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A slowly varying coastal Kelvin wave can force long Rossby waves at the same absolute frequency and alongshore wavenumber. In the regime $k^2+{\ell }^2\ll L_d^{-2}$, the offshore group velocity is westward with magnitude $\beta L_d^2$. With coastal phase velocity magnitude $c$, matching gives $|k|=c|\ell |/(\beta L_d^2)$. The zonal long-wave condition therefore additionally requires $|\ell |\ll \beta L_d/c$; $|\ell |L_d\ll 1$ alone is insufficient. This mechanism transmits coastal thermocline anomalies into an ocean basin over a much slower timescale than coastal propagation.

In 1992 the COBE DMR team, including George F. Smoot and Charles L. Bennett, used radiometer antennas to compare antenna temperatures between sky directions $60^{\circ }$ apart at $31.5$, $53$ and $90\mathrm{GHz}$, with two nearly independent channels per frequency. Repeated sky coverage supported least-squares solution map reconstruction.^[ref]

The analysis fitted magnetic response and removed the assumed kinematic angular dipole and its second-order Doppler effect contribution. Channel sums retained sky structure; channel differences estimated noise. Cross-channel correlations, different frequency maps and Milky Way latitude cuts tested the interpretation. Residual large-angle structure supported Cosmic microwave background anisotropy; Galactic contamination particularly limited the cosmic microwave background quadrupole. Brightness fluctuations alone were not sufficient evidence of primordial structure.^[ref]

The two differences serve different purposes. Comparing two sky directions measures angular variation. Comparing two independently calibrated receiver maps observing the same sky tests noise and consistency. A common sky signal cancels in the latter comparison if their antenna beam patterns and response scales match; a systematic error shared by both channels need not cancel. Separating observer motion, local emission and intrinsic Cosmic microwave background anisotropy requires these additional tests.

Define a stage as one storage increment of $2V_p$, with $V_p$ the alternating input's peak electric potential difference relative to its reference, not its peak-to-peak value. Assume ideal rectifiers, no load, no leakage or stray capacitance, a stiff alternating source and enough cycles to reach equilibrium. For $N$ such stages,The factor two follows from adding the input's next opposite excursion to an already stored peak. The subsequent series increments add. This ideal argument concerns equilibrium voltage; it supplies neither the charge-up time nor the useful load electric current.

To isolate the effect of drawing charge, consider the following original reservoir model. Each storage stage has equal capacitance $C$; a constant output electric current $I$ removes $\Delta q=I/f$ between replenishment events of frequency $f$. Every series reservoir loses this charge, hence its voltage falls by $a=I/(fC)$ per interval. Suppose replenishment removes a fixed fraction $1-\lambda$ of each reservoir's accumulated deficit, with $0\le \lambda <1$. This is an explicit phenomenological pump rule, not a solution of a particular multiplier's rectifier network. If $d_j$ is the deficit immediately after event $j$, thenHere $d_{+}$ and $d_{-}$ are the equilibrium deficits just after and just before replenishment. Linear discharge between events gives the average output and peak-to-peak variationFor chosen dimensionless settings $N=3$, $a/V_p=0.020$ and $\lambda =1/2$, the output varies between $5.88V_p$ and $5.94V_p$, averaging $5.91V_p$, instead of the unloaded $6V_p$. Starting with zero deficit, the recurrence approaches equilibrium geometrically with factor $\lambda$. Taking $I\to 0$ recovers the ideal ladder; increasing $C$ or $f$ reduces droop in this model. As $\lambda \to 1$, effective replenishment fails and the small-deficit assumption fails too.

Real multiplier stages have unequal charge-transfer burdens; their mean deficits need not be equal and depend on topology, input impedance and parasitic capacitance. Thus this reservoir model explains droop and voltage variation without claiming their exact dependence on stage count for a real Cockcroft-Walton generator. In particular, adding more stages cannot be judged solely by the unloaded $2N{V}_p$. For a beam sampling a slowly varying output, the electrostatic acceleration energy gain follows that instantaneous potential drop, so voltage variation also produces a spread of beam kinetic energies.

