $E_z=0$, not that the magnetic field is also wholly transverse. With real peak scale $E_0$, its complex amplitudes, including the common axial factor, can be writtenThe physical fields are the real parts after multiplication by $e^{-i\omega t}$. In particular, $E_y$ is zero at $x=0,a$, maximal at $x=a/2$, and independent of $y$. At $y=0,b$ it is normal to the metal, so it need not vanish there. The ${\mathrm{TE}}_{10}$ electric field crosses the narrow gap, while its strength varies across the broad dimension.

For an original example choose $a=30\mathrm{mm}$, $b=12\mathrm{mm}$ and $f=1.25f_c$, where $f_c=c/(2a)$. With $c=299792458m/s$, this givesHere ${\lambda }_0=c/f$ is vacuum wavelength and ${\lambda }_g=2\pi /\beta$ is guided wavelength. The next cutoff is ${\mathrm{TE}}_{20}$ at $2f_c$; ${\mathrm{TE}}_{01}$ starts at $(a/b)f_c=2.5f_c$. Thus only ${\mathrm{TE}}_{10}$ can propagate at the chosen frequency, although a source or discontinuity can additionally excite local evanescent waves. These are calculated values, not specifications or measurements of a commercial guide.

The redshift $z$ compares emitted and received local frequencies or wavelengths: $1+z={\lambda }_{\mathrm{rec}}/{\lambda }_{\mathrm{em}}={\nu }_{\mathrm{em}}/{\nu }_{\mathrm{rec}}$. Motion, gravitation, and cosmological expansion can all contribute.

For a stable isotropic solid in $y<0$, use $u_x={\varphi }_x+{\psi }_y$, $u_y={\varphi }_y-{\psi }_x$. An incident P wave with frequency $\omega$, tangential wavenumber $\kappa$ and upward normal wavenumber $a$ reflects into a P potential $A{e}^{i(\kappa x-ay-\omega t)}$ and an SV potential $B{e}^{i(\kappa x-by-\omega t)}$, where $b^2={\omega }^2/c_s^2-{\kappa }^2$. Vanishing of both displacements at the rigid plane gives $\kappa (1+A)-bB=0$, $a(1-A)-\kappa B=0$. Hence $A=(ab-{\kappa }^2)/(ab+{\kappa }^2)$ and $B=2a\kappa /(ab+{\kappa }^2)$. Because $c_p>c_s$ and ${\kappa }^2\le {\omega }^2/c_p^2$, $b^2>0$ and the converted S wave propagates rather than being evanescent.

Refraction changes the direction of a transmitted wave when it crosses between media with different phase velocities. For light at a planar interface, Snell's law relates the angles measured from the normal vector through the refractive indices. At normal incidence the direction remains unchanged even though the wavelength and phase velocity can change. The worked example under geometrical optics distinguishes transmission from total internal reflection.

For light, refraction explains why a submerged object looks nearer the surface and why an optical lens can form an image. A wavefront crossing a boundary must retain a consistent wave phase along that boundary: its normal spacing changes with phase velocity, and oblique light rays change direction. Normal incidence separates these two effects especially clearly.

Take stationary, transparent, homogeneous media with isotropy, linearity of their optical response, and real positive refractive indices, at one chosen frequency $f$. With $c$ the vacuum speed of light and ${\lambda }_0=c/f$ the vacuum wavelength, medium $j$ hasThe frequency is unchanged at a stationary boundary; the phase velocity, wavelength and wavenumber can change. Here $n_j$ is a phase refractive index, not the group refractive index. If there is optical dispersion, evaluate $n_j$ at this same frequency on both sides.^[ref]

For an original idealized air–water example, stipulate $n_{\mathrm{a}}=1$ and $n_{\mathrm{w}}=4/3$, with vacuum wavelength $600\mathrm{nm}$. Then $f=c/(600\mathrm{nm})\approx 4.99654\times 10^{14}\mathrm{Hz}$, the water phase velocity is $3c/4$, and its wavelength is $450\mathrm{nm}$. At an air angle of incidence $30^{\circ }$, Snell's law gives a water angle of refraction $\arcsin (3/8)\approx 22.0243^{\circ }$. These stipulated values illustrate the relations; actual refractive indices depend on material conditions and frequency. By reversibility of an optical ray, the same path emerging from water has outgoing air angle $30^{\circ }$.

