Phase velocity.

We now consider the evolution of the wave in both position and time together. We consider the wave

$$\tilde{a}(x,t)=A e^{i (\omega t - k x)}$$ (6.9)

which has phase $\phi(x,t)=\omega t - k x$. Let us follow the motion of the position where the phase has value $\phi(x,t)=0$ as time increases. We see that initially $\phi=0$ at $x=0,t=0$ and after a time $\Delta t$ this moves to a position

$$\Delta x= \left(\frac{\omega}{k} \right) \, \Delta t$$ (6.10)

shown in Figure 6.3.

Figure 6.3:

The point with phase $\phi=0$ moves at speed

$$v_p= \left(\frac{\omega}{k} \right) \,.$$ (6.11)

It is not difficult to convince oneself that this is true for any constant value of the phase, and the whole sinusoidal pattern propagates along the $+x$ direction (Figure 6.4) at the speed $v_p$ which is called the phase velocity of the wave.

Figure 6.4: