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Question

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


Solution

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.
$$y_1 = a \space sin2\pi(\frac{t}{T}-\frac{x}{\lambda})$$
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 $$-x$$
$$y_2 = a \space sin2\pi(\frac{t}{T}+\frac{x}{\lambda})$$
Now, according to principle of superposition, the resultant displacement will be: 
$$y = y_1+y_2$$
$$y = a \space sin2\pi(\frac{t}{T}-\frac{x}{\lambda}) + a \space sin 2 \pi(\frac{t}{T}+\frac{x}{\lambda}) $$
Using $$sinA + sinB = 2 sin[(A+B)/2] cos[(A-B)/2]$$, 
$$y=a [2sin (\frac{2\pi t}{T}) cos  (\frac{2\pi x}{λ})]$$
$$y = A sin(\frac{2 \pi t}{T})$$ where $$A = 2a \space cos(\frac{2 \pi x}{\lambda})$$
The above equation of $$y$$ is an equation of stationary wave and it's amplitude is $$A = 2a \space cos(\frac{2 \pi x}{\lambda})$$. 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. 



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Physics

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