Three sound sources S₁ , S₂ , and S₃ of equal intensity are arranged along a straight line such that S₁ S₂ = S₂ S₃ , as depicted in the figure. At a distant point P , the wave arriving from S₂ leads the wave from S₁ in phase by 120^ . Furthermore, the wave arriving from S₃ leads the wave from S₂ by 120^ . Determine the resultant intensity of the sound at P .

Step-by-step solution

Let the intensity of each individual sound source be I₀ , and the corresponding amplitude be A₀ . The wave arriving from S₂ leads the wave from S₁ by 120^ , and the wave from S₃ leads the wave from S₂ by 120^ . The equations of the three waves arriving at point P can be written as: y₁ = A₀ ( t) y₂ = A₀ ( t + 120^ ) y₃ = A₀ ( t + 240^ ) The resultant displacement at point P is given by the principle of superposition: y = y₁ + y₂ + y₃ y = A₀ [ ( t) + ( t + 120^ ) + ( t + 240^ )] Using the trigonometric identity C + D