For vacuum wavelength endpoints ${\lambda }_{\min }<{\lambda }_{\max }$, the corresponding frequency span is $\Delta \nu =c(1/{\lambda }_{\min }-1/{\lambda }_{\max })$. The reciprocal-bandwidth estimate of optical coherence time is ${\tau }_c\sim 1/\Delta \nu$, giving ${\ell }_c\sim c/\Delta \nu$. This endpoint conversion is valid even for a broad band, whereas ${\ell }_c\sim {\lambda }_0^2/\Delta \lambda$ requires a relatively narrow optical spectrum.

For constant spectral density per unit frequency in the band, the normalized first-order optical coherence is $g^{(1)}(\tau )=e^{-i2\pi {\nu }_0\tau }\sin (\pi \Delta \nu \tau )/(\pi \Delta \nu \tau )$. Its first zero occurs at $\tau =1/\Delta \nu$. A different spectral shape or a different contrast criterion changes the numerical optical coherence length; endpoints alone do not define an exact universal value.

Suppose the two input beams share frequency, electromagnetic polarization and overlapping transverse profiles at each output, with stable relative wave phase during the photodetector's averaging interval. For the ideal beam splitter scattering matrix above, expanding the squared complex amplitudes givesThe imaginary part terms describe wave interference. Applying the single-input power split to each input separately and adding the results would discard these terms. Their opposite signs ensure $P_1+P_2=|a_1{|}^2+|a_2{|}^2$: conservation of energy survives even when one output is dark.

For an original example, let $a_1=\sqrt{P}$, $a_2=\sqrt{P}{e}^{i\delta }$ and $P=1\mathrm{mW}$. Then $P_1=P(1-\sin \delta )$ and $P_2=P(1+\sin \delta )$:

At $+\pi /2$, $a_2=i\sqrt{P}$ makes $b_1=(\sqrt{P}-\sqrt{P})/\sqrt{2}=0$ and $b_2=i\sqrt{2P}$; at $-\pi /2$ the dark output swaps. Thus the two inputs can feed all their combined power into one output without gain. At zero relative input wave phase, the splitter's own relative wave phase leaves the contributions in quadrature, so neither output is dark.

For fluctuating fields, average the cross term as $\mathrm{Im}⟨a_1^{\ast }{a}_2⟩$. It vanishes for mutually uncorrelated inputs, making the independent power-splitting rule valid for the averaged output. Partial first-order optical coherence reduces it. Orthogonal linear polarizations eliminate the cross term in a polarization-insensitive power readout, and spatial mismatch reduces the overlap. Detector averaging can also erase rapid relative wave phase changes. A fixed relation between two inputs can be prepared by splitting one laser beam and recombining the two arms.

An actual teaching protocol in Michelson Interferometer & Precision Wavelength Measurement — Week 1, University of Colorado Boulder, §§4–6, starts with an ideal-model prediction, then builds a Michelson interferometer using a helium-neon laser, a nonpolarizing beam splitter, adjustable mirrors, a micrometer translation stage and a photodetector. Steering mirrors align beam overlap at near and far planes; far-plane interference fringes refine alignment. Moving the arm mirror changes the returning relative wave phase; detector voltage is watched with a live acquisition display or oscilloscope. The guide compares measured arm powers with 50:50 splitting and asks for a revised field model. It tests electromagnetic polarization dependence, optionally changing one arm with a quarter-wave plate, and estimates noise from repeated readings at fixed mirror position. These controls test whether fringe changes reflect coherent recombination rather than splitter imbalance or misalignment. The guide supplies a protocol, not a numerical dataset. Its detector noise, imperfect overlap and electromagnetic polarization mismatch limit displacement inference.^[ref]

Additional background subtraction controls follow from the field model: block both arms to record the detector offset and ambient light, then unblock each arm separately to establish its detected power at fixed gain. With both open, scan the mirror and compare extrema after subtracting that background. Keep the detector below saturation and repeat the scan to check drift. A changing signal alone cannot distinguish source optical coherence from motion or changing overlap. For a vacuum round trip, a mirror displacement $x$ changes relative wave phase by $4\pi x/\lambda$; the useful signal is its modulation after detection, not an optical-frequency oscillation that the detector must resolve.

