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

In summary, two identical 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 four times that of wave B. The formula for power (P = 1/2 * mu * omega^2 * A^2 * v) shows that the velocity and angular frequency are the only variables that differ between the two waves, resulting in wave A having four times the power of wave B. The answer to the given question is D.
  • #1
clairez93
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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



[tex]P = \frac{1}{2}\mu\omega^{2}A^{2}v[/tex]
[tex]v = \lambda[/tex][tex]f[/tex]

The Attempt at a Solution



[tex]V_{B} = \lambda[/tex][tex]f_{B}[/tex]
[tex]V_{A} = \lambda[/tex][tex](2f_{B})[/tex] = [tex]2V_{B}[/tex]
[tex]P_{B} = \frac{1}{2}\mu\omega^{2}A^{2}v_{B}[/tex]
[tex]P_{A} = \frac{1}{2}\mu\omega^{2}A^{2}2v_{B}[/tex] = [tex]2P_{B}[/tex]

Thus my answer is B.

However, the answer key says D. What did I do wrong?
 
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  • #2
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?
 
  • #3
Ah okay, so w = 2pi / f.
Therefore the Pa = 4Pb.
Thank you, that makes sense.
 

1. What are waves?

Waves are a disturbance or oscillation that travels through space and matter, transferring energy from one point to another without any actual movement of the medium itself.

2. What is frequency?

Frequency is the number of complete cycles or vibrations of a wave that occur in a given amount of time. It is measured in Hertz (Hz), which is equal to one cycle per second. Higher frequencies correspond to shorter wavelengths and more energy.

3. How is wave frequency related to power?

The power of a wave is directly proportional to its frequency. This means that as the frequency of a wave increases, so does its power. This is why higher frequency waves, such as visible light and gamma rays, are more energetic and have the potential to cause more damage than lower frequency waves, such as radio waves and microwaves.

4. How is the power of a wave calculated?

The power of a wave is calculated by multiplying the square of the amplitude (maximum displacement) of the wave by its frequency. This equation, P = A^2 * f, applies to all types of waves, whether they are sound waves, electromagnetic waves, or mechanical waves.

5. Can power be increased by increasing the amplitude of a wave?

Yes, increasing the amplitude of a wave will also increase its power. However, this increase in power is not directly proportional to the increase in amplitude. In fact, the power of a wave increases exponentially as the amplitude increases, which is why high amplitude waves can be potentially dangerous.

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