The refractive index compares the phase velocity of light in a medium with the vacuum speed of light $c$. For a transparent, uniform medium with isotropy and a positive-index propagation branch,Here $\omega >0$ is the angular frequency, $k>0$ is the wavenumber, and $v_{\mathrm{ph}}=\omega /k$ is the speed of a surface of constant wave phase. The refractive index is a dimensionless quantity; it is a property of a specified material response at a specified frequency, rather than a universal number attached to the word “glass”. More general absorbing or negative-index media require the conventions discussed in refractive index and electromagnetic response. ^[ref]

For an original model example, choose $n=3/2$ at vacuum wavelength ${\lambda }_0=600\mathrm{nm}$. The phase velocity is $v_{\mathrm{ph}}=2c/3\simeq 1.999\times 10^{8}m/s$, the frequency is $f=c/{\lambda }_0\simeq 4.997\times 10^{14}\mathrm{Hz}$, and the medium wavelength is $400\mathrm{nm}$. These are stipulated model values, not measured properties of glass or water. A value at this single frequency does not determine a pulse's group velocity.

The usefulness of refractive index is that one measured response connects several different observations: Snell's law predicts refraction, optical path length predicts accumulated wave phase, and changes with frequency describe optical dispersion. Even at normal incidence, where a light ray does not bend, a changed refractive index changes wavelength and wave phase. The following comparison isolates that effect; the oblique geometry is developed in wavefront.

For a uniform linear medium with isotropy and local constitutive response, Maxwell equations give $k^2={\omega }^2\epsilon (\omega )\mu (\omega )$. Here $\epsilon$ is permittivity and $\mu$ is magnetic permeability. Since $c^{-2}={\epsilon }_0{\mu }_0$, the relative permittivity ${\epsilon }_r=\epsilon /{\epsilon }_0$ and the relative magnetic permeability ${\mu }_r=\mu /{\mu }_0$ obeyIf both responses are real and positive with negligible loss, and the ordinary positive-index branch is intended, then $n=+\sqrt{{\epsilon }_r{\mu }_r}$. If additionally ${\mu }_r\simeq 1$, then $n\simeq \sqrt{{\epsilon }_r}$. All these responses must be evaluated at the electromagnetic wave's frequency; a static relative permittivity cannot simply be substituted at an optical frequency. ^[ref]

With loss, use a complex index $\tilde{n}=n^{\prime }+i\kappa$ and explicitly fix the phase convention for complex amplitudes. For $E\propto \exp [i(kz-\omega t)]$ with $k=\tilde{n}\omega /c$, its wave amplitude has factor $\exp (-\kappa \omega z/c)$. A forward attenuating branch has $\kappa \ge 0$, while its wave phase is controlled by $n^{\prime }$. The complex relation ${\tilde{n}}^2={\epsilon }_r{\mu }_r$ alone leaves a choice of sign: the physical solution also needs causality and the appropriate radiation or decay condition. In negative-index media the physically selected real part can be negative. The positive square root is therefore not a universal rule for complex or negative-index media. Direction and polarization of an electromagnetic wave also matter in a plane wave in an anisotropic dielectric, for which one scalar index need not describe every propagation branch.

