Question

A wave represented by the equation y = acos(kx - wt) is superposed with another wave to form a stationary
wave such that the point x = 0 is a node. The equation for the other wave is
(2) -acos(kx + wt)
(1) asin(kx + wt)
(3) -acos(kx-wt)
(4) -asin(kx - wt)
​

Answer 1

A wave represented by the equation y = acos(kx-ωt) is superimposed with another wave to form a stationary

wave such that the point x = 0 is a node. The equation for the other wave is -acos(kx + ωt)

Option (1) is correct.

Explanation:

The given wave equation:

$y=a\cos(kx-\omega t)$

It can be written as

$y=a[\cos \omega t.\cos kx+\sin \omega t.\sin kx]$

$\implies y=a\cos \omega t.\cos kx+a\sin \omega t.\sin kx$

If the superimposed wave has a node at x = 0 then the term containing cos kx should be zero

i.e. $a\cos\omega t\cos kx$ should be zero

Thus, the other wave can have the equation of the form

$y=-a\cos \omega t.\cos kx+a\sin \omega t.\sin kx$

or, $y=-a\cos(kx+\omega t)$

Therefore, option (1) is correct.

Hope this answer is helpful.

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