A wave represented by the equation y = A cos ` (ks - omega t)` superposes on another wave to produce a stationary wave with a node at s = 0 . What is the equation of this second wave ?

Explanation

As a stationary wave is produced . The second wave will have the same amplitude, frequency and velocity. But it will be oppositely directed. So its general equation will be .
`y. = A cos (ks + omega t + theta) ` , where `theta` = initial phase
According to the principle of superposition, the equation of the resultant stationary wave is
`y,= y + y. = A [cos (ks - omega t) + cos (ks + omega t + theta)]`
`y , = 0 ` at the point x = 0 due to the presence of a node .
So, ` 0 = A [ cos (-omega t ) + cos (omega t + theta)]`
or, `cos omega t + cos (omega t + theta ) = 0 `
or, cos `omega t = - cos (omega t + theta ) = cos (omega t + theta + pi)`
Hence , ` omega t = omega t + theta + pi or theta = - pi `
So , the required rquation is
` y . = A cos (ks + omega t - pi)`
or , ` y. = A - cos (ks + omega t)` .