What is Phase Velocity?
Phase velocity is the speed at which a specific point of constant phase (e.g., a crest or a trough) of a monochromatic (single frequency) wave propagates through a medium. It effectively describes the speed of an individual 'ripple' or a particular feature of the wave. For a simple wave, phase velocity (v_p) is calculated as the product of its wavelength (λ) and frequency (f), or as angular frequency (ω) divided by wave number (k).
What is Group Velocity?
Group velocity is the speed at which the overall shape of a wave's amplitude envelope, often referred to as a 'wave packet' or 'group' of waves, propagates through a medium. This velocity is crucial because it represents the speed at which energy or information is transported by the wave. It becomes particularly relevant when dealing with complex signals composed of multiple frequencies that interfere with each other.
Key Differences and Their Relationship
In a non-dispersive medium, where the phase velocity of a wave is constant regardless of its frequency, the group velocity and phase velocity are equal. However, in a dispersive medium (common in real-world scenarios, like light traveling through glass or water waves), the phase velocity depends on frequency. In such cases, the group velocity can be slower, faster, or even negative compared to the phase velocity, causing the wave packet to spread out or distort over distance.
Practical Applications and Importance
Understanding the distinction between group and phase velocity is foundational in various STEM fields. In fiber optics, for example, maintaining signal integrity over long distances requires careful management of dispersion, which affects group velocity. Similarly, in seismology, analyzing the arrival times of different components of seismic waves helps characterize Earth's internal structure, relying on the concept of group velocity for energy propagation.