= Crest crossings through a dispersive wave packet
{title2=$N_{\rm crossing}=N_{\rm snapshot}v_p/v_g$}
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.
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