Wave packet

A wave packet is a localized superposition of Fourier modes. In a narrow packet, nearby wavenumbers interfere to produce a slowly varying wave envelope around a rapid carrier wave. Its leading translation is governed by group velocity; the carrier's wave phase follows phase velocity. Localized does not necessarily mean compact support: a Gaussian function decays rapidly but extends over the whole line. The Gaussian envelope in a Klein-Gordon medium gives an original example with both translation and spreading.

For a nearly monochromatic packet of length $L$ travelling past a stationary observer, the snapshot count is $L/\lambda$ but the crossing count is $fL/v_g=(L/\lambda )(v_p/v_g)$. In the envelope frame the observer moves at $-v_g$ and the crests at $v_p-v_g$. For deep-water gravity waves, $v_p=2v_g$, so the observer counts twice as many crests as appear in an instantaneous snapshot. A sharply bounded monochromatic packet is an idealization; the count describes the approximately uniform central train.

A wave envelope is a slowly varying wave amplitude multiplying a rapidly oscillating carrier wave. For a complex wavefield $\psi =A(x,t)e^{i(k_{0}x-{\omega }_{c}t)}$, its nonnegative amplitude envelope is $|A|=|\psi |$; the complex $A$ can also encode a slowly varying wave phase. This choice distinguishes the amplitude envelope from a signed modulation factor, such as the cosine factor in a two-tone superposition. A slowly varying wave envelope is useful when the carrier's wavelength and period are short compared with its variation scales. Strong overlap between distinct packets or a broad frequency spectrum can spoil a unique carrier-envelope interpretation.

A carrier wave is the rapid reference oscillation in a modulated wave. In the narrow-packet representation $\psi =A(x,t)e^{i(k_{0}x-{\omega }_{c}t)}$, choose $k_0$ near the centre of the frequency spectrum and ${\omega }_c=\omega (k_0)$ on its dispersion relation. Its constant-wave phase reference moves at phase velocity ${\omega }_c/k_0$. This reference need not coincide with a local maximum of the complete real wavefield when the wave envelope varies. Moving the reference $k_0$ shifts some slow wave phase into $A$ without changing $\psi$.

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.