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)

1. Find the wavelength and the wave number k. 
2. 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

Given:
Length of the string = L
Velocity of wave is given as:
\[V = \sqrt{\frac{T}{m}}\]
(a) \[\text{Wavelength},   \lambda = \frac{\text{Velocity}}{\text{Frequency}}\]

\[\Rightarrow \lambda = \frac{\sqrt{\frac{T}{m}}}{\frac{1}{2L}\sqrt{\frac{T}{m}}} = 2L\] 

\[\text{ Wave  number,}   K = \frac{2\pi}{\lambda} = \frac{2\pi}{2L} = \frac{\pi}{L}\]
(b) Equation of the stationary wave:

We have `y = Asin (2pix)/lambda * cos(omegat + phi)`

Given, At t = 0

`x = L/2`

y = 0

∴` 0 = Asin  (2piL/2)/(2L)cos(0 + phi)`

or cos Φ = 0

or Φ = π/2

Now,

`y = Asin  (2pix)/(2L)cos(2 pi f t + pi/2)`

= `Asin(pix)/Lsin(2 pi f t)`