How Does Doubling Frequency Affect Wavelength on a Constant Tension String?

[BizzPhizz]

Homework Statement

This is a communication question, no variables given:

Predict what happens to the wavelength of a wave on a
string when the frequency is doubled. Assume that the
tension in the string remains the same. Confirm your
prediction mathematically.

The Attempt at a Solution

All I know is that my teacher said in a string the wave length is constant, so help me...

Please also show it mathematically c:

Thanks,
BizzPhizz

 

[haruspex]

What do you know about harmonics in strings, and how their wave lengths in the string relate to the string length?

 

[BizzPhizz]

Okay, Don't rage at me if I get this wrong, but

V=λ/τ or V=∫λ

∫ being frequency, I know the question stated there was no change in tension of the string so velocity must be constant.

I don't know why my teacher said wave length in a string is constant, if that's the case there would be no pitch.

So I can come to a conclusion that if frequency doubles, wave length halves..

 

[haruspex]

I think I may have misinterpreted the question before. I thought this was about standing waves, but now I suspect it's about traveling waves. If so, the scenario is that some source is generating waves at one end of the string, and its frequency is then doubled, right? (But note that if this is right then your teacher's remark about wave length being constant does not apply here.)
What you can be sure about is that the velocity of the waves is constant (because the tension is constant). That being all correct, I agree with your conclusion.

 

[Nickie]

I can provide a response to this question. When the frequency of a wave on a string is doubled, the wavelength will also be halved. This is known as the inverse relationship between wavelength and frequency. Mathematically, this can be represented by the equation:

λ = v/f

Where λ is the wavelength, v is the velocity of the wave, and f is the frequency. Since the velocity of the wave on a string remains constant, the only variable that changes is the frequency. Therefore, when we double the frequency, the wavelength will be halved.

To confirm this, let's assume that the frequency is initially f and the wavelength is λ. When we double the frequency, it becomes 2f. Plugging this into our equation, we get:

λ = v/2f

We can see that the wavelength has been halved.

I hope this helps answer your question, BizzPhizz. Let me know if you have any further questions or need clarification.