Question

Show that the group velocity and phase velocity are related by vgroup = vphase - dvphase/d.

a) vgroup = vphase + dvphase/d
b) vgroup = vphase * dvphase/d
c) vgroup = vphase / dvphase/d
d) vgroup = vphase - dvphase/d

Answer 1

The group velocity and phase velocity are related by the equation vgroup = vphase - dvphase/d.

Step-by-step explanation:

The group velocity and phase velocity are related by the equation vgroup = vphase - dvphase/d. This means that the group velocity is equal to the phase velocity minus the derivative of the phase velocity with respect to distance.

For example, if the phase velocity is constant, meaning it does not change with distance, then the group velocity will be equal to the phase velocity. On the other hand, if the phase velocity changes with distance, the group velocity will be different from the phase velocity.

Answer 2

The relation between group velocity (v_group) and phase velocity (v_phase) is given by v_group = v_phase - (dv_phase/d). Option D is answer.

Step-by-step explanation:

In understanding wave mechanics, the relationship between group velocity and phase velocity plays a crucial role. This relationship is expressed by a specific equation that involves the angular frequency, wavenumber, and their derivatives. The following explanation breaks down this equation step by step, providing clarity on the mathematical connection between group velocity and phase velocity in the context of wave propagation.

Group velocity (v_group): v_group = (dω/dk).

Phase velocity (v_phase): v_phase = ω/k.

Derivative of phase velocity: (dv_phase/dk).

Combining expressions: v_group = v_phase - (dv_phase/d).

Option D is the correct relation.