Gaussian envelope in a Klein-Gordon medium ($L(t)={\sigma }^{-1}\sqrt{1+{\beta }^2{t}^2{\sigma }^4}$)

For the stipulated Klein-Gordon packet velocities, take the original normalized spectral amplitudewith $k_0>0$ and $0<\sigma \ll k_0$. Here $\sigma$ is the width of the spectral amplitude, not the standard deviation of its squared magnitude. Initially $\psi (x,0)=e^{-{\sigma }^2{x}^2/2}{e}^{i{k}_{0}x}$, a Gaussian function multiplying a carrier wave. Taking $u=\mathrm{Re}\psi$ 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.

The dimensionless figure uses $c={\omega }_0=k_0=1$ and $\sigma =0.18$ at $t=0,4,8$. Blue curves are $\mathrm{Re}\psi$, orange bounds are $\pm |\psi |$, and the marked carrier wave reference is $x=v_{p}t$; it need not be a crest of the full wavefield. The dashed envelope-centre reference $x=v_{g}t$ 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 $v_{g}t$.

The first-order Taylor expansion translates the initial wave envelope without changing its shape. Retaining curvature instead givesThe 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 approximationand henceHere $L$ is the amplitude width: the standard deviation of the normalized intensity $|{\psi }_2{|}^2$ is $L/\sqrt{2}$. The peak falls as the packet widens, while $\int |{\psi }_2{|}^{2}dx$ stays constant. This conserved integral is not the complete physical wave energy of the Klein-Gordon equation.

Rigid translation requires $|\beta |t{\sigma }^2\ll 1$. 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 $(t/6){\sup }_{|q|\le 3\sigma }|{\omega }^{\prime \prime \prime }(k_0+q)|(3\sigma {)}^3\ll 1$. 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.