A guitar has two E-strings. On a particular guitar the lower frequency E-string has a radius of 1.45_mm and is made of a material with a density of 7,526kg/m3. The low frequency E-note is 164.8_Hz. The length of the string is 58.6_cm? What should the mass per unit length (in kg/m) of the other E string (two octaves above the lower one) be assuming the tension in each string is the same?

Question

jamesj · Answer

The fundamental relationship you need to use here is that the frequency, f, of a string varies with the tension T and linear density $ \mu $ as:
$$ f = \sqrt{\frac{T}{\mu}} $$
The frequency of the low E is 164.8 Hz.  The other E is two octaves higher and hence has frequency
$$ f = 164.8 \times 2^2 \ Hz = 659.2 \ Hz $$

If the tension T is invariant, that means the linear density must be some multiple of the original density.  Can you see what it must be?   If necessary, write out the above equation twice, one for each E string and see.

anonymous · Answer

thanks!