Coherent demodulation recovers a message by multiplying the received signal by a reference carrier wave of the same frequency and known relative wave phase, then passing the result through a low-pass filter. For the ideal AM model and reference $2\cos ({\omega }_{c}t+\phi )$,A unity-gain ideal low-pass filter passing $0$ and $\Omega$ but rejecting $2{\omega }_c$ and $2{\omega }_c\pm \Omega$ leaves $Aq(t)\cos \phi$. Such a separating cutoff exists because $\Omega <2{\omega }_c-\Omega$. At $\phi =0$, subtraction of $A$ recovers $Am\cos (\Omega t)$ exactly for any finite $m\ge 0$, including overmodulation, provided the transmitter and channel preserve this linear waveform. This is the signed factor, not its absolute value.

A constant phase error scales the recovered message by $\cos \phi$: a $\pi /3$ radian error halves it, and a $\pi /2$ error removes it. A known error with nonzero cosine can be compensated in gain; a frequency mismatch makes the relative wave phase drift and no longer gives constant attenuation. The lecture illustrates synchronous multiplication, filtering and phase error.^[ref]

In an isolated collinear completely inelastic capture, add incoming total energies and signed momenta. Their invariant fixes the composite rest mass, including internal excitation. If it later converts all energy to two opposite axial photons, the two conservation equations immediately give the displayed photon energies. The laboratory frequency ratio is $(E+Pc)/(E-Pc)$, not generally one; equality holds in the composite rest frame.

If phase-randomizing collisions form a Poisson process of mean interval ${\tau }_c$, field correlation survives only when no such collision occurs during the delay. Thus its magnitude decays as $e^{-|\tau |/{\tau }_c}$. The Wiener-Khinchin theorem gives the normalized Lorentzian line shapeIts full width at half maximum is $2/{\tau }_c$ in angular frequency or $1/(\pi {\tau }_c)$ in ordinary frequency. A population lifetime is a different time constant: lifetime-limited amplitude decay at half the population rate introduces the familiar factor of two. The definition of the supplied lifetime must therefore accompany any numerical linewidth.

Let the scale factor grow by $f$ from an initially thermal photon distribution. geodesic redshift gives ${\nu }_i=f\nu$, and conserving photons in corresponding comoving frequency bins gives $n_{\nu }(t)=f^{-2}{n}_{f\nu }(t_i)$. Inserting Planck's law leaves the same spectral form at temperature $T_i/f$. Total photon-number conservation alone would not prove that the entire radiation spectrum stays Planckian; frequency redshift and collisionless propagation are also required. Absorption, injection or spectral mixing can invalidate this one-temperature argument.

Comb self-referencing compares two parts of one optical frequency comb. Coherent broadening supplies an octave, a factor of two in frequency. Second-harmonic generation doubles a red tooth; a photodiode detects its beat with an original blue tooth.^[ref] Applying the tooth formula givesThe doubled branch carries $2f_0$, while the original high-frequency branch carries $f_0$: their difference retains exactly one offset.

The displayed difference uses the order doubled tooth minus original tooth. An ordinary electronic frequency counter returns a nonnegative beat magnitude; spectral selection and known electronic shifts must identify whether it represents $f_0$, $f_r-f_0$, or a shifted version. Coherent broadening, overlapping photon polarization, spatial alignment and temporal overlap are required. Merely producing a broad patch of colored light does not establish a usable offset signal.

One beat magnitude does not tell us the integer $n$. In the example, every proposed optical frequency $n(250\mathrm{MHz})+50\mathrm{MHz}$ produces the same signed beat with its corresponding tooth. A coarse independent optical measurement can select the integer: once the beat sign is known, an error bound below $f_r/2$ suffices if the candidate is selected by nearest distance.

