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Two waves are propagating along a taut string that coincides w

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Two waves are propagating along a taut string that coincides with the x-axis. The first wave has the wave function $y_{1} = A cos [k (x - v t)]$ and the second has the wave function $y_{2} = A cos [k (x + v t) + ϕ] .$

For constructive interference at

x = 0, ϕ = π .

For constructive interference at

x = 0, ϕ = 3 π .

For destructive interference at

x = 0, ϕ = π .

For destructive interference at

x = 0, ϕ = 2 π

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Two waves are propagating along a taut string that coincides with the x-axis. The first wave has the wave function

y_{1} = A cos [k (x - v t)]

and the second has the wave function

y_{2} = A cos [k (x + v t) + ϕ] .

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Text solutionVerified

Resultant wave function

y = y_{1} + y_{2}

y = A cos [k (x - v t)] + A cos [k (x + v t) + ϕ]

Using

cos A + cos B = 2 cos \frac{A + B}{2} cos \frac{A - B}{2}

y = 2 A cos [\frac{k ( x - v t ) + k ( x + v t ) + ϕ}{2}] cos [\frac{k ( x - v t ) - k ( x + v t ) + ϕ}{2}]

y = 2 A cos [k x + \frac{ϕ}{2}] cos [\frac{ϕ}{2} - v t]

Now for destructive interference ,

y = 0

i.e

cos [k x + \frac{ϕ}{2}]

must be equal to zero.

For

x = 0, ϕ = π ⟹ cos [k x + \frac{ϕ}{2}] = 0

For constructive interference ,

y

must be maximum i.e

cos [k x + \frac{ϕ}{2}] = 1

- 1

For

x = 0, ϕ = 2 π ⟹ cos [k x + \frac{ϕ}{2}] = - 1

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Question Text	Two waves are propagating along a taut string that coincides with the x-axis. The first wave has the wave function $y_{1} = A cos [k (x - v t)]$ and the second has the wave function $y_{2} = A cos [k (x + v t) + ϕ] .$
Answer Type	Text solution:1
Upvotes	150