Question

A string of length L fixed at both ends vibrates in its fundamental mode at a frequency ν 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. 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

Wavelength is ratio of twice of $\pi\ and\ \lambda$

Explanation:

Given, Length of string = L

Velocity of the wave is given by -

V = $\sqrt\frac {T}{m}$

a.) Wavelength, $\lambda = \frac {Velocity}{Frequency}$

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

Wave Number, K = $\frac {2\pi}{\lambda} = \frac {2\pi}{2L} = \frac {\pi}{L}$

(b) Equation of the stationary wave:

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

= $Acos(\frac {2\pi x}{2L}) sin(\frac {2\pi Vt}{2L}) (As\ \nu = (\frac {V}{2L}))$

= $Acos(\frac {\pi x}{L}) sin(\frac {2\pi}{ \nu t})$