Phase velocity in a waveguide
The phase velocity of a wave in a waveguide refers to the speed at which the phase of the wave propagates through the waveguide. It is denoted by 'vp'.

Explanation
In a waveguide, the condition that always holds true is that the phase velocity is greater than the speed of light (c). This can be explained as follows:

1. Definition of phase velocity: The phase velocity of a wave is given by the ratio of the wavelength (λ) to the time period (T) of the wave. Mathematically, it is expressed as vp = λ/T.

2. Propagation of waves in a waveguide: In a waveguide, electromagnetic waves propagate by bouncing off the walls of the guide. The waves can undergo multiple reflections and interference, resulting in a complex propagation pattern.

3. Effect of waveguide dimensions: The dimensions of the waveguide, such as its width and height, determine the allowed propagation modes and the cutoff frequencies for these modes. Each mode has a characteristic phase velocity associated with it.

4. Phase velocity and cutoff frequency: The cutoff frequency of a particular mode in a waveguide is the frequency below which the mode cannot propagate. At frequencies below the cutoff frequency, the phase velocity becomes imaginary, indicating that the wave cannot propagate in the waveguide.

5. Relation between phase velocity and speed of light: In a waveguide, the phase velocity can exceed the speed of light in vacuum (c). This is possible because the phase velocity is not a measure of the actual speed of energy propagation, but rather the speed at which the phase of the wave propagates.

6. Group velocity: The group velocity of a wave in a waveguide refers to the speed at which the energy of the wave propagates. It is denoted by 'vg'. The group velocity is always less than or equal to the speed of light in vacuum (c).

7. Comparison of phase and group velocities: The phase velocity and group velocity are related but not the same. While the phase velocity can exceed the speed of light in a waveguide, the group velocity cannot. This is because the group velocity represents the actual speed of energy transfer, which is limited by the speed of light.

Therefore, in a waveguide, the condition that always holds true is that the phase velocity is greater than the speed of light (c). This condition allows for the propagation of waves in the waveguide while ensuring that the group velocity remains within the limits of the speed of light.