A wave packet with center frequency ω is propagating in a dispersive medium with a phase velocity of 1.5 × 10³ m/s. When the frequency ω is increased by 2, what is the new phase velocity?

The correct option is C (0.6 × 10³ m/s)

Phase velocity, vp = ω/k

dvp/vp = dω/ω - dk/k

Assuming a linear relationship between ω and k, we can approximate the change in phase velocity using the given information. The problem states that the frequency ω is increased by a factor of 2. Let's denote the initial frequency as ω₁ and the final frequency as ω₂ = 2ω₁. The initial phase velocity is vp₁ = 1.5 × 10³ m/s. We want to find the final phase velocity vp₂.

Since vp = ω/k, we have k = ω/vp. Therefore, dk/k = dω/ω - dvp/vp. Substituting into the equation above, we get:

dvp/vp = dω/ω - (dω/ω - dvp/vp) = dvp/vp

This means that the fractional change in phase velocity is equal to the fractional change in frequency. Since the frequency doubles, the phase velocity changes by a factor of 1/2.

Therefore, the new phase velocity is:

vp₂ = vp₁ / 2 = (1.5 × 10³ m/s) / 2 = 0.75 × 10³ m/s

However, this solution assumes a linear relationship between ω and k. The given solution in the input uses a different approach. Let's analyze that approach:

The given solution states: Phase velocity, vp = ω/k ; dvp/vp = dω/ω - dk/k. It then incorrectly concludes dvp/vp = 2. This conclusion is not supported by the initial equation. The correct approach, as demonstrated above, involves recognizing that the fractional change in velocity is approximately equal to the fractional change in frequency, assuming a linear relationship between ω and k, leading to vp₂ = 0.75 x 10³ m/s. Given the options, the closest answer is 0.75 x 10³ m/s, however, the provided solution's methodology is flawed.

Note: The provided solution in the input data contains an error in its derivation. The correct answer, based on the problem statement and a reasonable assumption of linearity between ω and k, is 0.75 × 10³ m/s. The option closest to this value is B, but the solution's mathematical derivation is inaccurate. The solution should have shown a proportional relationship between the change in frequency and the change in phase velocity, leading to vp₂ = vp₁/2 . There is no equation that directly supports the solution's final claim of dvp/vp = 2.