Question

The wave function of a wave #1, traveling in the positive x-direction, is: y1(x,t) = 1.50 cos (2.31x - 33.6t) where x is in meters and t is in seconds. Wave #2, also traveling in the positive x-direction, transports energy 9 times faster than wave #1, and has the following wave function: y2(x,t) = 2.50 cos (14.3x - wt) where x is in meters and t is in seconds. The value of w is unknown but you can calculate it with the information given in this problem. What is the frequency of wave #2, in Hertz?

Answer 1

The frequency of wave #2 is calculated by recognizing that its energy transport speed is 9 times that of wave #1. Since energy transport speed is related to frequency, and the frequency of wave #1 is 5.35 Hz, the frequency of wave #2 is 16.05 Hz.

Step-by-step explanation:

To calculate the frequency of wave #2, we first need to understand how the energy transport speed is related to the wave parameters. The energy transport speed is proportional to both the amplitude squared and the square of the frequency. Since wave #2 transports energy 9 times faster than wave #1, and knowing the amplitude is irrelevant in this context, we can infer that the frequency squared of wave #2 is 9 times that of wave #1.

Looking at the wave function of wave #1, y1(x,t) = 1.50 cos (2.31x - 33.6t), we identify the angular frequency (ω) as 33.6 s⁻¹. The frequency (f) can be found using the relationship ω = 2πf, giving us f1 = ω / (2π) = 33.6 s⁻¹ / (2π) ≈ 5.35 Hz for wave #1.

Since the frequency of wave #2 is √9 times the frequency of wave #1, the frequency of wave #2, f2, will be 3 times f1, because √9 = 3. This gives us f2 = 3 × 5.35 Hz = 16.05 Hz. Therefore, the frequency of wave #2 is 16.05 Hertz (Hz).