Question

Including time variation, the phase expression for a wave propagating in the z-direction is ωt-ßz. For a constant phase point on the wave, this expression is constant, take the time derivative to derive velocity expression in (2-53) β= ω/c = 2πf/c

Answer 1

To derive the velocity expression using the given phase expression, we need to take the time derivative of the phase expression and equate it to zero since we are looking for a constant phase point on the wave.

Given the phase expression ωt - βz, let's take the time derivative:

d/dt (ωt - βz) = ω - 0 = ω

Since we want the time derivative to be zero, we have ω = 0.

Now, we can equate the obtained value of ω to the expression β = ω/c:

0 = ω/c

Solving for ω, we have ω = 0.

Substituting this value back into the expression β = ω/c:

β = 0/c = 0

Therefore, the velocity expression (2-53) is β = ω/c = 2πf/c, with β equaling zero in this case.