(a) Find the speed of waves on a violin string of mass $\mathbf{\text{800mg}}$and length$\mathbf{\text{22.0cm}}$if the fundamental frequency is$\mathbf{\text{920Hz}}$.

(b) What is the tension in the string? For the fundamental, what is the wavelength of (c) the waves on the string and (d) the wavelength of sound waves emitted by the string?

$\begin{array}{c}\mathrm{m}=800\text{\hspace{0.17em}mg}\\ =8\times {10}^{\text{-4}}\text{kg}\end{array}$

$\begin{array}{c}\mathrm{L}=22.0\mathrm{cm}\\ =0.2\text{2m}\end{array}$

$\mathrm{f}=920\text{Hz}$

Apply the formulas in terms of wavelength and frequency to find the velocity. Tension can be found from the formula for velocity in terms of tension and linear density.

The expression for the frequency is given by,

${\mathrm{f}}_0=\frac{1}{2\mathrm{L}}\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}$

Where,${\mathrm{f}}_0$is frequency, $L$is length, $\mathrm{T}$ is tension and $\mathrm{\mu }$is mass per unit length.

For fundamental frequency, length of string will be half of wavelength,

$\mathrm{L}=\frac{\mathrm{\lambda }}{2}$

$\begin{array}{c}\mathrm{\lambda }=2\mathrm{L}\\ =2\left(0.22\text{m}\right)\\ =0.44\text{m}\end{array}$

$\begin{array}{c}\text{speed}=\mathrm{v}\\ ={\mathrm{f}}_0\mathrm{\lambda }\\ =\text{920Hz}\left(\text{0.44m}\right)\\ =404.8\mathrm{m}/\mathrm{s}\end{array}$

Hence, the speed of waves on a violin string of mass$\text{800mg}$ and length $\text{22.0cm}$if the fundamental frequency is$\text{920Hz}$ is$404.8\mathrm{m}/\mathrm{s}$ .

Mass per unit length,

$\begin{array}{c}\mathrm{\mu }=\frac{\mathrm{m}}{\mathrm{L}}\\ =\frac{\left({\text{8×10}}^{\text{-4}}\text{kg}\right)}{\text{0.22m}}\\ =3.64\times {10}^{-3}\text{kg/m}\end{array}$

Fundamental frequency is defined as,

${\mathrm{f}}_0=\frac{1}{2\mathrm{L}}\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}$

$\begin{array}{c}\mathrm{T}={\mathrm{f}}_0^2\left(4{\mathrm{L}}^2\right)\mathrm{\mu }\\ ={\left(920\text{Hz}\right)}^2(4\times {\left(0.22\text{m}\right)}^2)\left(3.64\times \frac{{\text{10}}^{\text{-3}}\text{kg}}{\text{m}}\right)\\ =596\mathrm{N}\end{array}$

Hence, the tension in the string is $596\mathrm{N}$.

Wavelength,

$\begin{array}{c}\mathrm{\lambda }=2\mathrm{L}\\ =2\left(0.\text{22m}\right)\\ =0.44\text{\hspace{0.17em}m}\\ =44.\text{0\hspace{0.17em}cm}\end{array}$

Hence, the wave on the string is$44.0\mathrm{cm}$ .

Since, speed of sound,

$\mathrm{v}=343\mathrm{m}/\mathrm{s}$ (at, room temperature),

$\begin{array}{c}\mathrm{\lambda }=\frac{\mathrm{v}}{{\mathrm{f}}_0}\\ =\frac{343\mathrm{m}/\mathrm{s}}{\text{920Hz}}\\ =0.373\mathrm{m}\end{array}$

Hence, the wavelength of sound waves emitted by the string is $0.373m$.