For a linear stationary interface, phase matching at a planar wave interface preserves frequency: the incident and transmitted electric fields must oscillate together at each boundary point. Their phase velocities can differ, so their wavelengths differ. Writing $\omega =2\pi f$, the definitions giveThis explains the compressed wavefront spacing in the figure without changing the time between crests at the interface. Moving interfaces or time-dependent material responses need not preserve frequency. ^[ref]

The notation $n({\lambda }_0)$ labels refractive index by vacuum wavelength; it does not mean the wavelength inside the material. Dependence on ${\lambda }_0$, equivalently on frequency, is optical dispersion. The field driving the material induces oscillating electric charges; their emitted electromagnetic fields combine with the applied field to change the resulting wave phase. This collective response motivates refractive index, rather than a picture of independently delayed little pieces of light. ^[ref]

To distinguish three speeds, track a wavefront for phase velocity, a narrow wave envelope for group velocity, and the earliest newly introduced disturbance for front velocity. For a transparent differentiable propagation branch,The denominator is the group refractive index. It describes envelope transit when loss and reshaping are small. It need not equal $n$, and neither phase velocity nor a reshaped pulse peak generally measures signal arrival. The front velocity article explains the separate relativistic causality constraint and why a detector threshold is not an exact causal-front measurement.

Tikhonov regularization of deconvolution can minimize $|HF-G{|}^2+\lambda |F{|}^2$ separately at each frequency. Completing the square gives $F_{\lambda }=H^{\ast }G/(|H{|}^2+\lambda )$ for $\lambda >0$. The regularization parameter trades reduced noise amplification against bias: the reconstruction multiplies the true noiseless component by $|H{|}^2/(|H{|}^2+\lambda )$ and the noise by $H^{\ast }/(|H{|}^2+\lambda )$.

For a sample or finite biological population of positive size $n$, the relative frequency of a category represented $n_i$ times is $n_i/n$. It is a proportion and differs from an oscillation frequency.

The relative refractive index of medium two with respect to medium one is $n_{21}=n_2/n_1$, using their absolute refractive indices at the same frequency. For transparent media with isotropy and positive refractive indices, Snell's law gives $n_{21}=\sin i/\sin r$, with the angles measured from the interface normal. Thus a refraction measurement in air directly determines an index relative to air. Converting it to an absolute refractive index requires the refractive index of that air; replacing it by one is an approximation.

At larger kinetic energy, special relativity changes the relation between momentum and velocity. With $m$ the invariant mass, the Lorentz factor is $\gamma =1+K/(m{c}^2)$, and a relativistic cyclotron orbit in a fixed uniform magnetic field satisfiesThe cyclotron frequency decreases as the kinetic energy increases. If the drive stays at the classical value ${\omega }_0=|q|B/m$, the half-turn time is $\pi \gamma /{\omega }_0$. During it the drive advances through $\pi \gamma$ rather than $\pi$ radians. Its excess phase per half-turn is thereforeEven a small fractional frequency mismatch accumulates across many turns. A particle eventually reaches a weak or decelerating gap phase, so the classical fixed-field, fixed-frequency design has a practical kinetic energy limit. The limit depends on the kick size and acceptable accumulated phase error, rather than one universal cutoff energy. Changing drive frequency or shaping the magnetic field can restore synchronism; the argument does not forbid relativistic circular accelerators.

For the illustrative proton at $1.00\mathrm{MeV}$, $\gamma -1\simeq 0.00106579$: the frequency is lower by about $0.1065\%$, and the next half-turn adds about $0.19184^{\circ }$ of drive phase excess. To see cumulative sensitivity, prescribe an ideal ramp $K_j=jϵ$ with $ϵ=10\mathrm{keV}$ and sum the errors of the first $N$ coasts after kicks:This gives ${\Phi }_{100}\simeq 9.688^{\circ }$ after the coast following kick $100$; the arrival at kick $100$ instead has ${\Phi }_{99}\simeq 9.496^{\circ }$. The prescribed ramp diagnoses detuning but does not feed the resulting phase error back into the kicks. A self-consistent thin-gap model uses the cyclotron gap energy gain with $\cos {\varphi }_n$, then advances the phase with the new kinetic energy during the following coast.

The Lorentz transformation changes the photon four-momentum and therefore its frequency, producing the displayed formula for a boost along the first axis. Here the primed ray angle is measured in the emitting frame and $\beta =v/c$. The angle and frequency transforms must be used together: aberration of light changes the direction, and a photon-number angular fraction need not equal the corresponding radiant-energy fraction.

