Phase and phase velocity

If we examine a harmonic wave function

$LaTeX: \psi (x,t)=A \cos (k x-\omega t+\phi )$ $\psi (x,t)=A \cos (k x-\omega t+\phi )$

the entire argument of the sinusoid is known as the phase, $LaTeX: \varphi$ $\varphi$ (or total wave phase). In this case

$LaTeX: \varphi (x,t)=k x-\omega t +\phi$ $\varphi (x,t)=k x-\omega t +\phi$

the initial phase, or constant phase shift is the constant contribution to the phase arising at the generator and is independent of how far in space, or how long in time, the wave has traveled. The total phase of the wave $LaTeX: \varphi$ $\varphi$ is clearly a function of time and space. In fact, the partial derivatives taken in either domain give

$LaTeX: \left\vert \left(\frac{\partial \varphi (x,t)}{\partial t}\right)_x \right\vert =\omega$ $\left\vert \left(\frac{\partial \varphi (x,t)}{\partial t}\right)_x \right\vert =\omega$

where the subscript means "holding this variable constant". this says that the rate of change of the phase at any fixed location is the angular frequency of the wave; the rate at which the source oscillates. The source oscillation must go through the same number of cycles per second as the wave. For each cycle, $LaTeX: \varphi$ $\varphi$ changes by 2π.

Similarly, the rate-of-change of phase with distance, holding t constant, is

$LaTeX: \left\vert\left( \frac{\partial \varphi (x,t)}{\partial x}_t\right)\right\vert =k$ $\left\vert\left( \frac{\partial \varphi (x,t)}{\partial x}_t\right)\right\vert =k$

We can use these expressions to understand the following quantity

$LaTeX: \left(\frac{\partial x}{\partial t}\right)_{\varphi }=-\left(\frac{\partial \varphi (x,t)}{\partial t}\right)_x \left(\frac{\partial x}{\partial \varphi (x,t)}\right)_t=-\frac{\left(\frac{\partial \varphi (x,t)}{\partial t}\right)_x}{\left(\frac{\partial \varphi (x,t)}{\partial x}\right)_t}$ $\left(\frac{\partial x}{\partial t}\right)_{\varphi }=-\left(\frac{\partial \varphi (x,t)}{\partial t}\right)_x \left(\frac{\partial x}{\partial \varphi (x,t)}\right)_t=-\frac{\left(\frac{\partial \varphi (x,t)}{\partial t}\right)_x}{\left(\frac{\partial \varphi (x,t)}{\partial x}\right)_t}$ .

The term on the left hand equality represents the speed of propagation at constant phase. In other words, if we traveled along with the wave in order to maintain a fixed relationship relative to any given crest or trough, the ratio of $LaTeX: \partial x/\partial t$ $\partial x/\partial t$ would be the speed we are traveling at constant phase.

One more way to think about this is if we would like to travel along side the wave so that our position is fixed relative to a crest, the expression for phase (neglecting the constant $LaTeX: \phi$ $\phi$ )

$LaTeX: \varphi (x,t)=k x-\omega t=\text{const}.$ $\varphi (x,t)=k x-\omega t=\text{const}.$

which means that if time increases (as it does), then to stay aligned with that particular wave crest, we also must move to a larger x-position. In our previous example where we were in a frame translating to the right, we had the transformation

LaTeX: x=x'+v t $x=x'+v t$

inserting this into the phase relation

$LaTeX: \varphi \left(x',t\right)=k \left(x'+v t\right)-\omega t =k t v+k x'-\omega t$ $\varphi \left(x',t\right)=k \left(x'+v t\right)-\omega t =k t v+k x'-\omega t$

simplifying by using $LaTeX: k=2 \pi /\lambda$ $k=2 \pi /\lambda$ and $LaTeX: \omega =2 \pi f$ $\omega =2 \pi f$

$LaTeX: k t v=\frac{2 \pi t v}{\lambda }=2 \pi f t=\omega t$ $k t v=\frac{2 \pi t v}{\lambda }=2 \pi f t=\omega t$

we get upon inserting this result

$LaTeX: \varphi \left(x',t\right)=k x'+\omega t-\omega t=k x'$ $\varphi \left(x',t\right)=k x'+\omega t-\omega t=k x'$

i.e., the wave is constant in time. Now, going back to our expression for the wave speed

$LaTeX: \left(\frac{\partial x}{\partial t}\right)_{\varphi }=-\frac{\left(\frac{\partial \varphi (x,t)}{\partial t}\right)_x}{\left(\frac{\partial \varphi (x,t)}{\partial x}\right)_t}$ $\left(\frac{\partial x}{\partial t}\right)_{\varphi }=-\frac{\left(\frac{\partial \varphi (x,t)}{\partial t}\right)_x}{\left(\frac{\partial \varphi (x,t)}{\partial x}\right)_t}$

we can fill in the expressions on the right hand side

$LaTeX: \left(\frac{\partial x}{\partial t}\right)_{\varphi }=-\frac{\left(\frac{\partial \varphi (x,t)}{\partial t}\right)_x}{\left(\frac{\partial \varphi (x,t)}{\partial x}\right)_t}=\frac{\omega }{k}=v_{\varphi }$ $\left(\frac{\partial x}{\partial t}\right)_{\varphi }=-\frac{\left(\frac{\partial \varphi (x,t)}{\partial t}\right)_x}{\left(\frac{\partial \varphi (x,t)}{\partial x}\right)_t}=\frac{\omega }{k}=v_{\varphi }$

this is the speed that the wave moves and is known commonly as the phase velocity of the wave.