Question

Consider two waves x and y traveling in the same medium. The two waves carry the same amount of energy per unit time, but wave x has one-fifth the amplitude of wave y. What is the relationship between the frequencies of waves x and y?

1) The frequency of wave x is one-fifth the frequency of wave y.
2) The frequency of wave x is five times the frequency of wave y.
3) The frequency of wave x is the same as the frequency of wave y.
4) The frequency of wave x cannot be determined without additional information.

Answer 1

The frequency of wave x is five times the frequency of wave y because the energy of a wave is proportional to the square of both the amplitude and the frequency.

Step-by-step explanation:

Given two waves x and y traveling in the same medium with the same energy transmission rate, but wave x has one-fifth the amplitude of wave y, we want to determine the relationship between their frequencies. The energy of a mechanical wave is proportional to the square of both the amplitude and the frequency. Therefore, if wave x has one-fifth the amplitude of wave y, for them to carry the same amount of energy, the square of the frequency of wave x must be five squared, or 25 times the square of the frequency of wave y. This means the frequency of wave x must be 5 times the frequency of wave y.

The key relationship can be expressed by the formula E ≈ A^2f^2, where E is energy, A is amplitude, and f is frequency. In this scenario, as energy remains constant and amplitude for wave x is reduced, frequency must increase to compensate, leading us to conclude that the frequency of wave x is five times the frequency of wave y (option 2).