A nylon guitar string has a linear density of 7.20 g/mand is under a tension of 150 N.The fixed supports are distance D = 90.0 cmapart. The string is oscillating in the standing wave pattern shown in Fig.16-39. Calculate the (a) speed, (b) wavelength, and (c) frequency of the traveling waves whose superposition gives this standing wave.

The linear density of the guitar string is$\mu =7.20\mathrm{g}/\mathrm{m}\mathrm{or}7.20\times {10}^{-3}\mathrm{kg}/\mathrm{m}$ .

Tension in the guitar string is$T=150\mathrm{N}$

Distance between two fixed supports is L = 90 cm

We can find the wave speed from the tension and the linear density of the guitar string using the relation between them. From the figure, we can predict the wavelength of the wave in terms of the length of the string. Then from the above two quantities, we can easily find the frequency of the traveling waves.

Formula:

The formula of wave speed, $v=\sqrt{\frac{\tau }{\mu }}..........\left(1\right)$

The wavelength of standing wave in terms of length, $\lambda =\frac{2L}{n}......\left(2\right)$

The frequency of a wave, role="math" localid="1660981047869" $f=\frac{v}{\lambda }......\left(3\right)$

Using equation (1) and the given values, we get the speed of the wave as:

$$\begin{gathered} v=\sqrt{\frac{150\mathrm{N}}{7.20\times {10}^{-3}\mathrm{kg}/\mathrm{m}}} \\ =144.34\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the value of wave speed is 144 m/s

From the given figure, the wave is third harmonic wave. Thus, the wavelength of the standing wave using equation (2) for n = 3 is given as:

$$\begin{gathered} \lambda =\frac{2L}{3} \\ =\frac{2\times 90.0}{3} \\ =60.0\mathrm{cm} \end{gathered}$$

Hence, the value of wavelength of the standing wave is 60.0 cm

Using equation (3) and the given values, we get the frequency of the wave as given:

$$\begin{gathered} f=\frac{144}{0.600} \\ =240.6 \\ ~241\mathrm{Hz} \end{gathered}$$

Hence, the value of frequency of the wave is 241 Hz