Question

A 2 m long string fixed at both ends is set into vibrations in its first overtone. The wave speed on the string is 200 m s⁻¹ and the amplitude is 0⋅5 cm. (a) Find the wavelength and the frequency. (b) Write the equation giving the displacement of different points as a function of time. Choose the X-axis along the string with the origin at one end and t = 0 at the instant when the point x = 50 cm has reached its maximum displacement.

Answer 1

Given:
Length of the string (L) = 2.0 m
Wave speed on the string in its first overtone (v) = 200 m/s
Amplitude (A) = 0.5 cm

(a) Wavelength and frequency of the string when it is vibrating in its 1^st overtone (n = 2):
$L=\frac{n\lambda }{2}$
$$\begin{gathered} \Rightarrow \lambda =L=2\mathrm{m} \\ \Rightarrow f=\frac{\nu }{\lambda }=\frac{200}{2}=100\mathrm{Hz} \end{gathered}$$
(b) The stationary wave equation is given by:

$$\begin{gathered} y=2A\cos \frac{2\pi x}{\lambda }\sin \frac{2\pi vt}{\lambda } \\ =0.5\cos \frac{2\pi x}{2}\sin \frac{2\pi \times 200t}{2} \\ =\left(0.5\mathrm{cm}\right)\cos \left[\left(\pi {\mathrm{m}}^{-1}\right)x\right]\sin \left[\left(200\pi {\mathrm{s}}^{-1}\right)t\right] \end{gathered}$$