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# The standing waves set upon a string are given by $y = 4\sin \left( {\dfrac{{\pi x}}{{12}}} \right)\cos \left( {52\pi t} \right)$, If x and y are in centimeters and t is in seconds, what is the amplitude of the particle at x=2 cm?A) 12 cmB) 4 cmC) 2 cmD) 1 cm

Last updated date: 17th Jun 2024
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Hint: Standing wave is created when two oppositely travelling waves(with the same frequency) interfere with each other. Peaks of standing waves don’t move specially but oscillates w.r.t time. So, for a particular position maximum amplitude remains always the same.

Suppose, there are two oppositely travelling waves with the equations ${y_1} = A\sin (kx - \omega t)$ and ${y_2} = A\sin (kx + \omega t)$. Now, the resulting standing wave equation created by them is $y = 2A\sin (kx)\cos (\omega t)$ (1)
where, amplitude at any point x is given by: $2A\sin (kx)$.

Here given: Standing wave equation is given as: $y = 4\sin \left( {\dfrac{{\pi x}}{{12}}} \right)\cos \left( {52\pi t} \right)$ To find: Amplitude of the particle at x=2 cm.

Step 1
In the given equation of standing wave, put x=2 to get the equation as:
$y = 4\sin \left( {\dfrac{{\pi \times 2}}{{12}}} \right)\cos \left( {52\pi t} \right) \\ = 4 \times \dfrac{1}{2} \times \cos \left( {52\pi t} \right) = 2\cos \left( {52\pi t} \right) \\$ (2)

Now, comparing the standing wave equation of eq.(2) with eq.(1) we get the amplitude as 2cm.