11. Group Velocity versus Phase Velocity.


In previous simulations we have referred to the speed of a wave as determined by v = w/k. But what value do we use for speed if we add two waves together, each of which has a different value of  v = w/k? In cases where several waves add together to form a single wave shape (called the envelope) we can quantify the speed with two numbers, the group velocity of the combined wave and the phase velocity.
The phase velocity is the average velocity of the components, given by  . The group velocity is velocity of the envelope. For two waves group velocity is defined by  where  and . This expression for group velocity is the slope of a frequency versus wavenumber graph so in the case of adding many waves we can define the group velocity as .

  phase velocity:    m/s   group velocity:  m/s

   k1 =  rad/m    ω1 =  rad/s 

 

   k2 = rad/m   ω2 =  rad/s 

Since this simulation has waves traveling in the x-direction only we will talk about group speed and phase speed.

Questions:

11.1. The simulation starts with two identical waves. The bottom graph shows the sum of the two top graphs. Change both the wave number (k1) and the angular frequency (w1) for the first wave to 8.0, click 'set values' and 'play'. Describe what you see. Notice that the envelope moves to the right at the same speed as the components.

11.2. Two numbers appear in the bottom of the simulation window. These show the phase speed and the group speed. What are the group and phase speed for the case k1=8.0 rad/m, w1=8.0 rad/s and k2=8.4 rad/m, w2=8.4 rad/s?

11.3. What are the group and phase speeds for the case k1=8.0 rad/m, w1=8.4 rad/s with k2=8.4 rad/m, w2=8.4 rad/s? Describe what you see.

11.4. What are the group and phase speeds for the case k1=8.8 rad/m, w1=8.0 rad/s with k2=8.4 rad/m, w2=8.4 rad/s? Describe what you see.

11.5. For k1=8.0 rad/m, k2=8.4 rad/m, w2=8.4 rad/s try several values of w1 between 8.4 and 9.0rad/s. What can you conclude about the group velocity as w1 gets larger?

11.6. For k1=8.0 rad/m, k2=8.4 rad/m, w2=8.4 rad/s try several values of w1 between 8.4 and 7.6 rad/s. What can you conclude about the group velocity as w1 gets smaller?

11.7. For waves traveling in some material the speed is fixed by the medium in which the wave is traveling. In these cases the ratio of w/k is always the same number although w and k might be different be different. For k2=8.4 rad/m, w2=8.4 rad/s try several values of w1 and k1 such that the ratio w1/k2 is always equal to one (the same as w2/k2). What can you conclude about the group velocity as compared to the phase velocity in cases where all the components travel at the same speed?

Credits.
Go To: IUS Physics Top Page.
Contact Dr. K. Forinash, for comments/suggestions/corrections.