How Does Changing String Density Affect Piano Sound Frequency?

[Lavabug]

Homework Statement

The fundamental frequency of a piano string with mass of 7g and length 80cm is 261.63Hz.

What is the tension on the string?
If we coil the string duplicating its density(without changing length I presume), what is its new fundamental frequency?
If 16000Hz is the highest frequency audible to a listener, what is the highest harmonic he can hear?

Homework Equations

Wavelength of the nth harmonic: \lambda\ = 2L/n
v=\lambda\nu
v = sqrt(T/\rho)

The Attempt at a Solution

The fundamental wavelength is just twice the length of the string, so 160cm.
The given linear mass density is 7/80 g/cm.
using v/160 = \nu and v = sqrt(T/\rho)
I get a phase velocity of 418.6 m/s and solving for the tension I get 1533 N.

Duplicating the density of the string while keeping the tension the same gives me a slower phase velocity of 295.97m/s, is this correct? Sounds reasonable from what I know from instrument strings but I'm just making sure.

The new fundamental frequency should be at the new velocity/2L, so 295.97/1.6 = 184.88Hz.

For the last part, I keep the phase velocity the same (295.97m/s) and set it equal to lamda*nu. I know the wavelength of the nth harmonic is 2L/n, so I substitute that and nu = 16000Hz as its given, then I attempt to solve for n which gives me 86.49, so if I round up I would say the 87th harmonic is the highest one that the listener can hear. Did I do this step correctly?

 

[diazona]

Assuming that by "duplicating" you mean "doubling," then yes, that makes sense. But how do you justify rounding the harmonic number up instead of down?

 

[Lavabug]

Thanks for the reply. Yes I meant doubling and 86th harmonic. I'm mainly concerned about the last bit, is it right?

 

[diazona]

It seems like you did it the right way.