Question

A string of length L fixed at both ends vibrates in its fundamental mode at a frequency v and a maximum apmplitude A. (a) Find the wavelength and the wave number k. (b) Take the origin at one end of the string and the X-axis along the string. Take the Y-axis along the direction of the displacement. Take t = 0 at the instant when the middle point of the string passes through its mean position and is going towards the positive y-direction. Write the equation describing the standing wave.

Answer 1

Fundamental frequency,

$v=\frac{1}{2L}\sqrt{\frac{T}{m}}$

On $\sqrt{\frac{T}{m}}$ = velocity of wave

(a) Wavelength, $\lambda =\frac{\text{velocity}}{\text{frequency}}$

$=\frac{\sqrt{\frac{T}{m}}}{\frac{1}{2L}\sqrt{\frac{T}{m}}}=2L$

and wave number,

$K=\frac{2\pi }{\lambda }=\frac{2\pi }{L}=\frac{\pi }{L}$

(b) Therefore, equation of the stationary wave is

$y=Acos\left(\frac{2\pi x}{\lambda }\right)sin\left(\frac{2\pi Vt}{2L}\right)$

$=Acos\left(\frac{2\pi x}{2L}\right)sin\left(\frac{2\pi Vt}{2L}\right)$

$=Acos\left(\frac{\pi x}{L}\right)sin\left(\pi vt\right)v$

$=\frac{V}{2L}$ [because v $=\left(\frac{V}{2L}\right)$]