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.

 

[paranormal]

Your calculations and approach seem correct. The only suggestion I would make is to check your units and make sure they are consistent throughout the problem. For example, in your first calculation for tension, you used a linear mass density of 7/80 g/cm, but then in your calculation for the new fundamental frequency, you used a length of 1.6 meters. Make sure to convert all units to the same system (e.g. grams to kilograms, centimeters to meters) before plugging them into equations. Other than that, your solution looks good.