Phase velocity Definition and 82 Threads

The phase velocity of a wave is the rate at which the wave propagates in some medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as





v


p



=


λ
T


.


{\displaystyle v_{\mathrm {p} }={\frac {\lambda }{T}}.}
Equivalently, in terms of the wave's angular frequency ω, which specifies angular change per unit of time, and wavenumber (or angular wave number) k, which represents the proportionality between the angular frequency ω and the linear speed (speed of propagation) vp,





v


p



=


ω
k


.


{\displaystyle v_{\mathrm {p} }={\frac {\omega }{k}}.}
To understand where this equation comes from, consider a basic cosine wave, A cos(kx − ωt). After time t, the source has produced ωt/2π = ft oscillations. After the same time, the initial wave front has propagated away from the source through space to the distance x to fit the same number of oscillations, kx = ωt.
Thus the propagation velocity v is v = x/t = ω/k. The wave would have to propagate faster when higher frequency oscillations are distributed less densely in space unless the wave length is compensatorily shortened. Formally, Φ = kx − ωt is the phase, where







∂
x


∂
t



=
−



∂
Φ

/

∂
t


∂
Φ

/

∂
x



.


{\displaystyle {\frac {\partial x}{\partial t}}=-{\frac {\partial \Phi /\partial t}{\partial \Phi /\partial x}}.}
Since ω = −dΦ/dt and k = +dΦ/dx, the wave velocity is v = dx/dt = ω/k.

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