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A thermal diffusion wave is a periodically forced solution of the heat equation, with spatially decreasing amplitude and varying wave phase. In a semi-infinite rod $x\ge 0$, assume constant thermal diffusivity, negligible side losses, a nearly uniform temperature across each cross-section, and the boundary conditions $\theta (0,t)={\Theta }_0\cos \omega t$ and $\theta (x,t)\to 0$ as $x\to \infty$. A steady periodic thermal state then satisfiesChoose the square root in the complex numbers with $a>0$ so that the amplitude decays into the rod. Substitution of the phasor into the heat equation proves the formula: $i\omega \hat{\theta }=\alpha {\hat{\theta }}^{\prime \prime }$. Taking the real part gives the displayed cosine. In this convention $a$ is the spatial attenuation slope and $b$ is the spatial lag slope, both in ${\mathrm{m}}^{-1}$; $\gamma$ combines them into one complex number. If a complex wavenumber $K$ is defined by $e^{i\omega t-iKx}$ instead, $K=b-ia$.

With $\Delta x=x_2-x_1>0$, let $A_j=|\hat{\theta }(x_j)|$ and define a positive downstream lag $\Delta \varphi$. Thenand either measurement determines thermal diffusivity:The logarithm uses an amplitude ratio, not a ratio of squared amplitudes. The lag is in radians and must be tracked continuously across cycles: a measured phase difference modulo $2\pi$ leaves an integer-cycle ambiguity. Use closer sensor spacings or vary frequency continuously to resolve that ambiguity. Equal attenuation and lag slopes provide a useful diagnostic of the no-side-loss model.

The apparent phase velocity is $\omega /b=\sqrt{2\alpha \omega }$, which depends on frequency. A thermal diffusion wave therefore does not give a material-independent travel speed for heat; it is a forced pattern in a dissipative diffusion equation. The ideal heat equation still has infinite propagation speed, and the observed lag is not the arrival time of a sharp front.

Trigonometric functions turn an angle into a coordinate or a ratio of lengths. Start at $(1,0)$ on the unit circle and rotate counterclockwise through $\theta$ radians: the resulting point isThe cosine records horizontal position and the sine records vertical position. This definition continues around the whole circle, including negative angles, so the signs encode direction rather than requiring a separate rule for each right triangle.

The unit circle picture explains why these functions are useful beyond measuring triangles. The two coordinates of a point rotating at constant angular speed are periodic functions. Scaling the radius to $A$ and using $\theta =\omega t+\varphi$ gives $x=A\cos (\omega t+\varphi )$ and $y=A\sin (\omega t+\varphi )$. A circular motion becomes two simple repeating traces, connecting Euclidean geometry, simple harmonic motion, and Fourier series.

For an acute angle in a right triangle, rescaling its hypotenuse to one recovers the familiar ratios: sine is opposite side divided by hypotenuse, and cosine is adjacent side divided by hypotenuse. The tangent function is their ratio, $\tan \theta =\sin \theta /\cos \theta$, wherever the denominator is nonzero; the secant function, cosecant, and cotangent give the other reciprocal ratios.

The same names extend to complex numbers through the complex exponential:These definitions agree with the circle interpretation for real angles. For complex arguments, the values need not lie between $-1$ and $1$; the literal circle-coordinate picture is restricted to real arguments. ^[ref]

A series whose terms are multiples of sines and cosines of integer multiples of a variable. A Fourier series is the special case with coefficients extracted from a specified function by integration.

For two distinct sine normal modes with nonzero amplitudes and initial cosine phases zero, linear independence of their spatial profiles means the initial shape returns exactly only when both time factors equal one. A rational frequency ratio ${\omega }_2/{\omega }_1=p/q$ in lowest terms gives first positive recurrence $2\pi q/{\omega }_1$. An irrational ratio gives no exact positive recurrence, although arbitrarily close two-mode returns exist. These statements refer to the complete spatial profile, not a displacement at one observation point.

For a propagating branch of an ideal vacuum-filled rectangular electromagnetic waveguide, the phase velocity tracks constant wave phase along the axis, whereas the group velocity describes a sufficiently narrow, weakly distorted wave packet. Using ${\omega }^2=c^2{\beta }^2+{\omega }_c^2$ givesThus $v_{\mathrm{ph}}>c$, $0<v_g<c$, and $v_{\mathrm{ph}}{v}_g=c^2$. Differentiating the dispersion relation proves the second formula; dividing it by ${\beta }^2$ proves the first. This is the same mathematical structure as Klein-Gordon packet velocities. In the stipulated $f=1.25f_c$ example,Over $0.30\mathrm{m}$, the corresponding narrow-envelope delay is $(0.50\mathrm{m})/c$, approximately $1.668\mathrm{ns}$. Timing a carrier wave crest would give $0.6004\mathrm{ns}$; it does not time the arrival of a newly introduced message.

There is an independent energy check. For the forward ${\mathrm{TE}}_{10}$ fields, the average axial Poynting vector is $⟨S_z⟩=\beta |E_y{|}^2/(2{\mu }_0\omega )$. Integrating across the guide gives average power $P$ and average electromagnetic energy per unit length $U^{\prime }$:The transverse sine and cosine squared each integrate to $a/2$; ${\beta }^2+(\pi /a{)}^2={\omega }^2/c^2$ then makes the integrated electric and magnetic contributions to $U^{\prime }$ equal. The axial energy transport therefore agrees with the group velocity in this ideal model.

As $\omega \downarrow {\omega }_c$, $v_g\to 0$ and $v_{\mathrm{ph}}\to \infty$; exactly at cutoff $\omega /\beta$ is undefined. Strong wave dispersion close to cutoff also limits the narrow-packet approximation. Neither a large phase velocity nor a pulse peak establishes superluminal information velocity; front velocity asks a separate causal question. In a uniform filling with constant positive permittivity $\epsilon$ and magnetic permeability $\mu$, replace $c$ by $u=1/\sqrt{\epsilon \mu }$, giving $v_{\mathrm{ph}}{v}_g=u^2$. Frequency-dependent material response or significant loss requires differentiating the actual dispersion relation, so this product is not universal.

For the prescribed sine-mode model $\omega =k\sqrt{c_s^2+\alpha k^2}$, positive small stiffness shifts higher harmonics more strongly. Writing $\eta =\alpha k_1^2/c_s^2$ and $k_2=2k_1$ gives ${\omega }_2/{\omega }_1=2+3\eta +O({\eta }^2)$. At the original fundamental quarter-period, the fundamental displacement changes to first order, but the second-mode cosine is at an extremum and changes only to second order. If the quarter-period is redefined with the shifted fundamental frequency, that fundamental displacement is exactly zero. Comparing profiles requires specifying which clock convention is held fixed.