Question

A guitar string 60cm long has a linear mass density of 6.50x10^-3kg/m. If this string plays a fundamental frequency of 220Hz, determine the:

i) tension in the string

ii) frequency and wavelength of the 2nd, 3rd and 4th harmonics.

Answer 1

(a) We can find the tension in the string from the formula:

$$f_n=\dfrac{n}{2L}\sqrt{\dfrac{T}{\mu}},$$ $$f_1=\dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}},$$ $$T=4L^2f_1^2\mu,$$ $$T=4\cdot(0.6\ m)^2\cdot(220\ Hz)^2\cdot6.5\cdot10^{-3}\ \dfrac{kg}{m}=453\ N.$$

(b) Let's first find the frequency of the 2nd, 3rd and 4th harmonics:

$$f_2=\dfrac{2}{2\cdot0.6\ m}\sqrt{\dfrac{453\ N}{6.5\cdot10^{-3}\ \dfrac{kg}{m}}}=440\ Hz,$$ $$f_3=\dfrac{3}{2\cdot0.6\ m}\sqrt{\dfrac{453\ N}{6.5\cdot10^{-3}\ \dfrac{kg}{m}}}=660\ Hz,$$ $$f_4=\dfrac{4}{2\cdot0.6\ m}\sqrt{\dfrac{453\ N}{6.5\cdot10^{-3}\ \dfrac{kg}{m}}}=880\ Hz.$$

Let's find the velocity of the wave in the string:

$$v=\sqrt{\dfrac{T}{\mu}}=\sqrt{\dfrac{453\ N}{6.5\cdot10^{-3}\ \dfrac{kg}{m}}}=264\ \dfrac{m}{s}.$$

Finally, we can find the wavelength of the 2nd, 3rd and 4th harmonics:

$$\lambda_2=\dfrac{v}{f_2}=\dfrac{264\ \dfrac{m}{s}}{440\ Hz}=0.6\ m,$$ $$\lambda_3=\dfrac{v}{f_3}=\dfrac{264\ \dfrac{m}{s}}{660\ Hz}=0.4\ m,$$ $$\lambda_4=\dfrac{v}{f_4}=\dfrac{264\ \dfrac{m}{s}}{880\ Hz}=0.3\ m.$$