Waves/1D Examples

Examples
We now illustrate some examples of phase speed and group velocity by showing the displacement resulting from the superposition of two sine waves, as given by equation (1.38), in the $$x$$-$$t$$ plane. This is an example of a spacetime diagram, of which we will see many examples later on.

Figure 1.16: Net displacement of the sum of two traveling sine waves plotted in the $$x - t$$ plane. The short vertical lines indicate where the displacement is large and positive, while the short horizontal lines indicate where it is large and negative. One wave has $$k = 4$$ and $$\omega = 4$$, while the other has $$k = 5$$ and $$\omega = 5$$. Thus, $$\Delta k = 5 - 4 = 1$$ and $$\Delta \omega = 5 - 4 = 1$$ and we have $$u = \Delta \omega / \Delta k = 1$$. Notice that the phase speed for the first sine wave is $$c_1 = 4/4 = 1$$ and for the second wave is $$c_2 = 5/5 = 1$$. Thus, $$c_1 = c_2 = u$$ in this case.

Figure 1.16 shows a non-dispersive case in which the phase speed equals the group velocity. The regions with vertical and horizontal hatching (short vertical or horizontal lines) indicate where the wave displacement is large and positive or large and negative. Large displacements indicate the location of wave packets. The positions of waves and wave packets at any given time may therefore be determined by drawing a horizontal line across the graph at the desired time and examining the variations in wave displacement along this line. The crests of the waves are indicated by regions of short vertical lines. Notice that as time increases, the crests move to the right. This corresponds to the motion of the waves within the wave packets. Note also that the wave packets, i. e., the broad regions of large positive and negative amplitudes, move to the right with increasing time as well.

Since velocity is distance moved $$\Delta x$$ divided by elapsed time $$\Delta t$$, the slope of a line in figure 1.16, $$\Delta t / \Delta x$$, is one over the velocity of whatever that line represents. The slopes of lines representing crests (the slanted lines, not the short horizontal and vertical lines) are the same as the slopes of lines representing wave packets in this case, which indicates that the two move at the same velocity. Since the speed of movement of wave crests is the phase speed and the speed of movement of wave packets is the group velocity, the two velocities are equal and the non-dispersive nature of this case is confirmed.

Figure 1.17: Net displacement of the sum of two traveling sine waves plotted in the $$x$$-$$t$$ plane. One wave has $$k = 4.5$$ and $$\omega = 4$$, while the other has $$k = 5.5$$ and $$\omega = 6$$. In this case $$\Delta k = 5.5 - 4.5 = 1$$ while $$\Delta \omega = 6 - 4 = 2$$, so the group velocity is $$u = \Delta \omega / \Delta k = 2/1 = 2$$. However, the phase speeds for the two waves are $$c_1 = 4/4.5 = 0.889$$ and $$c_2 = 6/5.5 = 1.091$$. The average of the two phase speeds is about $$0.989$$, so the group velocity is about twice the average phase speed in this case.

Figure 1.18: Net displacement of the sum of two traveling sine waves plotted in the $$x$$-$$t$$ plane. One wave has $$k = 4$$ and $$\omega = 5$$, while the other has $$k = 5$$ and $$\omega = 4$$. Can you figure out the group velocity and the average phase speed in this case? Do these velocities match the apparent phase and group speeds in the figure?

Figure 1.17 shows a dispersive wave in which the group velocity is twice the phase speed, while figure 1.18 shows a case in which the group velocity is actually opposite in sign to the phase speed. See if you can confirm that the phase and group velocities seen in each figure correspond to the values for these quantities calculated from the specified frequencies and wavenumbers.