Question

What are stationary waves? Discuss the formation of stationary waves at a free boundary, and hence explain their changes with respect to the position and time. 

Answer 1

The stationary wave, is a wave which oscillates in time but whose peak amplitude profile does not move in space. The equation of standing wave on an infinite string is:

$$y(x,t) = A\sin\left( \dfrac{2\pi x}{\lambda} \right)\cos(\omega t)$$

where $A$ is an amplitude, $\lambda$ is a wavelength, $\omega$ is an angular frequency, and $x,t$ are position and time coordinates respectively. Considering the string with one fixed $x = 0$ and one free $x = L$ ends. Thus, at the fixed end we have the node $y(0,t) = 0$, at the free end we have tha anti-node $y = y_{max}$. Thus, it should be:

$$\sin\left( \dfrac{2\pi L}{\lambda} \right) = 1$$

Expressing the wavelength, obtain:

$$\lambda= \dfrac{4L}{n}, \space \space n = 1,3,5...$$

Answer. Thus, we have antinode at the free end, that oscillates in time as $\cos(\omega t)$.