Speed of advance of a wave train; for deep-water waves, half the speed of individual waves within the group is known as what?

• A. Phase Velocity
• B. Wave Speed
• C. Group Velocity
• D. Propagation Speed

Answer: (C)

The idea being tested is the difference between how fast the moving packet of waves (the group) travels versus how fast the individual wave crests move inside it. For deep-water waves, the dispersion relation gives the group velocity as half the phase velocity. That means the speed at which the overall wave train or envelope advances—carrying the wave energy and information—is half the speed of the individual wave crests within the group. Therefore, the speed of advance of the wave train is Group Velocity.

In deep water, the phase velocity is c = ω/k = sqrt(g/k), and the group velocity is dω/dk = (1/2) sqrt(g/k) = (1/2) c, showing why the envelope moves more slowly than the crests themselves.