= Klein-Gordon packet velocities
{c}
{title2=$v_pv_g=c^2$}
A useful stipulated dispersive medium obeys the <Klein-Gordon equation> $u_{tt}-c^2u_{xx}+\omega_0^2u=0$, with $c>0$ and $\omega_0>0$. Its positive-<angular frequency> <dispersion relation> is
$$
\omega(k)=\sqrt{\omega_0^2+c^2k^2},\qquad
v_p=\frac{\omega(k)}{k},\qquad
v_g=\omega'(k)=\frac{c^2k}{\omega(k)}.
$$
Here $\omega_0$ is the gap at zero <wavenumber>, not the central <angular frequency> $\omega_c=\omega(k_0)$ of a <wave packet>; the parameter in the preceding formula is $A=\omega_0/c$. For $k>0$, the <phase velocity> satisfies $v_p>c$, the <group velocity> satisfies $0<v_g<c$, and $v_pv_g=c^2$. In this stipulated medium $c$ is the limiting propagation speed of the model; it need not mean the vacuum <speed of light>. The same square-root <dispersion relation> appears in <waveguide phase and group velocities>, with a <waveguide cutoff frequency> playing the role of $\omega_0$.<Pulse Propagation by Richard Fitzpatrick>[equation 9.6 and equations 9.10–9.12: square-root dispersion and local group velocity]{ref}
For the original dimensionless choice $c=\omega_0=k_0=1$, a <carrier wave> has <phase velocity> $v_p=\sqrt2$, while a narrow <wave packet> has leading <group velocity> $v_g=1/\sqrt2$. After time $t=8$, the carrier's zero-<wave phase> reference has moved to $x=8\sqrt2\simeq11.31$, whereas the leading <wave envelope> centre is at $x=8/\sqrt2\simeq5.66$. A carrier <wave crest> therefore moves through the <wave envelope>; it does not carry an identifiable piece of the envelope with it.
\Image[/klein-gordon-dispersion-and-velocity-scales.png]
{title=Stipulated Klein-Gordon medium: secant phase velocity, tangent group velocity and front speed}
The origin-to-curve secant on the <dispersion diagram> gives <phase velocity>, while its local tangent gives <group velocity>. As $k\to0^+$, $v_p\to\infty$ and $v_g\to0$; at $k=0$ the <phase velocity> itself is undefined. As $k\to\infty$, both approach $c$, from opposite sides. On the same positive-<angular frequency> branch with $k<0$, both signed velocities are negative and their product remains $c^2$. These conventions distinguish direction from <phase speed>. Reversing both $k$ and $\omega$ represents the same real <plane wave>; changing just one sign changes its direction.
For an original finite-bandwidth check, take equal-amplitude <plane waves> at $k_1=0.8$ and $k_2=1.2$, still with $c=\omega_0=1$. The exact two-tone <wave envelope> speed is
$$
v_{\rm beat}=\frac{\sqrt{1+1.2^2}-\sqrt{1+0.8^2}}{1.2-0.8}\simeq0.703563,
$$
while its <carrier wave> has <phase velocity> $(\sqrt{1+0.8^2}+\sqrt{1+1.2^2})/2\simeq1.421337$. Their product is exactly $1$: rationalizing the difference gives $v_{\rm beat}=2/(\omega_1+\omega_2)$. Nevertheless $v_{\rm beat}\ne\omega'(1)$; the derivative describes the limit of vanishing separation, not an arbitrary pair of tones. The two-tone <wave envelope> repeats indefinitely, so it has no isolated first arrival and cannot measure <front velocity>.
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