Question

Explain the formation of stationary waves by analytical method. Show the formation of the stationary wave diagramatically.

Answer 1

When two progressive waves of same amplitude and wavelength travelling along a straight line in opposite directions, they superimpose on each other which results in formation of stationary waves.

Consider a progressive wave of amplitude $a$ and wavelength $\lambda$ travelling in the x-axis direction.
$y_1=asin2\pi (\frac{t}{T}-\frac{x}{\lambda })$

This wave is reflected from a free end and it travels in the negative x-axis direction. It will have same characteristics expect $x$ changes to $-x$

$y_2=asin2\pi (\frac{t}{T}+\frac{x}{\lambda })$

Now, according to principle of superposition, the resultant displacement will be:

$y=y_1+y_2$
$y=asin2\pi (\frac{t}{T}-\frac{x}{\lambda })+asin2\pi (\frac{t}{T}+\frac{x}{\lambda })$
Using $sinA+sinB=2sin\left[\right(A+B\left)/2\right]cos\left[\right(A-B\left)/2\right]$,
$y=a\left[2sin\right(\frac{2\pi t}{T}\left)cos\right(\frac{2\pi x}{\lambda }\left)\right]$
$y=Asin\left(\frac{2\pi t}{T}\right)$ where $A=2acos\left(\frac{2\pi x}{\lambda }\right)$
The above equation of $y$ is an equation of stationary wave and it's amplitude is $A=2acos\left(\frac{2\pi x}{\lambda }\right)$. This represents that at some values of $x$ the resultant amplitude is maximum known as antinodes and for some values of $x$ it will be minimum (zero) known as nodes.