Questions:

12.1. 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 (ω₁) 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.

12.2. 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, ω₁ = 8.0 rad/s and k₂ = 8.4 rad/m, ω₂ = 8.4 rad/s?

12.3. What are the group and phase speeds for the case k₁ = 8.0 rad/m, ω₁ = 8.4 rad/s with k₂= 8.4 rad/m, ω₂ = 8.4 rad/s? Describe what you see.

12.4. What are the group and phase speeds for the case k₁ = 8.8 rad/m, ω₁ = 8.0 rad/s with k₂ = 8.4 rad/m, ω₂ = 8.4 rad/s? Describe what you see.

12.5. For k₁ = 8.0 rad/m, k₂ = 8.4 rad/m, ω₂ = 8.4 rad/s try several values of ω₁ between 8.4 and 9.0 rad/s. What can you conclude about the group velocity as ω₁ gets larger?

12.6. For k₁ = 8.0 rad/m, k₂ = 8.4 rad/m, ω₂ = 8.4 rad/s try several values of ω₁ between 8.4 and 7.6 rad/s. What can you conclude about the group velocity as ω₁ gets smaller?

You will have noticed by now that the phase velocity can be greater than the group velocity. In the case of electromagnetic waves the phase velocity can be greater than c = 3 × 10⁹ m/s, the speed of light. This does not break the rules of special relativity because information is transmitted at the group velocity, which is never greater than c.

12.7. Based on the previous few questions, what is the general rule for when group velocity is larger than the phase velocity? Under what conditions is phase velocity greater than the group velocity?

12.8. For some material the wave speed is fixed by the medium in which the wave is traveling (see simulation 3). In these cases the ratio of ω / k is always the same number although ω and k might be different for different waves. For k₂ = 8.4 rad/m, ω ₂ = 8.4 rad/s try several values of ω₁ and k₁ such that the ratio ω₁ / k₁ is always equal to one (the same as ω₂ / 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?