Question

A string of length L fixed at both ends vibrates in its fundamental mode at a frequency v and a maximum amplitude 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 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

$V=\frac{1}{2L}\sqrt{\frac{T}{m}}\Rightarrow \sqrt{\frac{T}{m}}=V\times 2L$ [$\sqrt{\frac{T}{m}}=velocityofwave\times wavelength$ ]

(a) Wavelength, $\lambda$= $\frac{velocity}{frequency}$= $\frac{V2L}{V}=2L$

and wave number $=K=2\pi /\lambda =2\pi /2L=\pi /l$

(b) Therefore, equation of the stationary wave is

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

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

$=A\cos \left(\frac{\pi x}{L}\right)\sin \left(\pi vt\right)$ [because $v=\left(V/2l\right)$]