Here $c$ is the vacuum speed of light, $\gamma$ is the Lorentz factor, and the two frequencies are measured in different inertial reference frames. Unlike the classical Doppler effect for sound, there is no material rest reference frame for light in vacuum. For small radial recession speed $v_r\ll c$, the wavelength shift satisfies $\Delta \lambda /\lambda \simeq v_r/c$. This small-speed limit underlies Doppler spectroscopy; it is not an exact sound formula.

For a passing optical trap on a circle of circumference $L$ with nonoverlapping support, a particle that is stationary between encounters receives displacement $\Delta x$ in each encounter. The trap revolution frequency $f_T$ and particle revolution frequency obey $L{f}_p=\Delta x(f_T-f_p)$. This corrects the naive independent-kick estimate when the particle motion changes the encounter rate.

Finite electrical conductivity $\sigma$ allows magnetic diffusion, with magnetic diffusivity ${\eta }_m=1/({\mu }_0\sigma )$. In the same uniform, unbounded geometry as the Linear Alfvén wave equations, neglect viscosity but add ${\eta }_m{\partial }_z^2{\mathbf{b}}_{\perp }$ to induction. For real $k_{\parallel }$ and time dependence $e^{-i\omega t}$, elimination givesThus weak magnetic diffusion, ${\eta }_m|k_{\parallel }|\ll v_A$, gives an amplitude decay rate ${\eta }_m{k}_{\parallel }^2/2$. For ${\eta }_m|k_{\parallel }|>2v_A$, both temporal roots are purely decaying; they no longer describe an oscillating traveling mode. This temporal calculation at fixed real wavenumber is different from a driven vessel at fixed real frequency, where a complex spatial wavenumber describes penetration and attenuation. Viscosity also dissipates kinetic energy, and walls introduce transverse gradients and additional losses.

For an axial length $L$, comparing the magnetic diffusion time $L^2/{\eta }_m$ with the transit time $L/v_A$ gives the ratio $L{v}_A/{\eta }_m$. A large value favors propagation before diffusion, but a narrow vessel can introduce a shorter transverse diffusion scale. In the actual 1949 analysis, finite electrical conductivity and vessel dimensions restrict the useful frequency interval; above its upper limit the response becomes a skin wave.^[ref]

Connect one physical resistor $R$ across an ideal lossless capacitor $C$, with no applied source and an ideal readout that draws no electric current. Represent the resistor by its noisy Thévenin equivalent. The resulting RC low-pass frequency response gives the stationary capacitor voltage power spectral densityIts mean square follows by the effective noise bandwidth calculation:This is also the equipartition theorem: the mean capacitor energy is $\frac{1}{2}C⟨v_C^2⟩=\frac{1}{2}{k}_{\mathrm{B}}T$. Raising $R$ makes the low-frequency spectrum taller and its width narrower in exactly compensating proportions. It changes the equilibration time $RC$, not the final variance.

The integral to infinity belongs to the ideal classical model. For a physical electrical circuit, assume thermal equilibrium, linear components with negligible extra dissipation, and a corner far below the frequencies where parasitic effects, frequency-dependent electrical resistance, or $hf\sim k_{\mathrm{B}}T$ invalidate the model; the omitted high-frequency tail must be negligible. A finite-bandwidth instrument measures only part of this variance. Its input capacitance contributes to the total $C$, and its finite input electrical resistance introduces both loading and its own thermal electrical noise.

For $T=300\mathrm{K}$ and $C=1\mathrm{nF}$, the predicted equilibrium variance is $4.141947\times 10^{-12}{\mathrm{V}}^2$ and the root mean square voltage is $2.0352\mu \mathrm{V}$. At $R=10\mathrm{k}\Omega$, $RC=10\mu \mathrm{s}$ and $B_{\mathrm{n}}=25\mathrm{kHz}$. At $R=100\mathrm{k}\Omega$, $RC=100\mu \mathrm{s}$ and $B_{\mathrm{n}}=2.5\mathrm{kHz}$; the equilibrium root mean square voltage is identical.

