Questions: Group Velocity and Dispersion Relations

Question 1

A radar pulse travels through a dispersive medium. An observer notices that the individual wave crests inside the pulse appear to move faster than the pulse envelope as a whole. This means:

• A. There is a measurement error — wave crests must travel at the same speed as the envelope
• B. The phase velocity exceeds the group velocity in this medium
• C. The group velocity exceeds the phase velocity in this medium
• D. The medium is non-dispersive, and the two velocities are equal

Answer: (B)

The crests (individual wavefronts) move at the phase velocity vₚ = ω/k; the envelope (the pulse as a whole) moves at the group velocity vg = dω/dk. When crests visibly slide through the envelope — moving faster than the bump — this means vₚ > vg. This is common in water waves, where ripples (phase) travel faster than the wave group (envelope). It is not an error; it is the expected behavior in a dispersive medium where vg ≠ vₚ.

Question 2

A narrow pulse of light travels through a long optical fiber. After propagating a great distance, the pulse has broadened significantly. The primary cause is:

• A. Energy absorption at the fiber walls gradually removes the pulse's outer frequencies
• B. The pulse contains multiple frequency components that travel at slightly different phase velocities, causing them to drift apart over time
• C. Group velocity is zero in dispersive media, so pulses inevitably stop
• D. Diffraction causes the pulse to spread transversely, reducing its longitudinal coherence

Answer: (B)

Pulse spreading (group velocity dispersion) occurs because a real pulse is a superposition of many frequency components, not a single sinusoid. In a dispersive medium, different frequencies have different phase velocities, so they drift apart over time — the high-frequency components arrive at a different time than the low-frequency ones. This is the fundamental limit on data rates in fiber-optic communications: if pulses spread into each other, adjacent bits become indistinguishable.

Question 3

In a non-dispersive medium, the group velocity and phase velocity are equal, and a wave packet travels without spreading.

• T. True
• F. False

Answer: True

A non-dispersive medium has a linear dispersion relation ω = ck, so vₚ = ω/k = c and vg = dω/dk = c — they are equal. All frequency components travel at the same speed, so a superposition of frequencies (a wave packet) maintains its shape indefinitely. Vacuum is non-dispersive for electromagnetic waves: any pulse shape travels at c without distortion. Dispersion requires the dispersion relation to be nonlinear.

Question 4

The phase velocity of a wave describes how quickly a wave packet (like a signal or pulse) travels through a medium.

• T. True
• F. False

Answer: False

The group velocity, not the phase velocity, describes the motion of a wave packet or pulse. The phase velocity describes the motion of individual wave crests — the speed at which surfaces of constant phase move. These can differ significantly in dispersive media, and in some media the phase velocity can even exceed c without violating relativity, because phase carries no information. Energy and information travel at the group velocity (in normal dispersion), which is why it is physically the more important quantity.

Question 5

Explain why wave packets spread as they propagate through a dispersive medium, using the concepts of phase velocity and group velocity.

Model answer: A wave packet is built from a superposition of sinusoidal components with different frequencies. In a dispersive medium, different frequencies travel at different phase velocities (because ω/k varies with k). This means the various frequency components that make up the packet drift apart as they propagate — faster components run ahead while slower ones fall behind. The group velocity dω/dk gives the speed of the peak of the packet, but because dω/dk itself varies across the packet's frequency range, the packet broadens over time. The rate of spreading is governed by d²ω/dk² (group velocity dispersion): larger dispersion means faster broadening.

The contrast with a non-dispersive medium makes this concrete: in vacuum (ω = ck), all components travel at exactly c regardless of frequency, so the packet shape is preserved. Any deviation from linearity in ω(k) introduces spreading.