Next: Waves in three dimensions. Up: Sinusoidal Waves. Previous: Angular frequency and wave Contents

Phase velocity.

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

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

(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

and after a time $\Delta t$ this moves to a position

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

(6.10)

shown in Figure 6.3.

**Figure 6.3:**
$\begin{figure} \epsfig{file=chapt6//vphase.eps,height=2.0in} \end{figure}$

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

$\begin{displaymath} v_p= \left(\frac{\omega}{k} \right) \,. \end{displaymath}$

(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

direction (Figure 6.4) at the speed

which is called the phase velocity of the wave.

**Figure 6.4:**
$\begin{figure} \epsfig{file=chapt6//wave.eps,height=2.0in} \end{figure}$

Next: Waves in three dimensions. Up: Sinusoidal Waves. Previous: Angular frequency and wave Contents

Physics 1st Year 2009-01-06