There is also a useful two-spacing check. Keep the unknown laser and $f_0$ fixed, track the same tooth continuously, and increase $f_r$ by a stipulated $5\mathrm{Hz}$. Its frequency rises by $10\mathrm{MHz}$, so the signed beat falls from $30$ to $20\mathrm{MHz}$. HenceThis inference fails if the unknown drifts appreciably, the offset changes unnoticed, the beat sign is lost, or the tracker jumps to another tooth. The assumed tooth motion stays well below half the spacing in this example. If $f_0$ also changes by a measured amount, use $n=-(\Delta b+\Delta f_0)/\Delta f_r$.

For a fixed verified tooth number, the first-order frequency error is $\delta f_L=n\delta f_r+\delta f_0+\delta b$. Errors in these readings can be correlated because they share a reference. A small beat uncertainty therefore does not imply the same absolute optical accuracy: the reference's calibration and the factor $n$ remain in the error budget. A wrong tooth number is a separate discrete error, not a small correction. The beauty of the comb is that two controlled parameters organize millions of optical reference points, while the unknown's position is read electronically.

Comoving spectral emissivity is emitted spectral luminosity per comoving volume, measured using source-rest-frame time and frequency. For identical sources with comoving number density $n_c$, it is ${ϵ}_{\nu }=n_c{L}_{\nu }$. For a population with a luminosity function, integrate its rest-frame spectral luminosity over that distribution. Multiplying a proper-volume emissivity by $a^3$ gives this comoving convention for dimensionless scale factor $a$.

For positive real mass density and shear modulus, real $k$ and $\mathrm{Im}\omega >0$, the displayed derivative of $S_z=-\frac{1}{2}\mathrm{Re}(V^{\ast }T)$ follows from the two first-order shear equations. Every nontrivial solution has strictly decreasing $S_z$. Since $S_z=\frac{1}{2}\mathrm{Re}Z|V{|}^2$, positive-real impedance at a deeper point remains positive at all shallower points; negative-real impedance at a shallower point remains negative deeper down. A zero field is the necessary exception to strictness. At complex frequency this is a spectral flux diagnostic, reducing to a physical cycle average on the real axis.

For a weak force depending only on position, the amplitude average is a periodic total derivative and vanishes. The leading harmonic amplitude is constant, while the phase may drift. The method of multiple scales agrees with the exact conservation of energy for bounded orbits in the effective potential. A cubic position force on the right side of the oscillator equation lowers the frequency at first order when its coefficient is positive.

Coherent Thomson scattering by cold electrons changes a photon's direction without changing its frequency in the electron rest frame. A monochromatic incident radiation field therefore remains monochromatic after arbitrarily many such scatterings. Large scattering optical depth can isotropize radiation but does not imply a Planck photon distribution. True absorption and emission, or suitable additional inelastic processes and photon production, are needed to establish a blackbody spectrum independently of the initial radiation field.

The continuous-time frequency response is the frequency response of a continuous-time linear time-invariant system. From zero initial state, a BIBO stable system driven by a sustained sinusoid $u(t)=A\cos (\omega t)$ approaches the steady responseAny decaying initial or switching transient is separate. The modulus $|G(i\omega )|$ is the amplitude gain; the complex argument is the shift in oscillation phase. Thus evaluating a transfer function at $s=i\omega$ measures both how much of an oscillation passes through and how far it lags.

For the first-order low-pass filter, with $K>0$, direct complex number arithmetic givesThe phase uses the inverse tangent branch between $-\pi /2$ and $\pi /2$. Low angular frequencies have gain close to $K$ and little lag. High angular frequencies have gain approximately $K/(\omega \tau )$ and lag approaching $\pi /2$. This is why the model is a first-order low-pass filter: it suppresses fast variation compared with slow variation. Rowell relates frequency response gain and phase to poles and then introduces Bode plots in MIT 2.004, lecture 32, sections 1 and 2.

At the corner angular frequency,For $K=2$ and $\tau =3\mathrm{s}$, ${\omega }_c=1/3\mathrm{rad}{\mathrm{s}}^{-1}$, the gain is $\sqrt{2}\approx 1.4142$, and the lag is $45$ degrees. The angular frequency $\omega$ is measured in radians per second; ordinary frequency $f$ is measured in hertz (cycles per second), with $\omega =2\pi f$. Consequently the same corner is $f_c=1/(2\pi \tau )\approx 0.0531\mathrm{Hz}$, not $1/3\mathrm{Hz}$.

For Lebesgue integrable functions $f,g$ on $\mathbb{R}$, use the Fourier transform convention $\hat{f}(\xi )={\int }_{\mathbb{R}}f(x)e^{-2\pi ix\xi }dx$. ThenThus the integral combining all translated products becomes an ordinary multiplication at each frequency. Different Fourier transform normalizations can place a constant in this identity; fixing the convention prevents that ambiguity. Its use in filtering is illustrated by ^[ref].

For the indicator function $q=1_{[0,1]}$,with $\mathrm{sinc}(0)=1$. Therefore the triangular function $q\ast q$ has Fourier transform $e^{-2\pi i\xi }{\mathrm{sinc}}^2(\xi )$. The extra factor of the sinc function improves the large-frequency bound from a constant times $1/|\xi |$ to one times $1/|\xi {|}^2$, matching the smoother output in the overlap picture. Fourier decay of repeated interval convolutions extends this calculation to further repetitions.

For a proof outline, Fubini's theorem permits exchanging the two integrals because the integral of $|f(y)g(x-y)|$ over $(x,y)$ is $\Vert f{\Vert }_1\Vert g{\Vert }_1<\infty$. Set $u=x-y$ in the inner integral:The factorization of the complex exponential is the mechanism behind the simplification. The discrete convolution version similarly turns coefficient sums into products of generating functions.

A rotating gravitational pattern resonates with a stellar orbit when their azimuthal frequencies agree. In action-angle variables, this is the $k_R=0$ resonance of an azimuthal harmonic. Its torque changes azimuthal angular momentum without changing radial action to leading order, allowing stellar radial migration with little heating. The circular-orbit location is the corotation radius.

The blackbody temperature of freely propagating cosmic microwave background photons scales inversely with the scale factor: cosmological redshifting changes every frequency by the same factor and preserves the Planck spectrum shape. Its present value is approximately $2.73\mathrm{K}$. Measuring that absolute spectral temperature is distinct from measuring temperature fluctuations about it.

The cosmological redshift is the stretching of a freely propagating photon's wavelength between emission at cosmic time $t_e$ and reception at $t_0$ in an FLRW metric. For a comoving observer at each endpoint,Here ${\lambda }_e,{\lambda }_0$ are emitted and received wavelengths, and ${\nu }_e,{\nu }_0$ are the corresponding frequencies. A positive cosmological redshift means that the scale factor grew while the photon travelled. The name describes a shift toward longer wavelengths; it applies to light beyond the visible range as well.

For a computed example, let the scale factor double from $a(t_e)=0.5$ to $a(t_0)=1$. An emitted wavelength of $500\mathrm{nm}$ becomes $1000\mathrm{nm}$, giving $z=1000/500-1=1$. The received frequency is half the emitted frequency. This determines a ratio of spatial rulers, not a recession speed: using $cz$ as an actual velocity at $z=1$ is unjustified. The wavelength panel in the illustration under Hubble's law shows this doubling.

A short derivation follows successive wave crests along a radial null geodesic in FLRW spacetime. Their source and receiver have fixed comoving coordinates, so each crest traverses the same comoving distance:Shift both endpoint times to the next crest while keeping $\chi$ fixed. To first order in the very short periods $\delta t_e,\delta t_0$, this gives $\delta t_0/a(t_0)=\delta t_e/a(t_e)$. Locally, wavelength equals the speed of light times the crest period, yielding the displayed cosmological redshift relation. These periods are much shorter than the timescale on which the scale factor changes.

The relation isolates the smooth expansion contribution. An observed redshift can also contain a Doppler effect from source or observer peculiar velocity and a gravitational redshift. See combined kinematic and cosmological redshift for the multiplicative combination of endpoint motion and expansion. Treating every observed redshift as pure cosmological redshift can therefore misassign a nearby source's distance.

The useful insight is that a received wavelength ratio records how much the scale factor changed during the photon's journey. Turning that ratio into a distance requires an expansion history, through the Hubble parameter, and a specified convention such as luminosity distance or proper distance in cosmology; the ratio alone supplies neither.