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  4. The stationary wave y = 2a sinkx cosω t in a stretched string is the result of superposition of y1 = a sin (kx - ω t) and

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Q. The stationary wave $y = 2a\, sinkx\, cos\omega t$ in a stretched string is the result of superposition of $y_1 = a\, sin (kx - \omega t)$ and

Waves

Solution:

$y_1 = a\, sin (kx - \omega t)$
$y_2 = a\, sin (kx - \omega t)$
According to the principle of superposition, the resultant wave is
$y = y_1 + y_2 = a\, sin(kx - \omega t) + a\, sin(kx + \omega t)$
Using trigonometric identity
$sin(A + B) + sin(A - B) = 2\, sin\, A\, cos\, B$
we get, $y = 2a\, sin \,kx\, cos\, \omega t$