To obtain the growth curve, prepare $v_C(0)=0$ exactly and connect the resistor at $t=0$ while maintaining its temperature. The RC low-pass filter's causal impulse response is $h(t)=e^{-t/(RC)}/(RC)$ for $t\ge 0$. Classical ideal white noise with two-sided power spectral density $2k_{\mathrm{B}}TR$ givesThe factor $2k_{\mathrm{B}}TR$ here is the two-sided time-domain noise intensity; the equilibrium spectral integral above used the one-sided $4k_{\mathrm{B}}TR$. They give the same long-time limit. The variance approaches equilibrium with time constant $RC/2$, although a deterministic initial voltage decays with time constant $RC$. If the capacitor already starts in thermal equilibrium, its initial variance supplies the missing exponentially decaying term and the total variance stays constant.

A frequency at which a specified driven response is resonant. The value can depend on the observable and damping: the electric current amplitude of a driven series RLC circuit peaks at $f_0=1/(2\pi \sqrt{LC})$, while a different circuit readout need not peak at the same frequency.

Drive a series RLC circuit with electric potential difference $V(t)=V_0\cos (\omega t)$, where $V_0>0$ and $\omega >0$. After the free transient decays, the electric current has the same angular frequency as the source:For fixed positive $R,L,C,V_0$, the electric current amplitude is largest exactly at $\omega ={\omega }_0=1/\sqrt{LC}$. The two electrical reactance contributions cancel there, leaving only the drop from electrical resistance. In particular,Every solution approaches this periodic response because its difference from the displayed solution is a decaying free response. The peak remains at ${\omega }_0$ even for an overdamped RLC response; free ringing is not required for a maximum of driven electric current. ^[ref].

For example, stipulate $L=10\mathrm{mH}$, $C=1\mu \mathrm{F}$, $R=20\Omega$, and $V_0=1\mathrm{V}$. Then ${\omega }_0=10^4{\mathrm{s}}^{-1}$, $f_0={\omega }_0/(2\pi )\approx 1591.55\mathrm{Hz}$, and the peak electric current amplitude is $50\mathrm{mA}$. Holding the source amplitude fixed is essential when comparing responses at different frequencies.

Three frequencies must be distinguished. The free underdamped RLC response rings at ${\omega }_d=\sqrt{{\omega }_0^2-{\alpha }^2}$. Driven electric current peaks at ${\omega }_0$. Driven electric charge has amplitudewhose peak occurs at $\sqrt{{\omega }_0^2-2{\alpha }^2}$ only if $\alpha <{\omega }_0/\sqrt{2}$. Otherwise it decreases from its zero-frequency limit $C{V}_0$, with no positive-frequency peak. The different peak locations follow because electric current is the time derivative of electric charge, multiplying a sinusoidal amplitude by $\omega$. The limiting responses are $I_{\mathrm{amp}}\sim V_{0}C\omega$ as $\omega \to 0^{+}$ and $I_{\mathrm{amp}}\sim V_0/(L\omega )$ as $\omega \to \infty$.

The Q factor measures selectivity:The average Joule heating power is $R{I}_{\mathrm{amp}}^2/2$. Its half-maximum angular frequencies and their separation are exactlyThese follow by setting $|\omega L-1/(\omega C)|=R$. In the numerical example, $Q=5$ and $\Delta \omega =2000{\mathrm{s}}^{-1}$. Reducing $R$ makes a taller, narrower electric current peak; varying $C$ tunes the selected frequency. Measuring the drop $v_R=Ri$ gives a band-pass filter that suppresses very low and very high frequencies while passing a range near resonance. This is useful for selecting a radio signal, though the range must be broad enough to pass its modulation. ^[ref].

A resonator stores oscillatory energy in one or more normal modes. Driving near a mode's frequency can produce resonance. Losses limit the response and determine its quality factor. After the drive is removed, an isolated weakly damped mode decays approximately exponentially; an observed room response may combine many modes.