Question

The resultant amplitude, when two waves of same frequency but with amplitudes $a_{1}$ and $a_{2}$ superimpose with a phase difference of $\pi / 2$ will be
(A) $a_{1}^{2}+a_{2}^{2}$
(B) $\sqrt{a_{1}^{2}+a_{2}^{2}}$
(C) ${{a}_{1}}-{{a}_{2}}$
(D) ${{a}_{1}}+{{a}_{2}}$

Answer 1

Hint: We know that the amplitude of a wave refers to the maximum amount of displacement of a particle on the medium from its rest position. In a sense, the amplitude is the distance from rest to crest. Similarly, the amplitude can be measured from the rest position to the trough position. The amplitude of a variable is the measure of how far, and in what direction, that variable differs from zero. Thus, signal amplitudes can be either positive or negative. Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting messages with a radio carrier wave. In amplitude modulation, the amplitude (signal strength) of the carrier wave is varied in proportion to that of the message signal, such as an audio signal.

Complete step by step answerWe know that the equation of the two superimposing waves can be given by
$\mathrm{y}_{1}=\mathrm{a}_{1} \sin (\omega \mathrm{t})$ and $\mathrm{y}_{2}=\mathrm{a}_{2} \sin (\omega \mathrm{t}+\pi / 2)=\mathrm{a}_{2} \cos (\omega \mathrm{t})$
So, the equation of the resultant wave is $\mathrm{y}=\mathrm{y}_{1}+\mathrm{y}_{2}=\mathrm{a}_{1} \sin (\omega \mathrm{t})+\mathrm{a}_{2} \cos (\omega \mathrm{t})$
Assuming $\mathrm{a}_{1}=\mathrm{a} \cos (\phi)$ and $\mathrm{a}_{2}=\mathrm{a} \sin (\phi),$ we get
$\mathrm{y}=\mathrm{a} \sin (\omega \mathrm{t}+\phi)$
Here $a_{1}^{2}+a_{2}^{2}=a^{2}\left(\sin ^{2}(\phi)+\cos ^{2}(\phi)\right)=a^{2} \Rightarrow a=\sqrt{a_{1}^{2}+a_{2}^{2}}$

Therefore, the correct answer is Option B.

Note: We know that the principle of superposition may be applied to waves whenever two (or more) waves travelling through the same medium at the same time. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave displacements. The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. Superposition is the combination of two waves at the same location. Constructive interference occurs when two identical waves are superimposed in phase.