Phase velocity and group velocity are concepts in wave propagation, especially relevant in the study of waves such as electromagnetic waves, sound waves, and matter waves (e.g., quantum mechanics).
- Phase Velocity: Phase velocity refers to the speed at which a specific phase of a wave propagates through a medium. A wave consists of repeating patterns or cycles, called phases. For example, in a sinusoidal wave, one complete cycle from the starting point to a maximum value, back to the starting point, and then to a minimum value constitutes a phase. The phase velocity is the rate at which these phases advance or move forward in space.
Mathematically, the phase velocity (v_phase) of a wave with wavelength (λ) and frequency (f) is given by the formula: v_phase = λ * f
It's important to note that phase velocity can exceed the speed of light in certain media, like in the case of light waves passing through a medium with a refractive index greater than 1. This can lead to phenomena like Cherenkov radiation.
- Group Velocity: Group velocity refers to the speed at which the overall envelope or modulation of a wave propagates through a medium. It describes the velocity at which energy or information is transported by the wave. When a wave consists of multiple frequencies or wavelengths (e.g., wave packets), the group velocity represents the speed at which the wave packet moves.
In certain cases, such as dispersion, different frequency components of a wave may travel at different speeds. This causes the wave packet to spread out over time, which is an effect known as dispersion. The group velocity can be different from the phase velocity when dispersion occurs.
Group velocity can be calculated by taking the derivative of the wave's angular frequency (ω) with respect to its wave vector (k): v_group = dω/dk
For non-dispersive media, where the relationship between ω and k is linear, the group velocity will be the same as the phase velocity.
In summary, phase velocity relates to the propagation of individual phases of a wave, while group velocity describes the speed at which wave packets or energy propagate through a medium. Both velocities are fundamental in understanding the behavior of waves in various contexts.