The resultant amplitude, when two waves of same frequency but with amplitudes $a_{1}$ and $a_{2}$ superimpose with a phase difference of $π / 2$ will be

a_{1}^{2} + a_{2}^{2}

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\sqrt{a_{1}^{2} + a_{2}^{2}}

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a_{1} - a_{2}

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a_{1} + a_{2}

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Solution

The correct option is A $\sqrt{a_{1}^{2} + a_{2}^{2}}$
The equation of the two superimposing waves can be give by
$y_{1} = a_{1} sin (ω t)$ and $y_{2} = a_{2} sin (ω t + π / 2) = a_{2} cos (ω t)$ .
So, the equation of the resultant wave is $y = y_{1} + y_{2} = a_{1} sin (ω t) + a_{2} cos (ω t)$ .
Assuming $a_{1} = a cos (ϕ)$ and $a_{2} = a sin (ϕ)$ , we get
$y = a sin (ω t + ϕ)$ .
Here $a_{1}^{2} + a_{2}^{2} = a^{2} ({sin}^{2} (ϕ) + {cos}^{2} (ϕ)) = a^{2} \Rightarrow a = \sqrt{a_{1}^{2} + a_{2}^{2}}$ .

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