Changing string length -> changing fundamental freq.

Lavabug

1. The problem statement, all variables and given/known data
Don't have the solution, just want to check if I did this properly.

A 0.5 m violin string fixed at both ends has its first harmonic or fundamental frequency at 440 Hz. Assuming the string is non dispersive, calculate the length it should have so its new fundamental frequency will be 528 Hz.


2. Relevant equations
The nth harmonic's wavelength is 2L/n, where L is the length of the string.
[tex]\lambda[/tex][tex]\upsilon[/tex] = v

3. The attempt at a solution
Apparently the phase velocity of the given fundamental harmonic is 440m/s.

If I want a new [tex]\upsilon[/tex] of 528 Hz, I'd need a string of length = v/2[tex]\upsilon[/tex], so 440/2(528) = 0.416m. A shorter string = higher frequency which makes sense, but my question is: how is the velocity of the wave still 440 m/s? I just shortened the string, wouldn't the tension be greater necessarily(or the linear density decreased as a consequence), hence the velocity would change?

For the sake of simplicity: does the new fundamental harmonic at 528Hz still propagate at 440m/s along the string?

ehild

You can choose the tension in the shorter string the same as in the longer one. (It is not stretched to the original length). So the linear density stays the same.




ehild