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Expression for Standing Waves

The stationar...

Question

The stationary wave $y = 2 a$ $s i n k x$ $c o s ω t$ in a stretched string is the result superposition of $y_{1} = a s i n (k x - ω t)$ and

A

y_{2} = a c o s (k x + ω t)

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B

y_{2} = a s i n (k x + ω t)

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C

y_{2} = a c o s (k x - ω t)

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D

y_{2} = a s i n (k x - ω t)

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Solution

The correct option is B $y_{2} = a s i n (k x + ω t)$
$y_{1} = a s i n (k x - ω t)$
$y_{2} = a s i n (k x + ω t)$
According to the principle of superposition, the resultant wave is
$y = y_{1} + y_{2} = a s i n (k x - ω t)$
Using trigonomatric identity
$s i n (A + B) + s i n (A - B) = 2 s i n A c o s B$
we get, $y = 2 a s i n (k x) \times c o s (ω t)$

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