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. Physics

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