Consider two waves of the same frequency, different amplitudes A1 and A2 and differing in phase by φ. Let these two waves interfere at x = 0.
The displacement of each wave at x = 0 are y1 = A1 sin ωt y2 = A2 sin (ωt + φ) According to the principle of superposition of waves, the resultant displacement at that point is y1 = y1 + y2 = A1 sin ωt + A2 sin (ωt + φ) Using the trigonometrical identity, sin (C + D) = sin C cos D + cos C sin D, y = A1 sin ωt + A2 ωt cos φ + A2 cos ωt sin φ y = (A1 + A2 cos φ) sin ωt + A2 sin φ cos ωt … (1) Let (A1 + A2 cos φ) = A cos θ … (2) and A2 sin φ = A sin θ … (3) Substituting Eqs. (2) and (3) in EQ. (1), we get the equation of the resultant wave as y = A cos θ sin ωt + A sin θ cos ωt = A sin (ωt + θ) … (4) It has the same frequency as that of the interfering waves. The amplitude A of the resultant wave is given by squaring and adding Eqs. (2) and (3).
Case (1) : When the two interfering waves are in phase, φ = 0. Then, the amplitude of the resultant wave is Thus, the amplitude of the resultant wave is maximum when the two interfering waves are in phase.
Case (2) : When the two interfering waves are out of phase, ivarphi = ipi. Then, the amplitude of the resultant wave is, Thus, the amplitude of the resultant wave is minimum when the two interfering waves .
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Consider two waves of the same frequency, different amplitudes A1 and A2 and differing in phase by φ. Let these two waves interfere at x = 0.
The displacement of each wave at x = 0 are y1 = A1 sin ωt y2 = A2 sin (ωt + φ) According to the principle of superposition of waves, the resultant displacement at that point is y1 = y1 + y2 = A1 sin ωt + A2 sin (ωt + φ) Using the trigonometrical identity, sin (C + D) = sin C cos D + cos C sin D, y = A1 sin ωt + A2 ωt cos φ + A2 cos ωt sin φ y = (A1 + A2 cos φ) sin ωt + A2 sin φ cos ωt … (1) Let (A1 + A2 cos φ) = A cos θ … (2) and A2 sin φ = A sin θ … (3) Substituting Eqs. (2) and (3) in EQ. (1), we get the equation of the resultant wave as y = A cos θ sin ωt + A sin θ cos ωt = A sin (ωt + θ) … (4) It has the same frequency as that of the interfering waves. The amplitude A of the resultant wave is given by squaring and adding Eqs. (2) and (3).
Case (1) : When the two interfering waves are in phase, φ = 0. Then, the amplitude of the resultant wave is Thus, the amplitude of the resultant wave is maximum when the two interfering waves are in phase.
Case (2) : When the two interfering waves are out of phase, ivarphi = ipi. Then, the amplitude of the resultant wave is, Thus, the amplitude of the resultant wave is minimum when the two interfering waves .
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