= Front velocity
{title2=$v_f$}
The <front velocity> is the propagation speed of the earliest boundary of a newly introduced disturbance's <support>. It asks when an initially unaffected observer can first respond, whereas <group velocity> asks how a narrow <wave envelope> translates. A detector threshold defines a practical arrival time, but can miss arbitrarily small earlier responses; a packet peak is another distinct timing convention. In a causal dielectric model, a high-<angular frequency> contour argument gives exactly zero field for $t<x/c$, even when <phase velocity> exceeds $c$. This is a statement about that model's <relativistic causality>, not a universal identification of <group velocity> with <front velocity>.<Wave-Front Propagation by Richard Fitzpatrick>[equations 871–873 and the paragraph following equation 887: zero field ahead of the causal front]{ref}
For the stipulated <Klein-Gordon equation> with compactly supported real <Cauchy data>, <front velocity> is bounded by $c$, and $c$ is the model's maximal speed of influence. A short <energy method> outline uses
$$
e=\frac12\left(u_t^2+c^2u_x^2+\omega_0^2u^2\right),\qquad
J=-c^2u_tu_x,\qquad
\partial_te+\partial_xJ=0,\qquad |J|\le ce.
$$
Integration outside an expanding interval shows that zero initial <energy> there remains zero. Thus initial <support> in $[-a,a]$ cannot influence $x>a$ before $t=(x-a)/c$. The positive <energy density> includes the restoring term $\omega_0^2u^2/2$: this adds oscillation behind the front without extending its <domain of dependence>. A particular disturbance may fail to excite the leading boundary, so the maximal speed is not a promise of a detectable response for every source.
A <Gaussian function> is nonzero arbitrarily far away, so the <Gaussian envelope in a Klein-Gordon medium> has no sharp initial <support> boundary. Its small distant tails are already in the initial data. Likewise, keeping only a narrow interval of the <frequency spectrum> cannot reproduce an exact abrupt onset. Cutting off a Gaussian tail changes its <Fourier transform> and the accompanying initial <velocity>; it is a different initial-value problem. The <exponential initial-velocity tail for the Klein-Gordon equation> is another concrete example of a pre-existing tail outside the main disturbance.
The water-tank measurement in <phase velocity> records <free surface> elevation with calibrated resistance probes and uses a ruler and stopwatch for an independent check. The inspected protocol calls the front or back of a group a timing feature. That visible edge establishes an envelope transit estimate, not the first nonzero disturbance in the sense of <front velocity>.<Surface Waves Kinematics (MIT 2.016 Hydrodynamics) by Massachusetts Institute of Technology, 2.016 course staff>[p. 1, Equipment; p. 2, task 2; p. 3, tasks 5–6: apparatus, calibration and group-feature timing]{ref} A proposed additional control would repeat the transit fit at several detection thresholds and train durations, using matched probe calibration and the same <sampling rate>. Dependence on the threshold diagnoses <wave envelope> reshaping or limited sensitivity; records should be compared before reflections return. Such a test could expose limitations of a <group velocity> estimate, but would not establish an exact causal front or validate the stipulated <Klein-Gordon equation> for water waves.
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