Assertion :Function $ε = a e^{- α x} cos (ω t - k x)$ where $a, α, ω$ and $k$ are constants, represents a plane wave. Reason: Any differentiable function $f (t - α x)$ provides a solution of wave equation.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Both Assertion and Reason are incorrect

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Solution

The correct option is D Both Assertion and Reason are incorrect
Let us take $y_{1} = a c o s (k x - w t)$
Let the other wave equation be $y_{2} = - a c o s (k x + w t)$
On superposition
$y = y_{1} + y_{2} = a c o s (k x - w t) - a c o s (k x + w t)$
$= a [c o s k x c o s w t + s i n k x s i n w t - c o s k x c o s w t + s i n k x s i n w t]$
$y = 2 a s i n k x s i n w t$
At $x = 0$ we get $s i n k x = 0$
$\Rightarrow$ $y = 0$
Hence, $x = 0$ is a node.

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