Standing waves on string with increasing tension

[echoi11]

1. The problem statement, all variables, and given/known data
Consider a two-loop standing wave on a string. If we increase the tension without changing the frequency, what kind of standing wave can we obtain?
(a) one-loop (b) three-loop

Homework Equations

Velocity = square root of(T/U)
Wavelength = velocity/ frequency.
f= 2L/(n)

The Attempt at a Solution

I feel that this will become a one loop standing wave because if I plug in 2 in the frequency equation it becomes 2L. Also, it seems if you increase tension, you also increase velocity which increases wavelength, so does that mean it becomes a three loop standing wave instead. I am confused through the process.

 

[scottdave]

I had not hear the term "loop" before. I did some digging. It was as I had guessed. A loop is a half-wavelength, like this picture that I found:

If the number of loops stays constant, what happens to the frequency as tension increases? How are wavelength and frequency related?

 

[haruspex]

if I plug in 2 in the frequency equation it becomes 2L.

I don't follow what your procedure is there. Please explain in more detail.

if you increase tension, you also increase velocity which increases wavelength, so does that mean it becomes a three loop standing wave instead.

The overall length is fixed. How would increasing the length of each loop increase the number of loops?

 

[scottdave]

I play a stringed instrument, so I can intuitively figure this out, from my experience. Try wrapping a rubber band around a pencil. Strum it and notice the pitch. If you can rotate the pencil, such that the length stays the same, and the tension increases, what happens when you strum it? Depending on the length of your band, you most likely will only be able to get one loop mode. If you have access to a guitar, you can try touching the string lightly with a finger in the middle, to create a two loop mode. You could try at 1/3 of the length of the string. What happens to the frequency, when you change modes, and the tension stays the same?