= Gaussian envelope in a Klein-Gordon medium
{c}
{title2=$L(t)=\sigma^{-1}\sqrt{1+\beta^2t^2\sigma^4}$}
For the stipulated <Klein-Gordon packet velocities>, take the original normalized spectral amplitude
$$
a(k)=\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left[-\frac{(k-k_0)^2}{2\sigma^2}\right],\qquad
\psi(x,t)=\int_{\mathbb R}a(k)e^{i[kx-\omega(k)t]}\,dk,
$$
with $k_0>0$ and $0<\sigma\ll k_0$. Here $\sigma$ is the width of the spectral amplitude, not the <standard deviation> of its squared magnitude. Initially $\psi(x,0)=e^{-\sigma^2x^2/2}e^{ik_0x}$, a <Gaussian function> multiplying a <carrier wave>. Taking $u=\operatorname{Re}\psi$ gives a real solution of the <Klein-Gordon equation> with the corresponding initial <velocity>. The tiny negative-$k$ tail is retained in this exact integral; the statement that it is mainly right-going is a narrow-spectrum approximation, not a claim of strictly positive <wavenumbers>.
\Image[/klein-gordon-gaussian-carrier-envelope-evolution.png]
{title=Original Klein-Gordon packet evolution: exact spectral integral, slower envelope and carrier phase reference}
The dimensionless figure uses $c=\omega_0=k_0=1$ and $\sigma=0.18$ at $t=0,4,8$. Blue curves are $\operatorname{Re}\psi$, orange bounds are $\pm|\psi|$, and the marked <carrier wave> reference is $x=v_pt$; it need not be a crest of the full wavefield. The dashed envelope-centre reference $x=v_gt$ is a leading approximation. These are calculated curves, not experimental records. The exact <dispersion relation> makes the packet slightly asymmetric; a quadratic approximation predicts symmetric broadening about $v_gt$.
The first-order <Taylor expansion> translates the initial <wave envelope> without changing its shape. Retaining curvature instead gives
$$
\omega(k)\simeq\omega_c+v_gq+\frac12\beta q^2,\qquad q=k-k_0,\qquad
\beta=\omega''(k_0)=\frac{c^2\omega_0^2}{\omega_c^3}>0.
$$
The distinction between translation and curvature-driven spreading is the same one exposed by the Gaussian pulse calculation in <Pulse Propagation by Richard Fitzpatrick> (equations 9.8–9.18). Its first-order approximation is not an assertion that a dispersive packet keeps its shape indefinitely.
Evaluating the quadratic <Gaussian integral> gives the original approximation
$$
\psi_2(x,t)=\frac{e^{i(k_0x-\omega_ct)}}{\sqrt{1+i\beta t\sigma^2}}
\exp\!\left[-\frac{\sigma^2(x-v_gt)^2}{2(1+i\beta t\sigma^2)}\right],
$$
and hence
$$
|\psi_2(x,t)|=(1+\beta^2t^2\sigma^4)^{-1/4}
\exp\!\left[-\frac{(x-v_gt)^2}{2L(t)^2}\right],\qquad
L(t)=\frac{\sqrt{1+\beta^2t^2\sigma^4}}{\sigma}.
$$
Here $L$ is the amplitude width: the <standard deviation> of the normalized intensity $|\psi_2|^2$ is $L/\sqrt2$. The peak falls as the packet widens, while $\int|\psi_2|^2dx$ stays constant. This conserved integral is not the complete physical <wave energy> of the <Klein-Gordon equation>.
Rigid translation requires $|\beta|t\sigma^2\ll1$. The quadratic approximation can allow that quantity to be order one, but still requires the accumulated neglected cubic <wave phase> to be small over the appreciable spectrum, for example $(t/6)\sup_{|q|\le3\sigma}|\omega'''(k_0+q)|(3\sigma)^3\ll1$. This sufficient check controls only the stated three-width spectral window; the Gaussian tails must also be negligible at the required accuracy. Higher <derivatives> or very long times require returning to the exact spectral integral. The width $L(t)$ is therefore a controlled local approximation; it neither predicts <front velocity> nor establishes a universal identity between <group velocity> and <information velocity>.
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