What is the resultant wave obtained for ϕ=(π)/(2) rad when two harmonic waves are y1(x, t)=0.2 s

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What is the resultant wave obtained for ϕ=(π)/(2) rad when...

What is the resultant wave obtained for $\phi=\frac{\pi}{2}$ rad when two harmonic waves are $y_{1}(x, t)=0.2 \sin (x-3 t)$ $y_{2}(x, t)=0.2 \sin (x-3 t+\phi) ?$ (a) $y=0.28 \sin \left(x-3 t+\frac{\pi}{4}\right)$ (b) $y=3 \sin (x-3 t)$ (c) $y=0.28 \sin \left(x-3 t-\frac{\pi}{4}\right)\left(\right.$ d) $y=0.28 \sin \left(x+3 t-\frac{\pi}{4}\right)$

What is the resultant wave obtained for $\phi=\frac{\pi}{2}$ rad when two harmonic waves are $y_{1}(x, t)=0.2 \sin (x-3 t)$
$y_{2}(x, t)=0.2 \sin (x-3 t+\phi) ?$
(a) $y=0.28 \sin \left(x-3 t+\frac{\pi}{4}\right)$ (b) $y=3 \sin (x-3 t)$
(c) $y=0.28 \sin \left(x-3 t-\frac{\pi}{4}\right)\left(\right.$ d) $y=0.28 \sin \left(x+3 t-\frac{\pi}{4}\right)$

Key Concepts

Resultant Amplitude and Phase Shift

When two harmonic waves are added, the resultant wave has an amplitude and phase that depend on the individual amplitudes and their phase difference. The amplitude of the resultant wave can be determined using vector addition (often via phasor representation), and the phase shift is found by considering the relative shift introduced by the phase difference between the waves.

Superposition Principle

This concept involves adding individual wave functions to obtain a resultant wave. In general, when two or more waves overlap in space and time, their amplitudes add algebraically, leading to interference effects. This principle is fundamental in understanding how complex wave patterns arise from simpler ones.

Phase Difference

Phase difference refers to the angular difference between the phases of two waves at a given point in time or space. It plays a critical role in determining the interference pattern when the waves are superposed, affecting both the resulting amplitude and the phase shift of the combined wave.

Trigonometric Identities in Wave Addition

The addition of sine waves of the same frequency but different phases can be simplified using trigonometric identities, such as the sine addition formula. This identity enables the resultant wave to be expressed in a compact form, revealing how the original amplitudes and phase difference combine to affect the overall wave properties.

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