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 $200m{s}^{-1}$ 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

$V=200m/s,2A=0.5m$

(a) The string is vibrating in its $1^{st}$ overtone

$\Rightarrow \lambda =L=2m$

$\Rightarrow f=\frac{v}{\lambda }=\frac{200}{2}$

$=100Hz$

(b) The stationary wave equation is given by

$y=2Acos\frac{2\pi x}{\lambda }sin\frac{2\pi Vt}{\lambda }$

$=0.5cos\frac{2\pi x}{2}sin\frac{2\pi \times 200t}{2}$

$=\left(0.5cm\right)cos$

$\left[\right(\pi m^{-1}\left)x\right]sin\left[\right(200\pi s^{-1}\left)t\right]$