Two sound sources, S₁ and S₂ , emitting waves of identical wavelength 20.0 cm , are positioned with a separation of 20.0 cm between them. A detector can move along a line parallel to S₁ S₂ and at a distance of 20.0 cm from it. Initially, the detector lies equidistant from both sources. Assuming the waves emitted by the sources are in phase, determine the minimum distance the detector must be shifted to detect a sound

Step-by-step solution

Given wavelength = 20.0 cm , distance between sources d = 20.0 cm , and distance to the line of movement D = 20.0 cm . For the first minimum, the path difference must be x = 2 = 10.0 cm . Let the sources be located at (-10, 0) and (10, 0) . The detector is initially at (0, 20) and moves to a new position (x, 20) . The distances from the sources to the detector are S₁ P = (x + 10)^2 + 20^2 and S₂ P = (x - 10)^2 + 20^2 . The path difference is x = S₁ P - S₂ P = 10 . (x + 10)^2 + 400 - (x - 10)^2 + 400 = 10 (x + 10)^2