How does the frequency of a wave affect its power transmission?

[clairez93]

Homework Statement

Two identical but separate strings, with the same tension, carry sinusoidal waves with the same amplitude. Wave A has a frequency that is twice that of wave B and transmits energy at a rate that is ____ that of wave B.
A) half
B) twice
C) one fourth
D) four times
E) eight times

Homework Equations

$$P = \frac{1}{2}\mu\omega^{2}A^{2}v$$
$$v = \lambda$$$$f$$

The Attempt at a Solution

$$V_{B} = \lambda$$$$f_{B}$$
$$V_{A} = \lambda$$$$(2f_{B})$$ = $$2V_{B}$$
$$P_{B} = \frac{1}{2}\mu\omega^{2}A^{2}v_{B}$$
$$P_{A} = \frac{1}{2}\mu\omega^{2}A^{2}2v_{B}$$ = $$2P_{B}$$

Thus my answer is B.

However, the answer key says D. What did I do wrong?

 

[Dick]

If the strings are identical (same mu) and have the same tension, the velocity v is also identical. You can't assume lambda is the same, it can't be. What changes in your power formula is omega. How much does it change?

 

[clairez93]

Ah okay, so w = 2pi / f.
Therefore the Pa = 4Pb.
Thank you, that makes sense.