A sound source is fixed at the origin and continuously emits sound at a frequency of 660 Hz . The sound propagates through air at a speed of 330 m s ⁻¹ . A listener travels along the line x = 336 m with a constant speed of 26 m s ⁻¹ . Calculate the frequency of the sound observed by the listener when he is positioned at y = -140 m .

Step-by-step solution

The position of the source is S(0, 0) and the listener is at L(336, -140) . The distance between the source and the listener is d = 336^2 + (-140)^2 = 112896 + 19600 = 364 m . Assuming the listener moves in the positive y -direction, the velocity of the listener is v _L = 26 j m s ⁻¹ . The unit vector from the listener to the source is u = -336 i + 140 j 364 = - 12 13 i + 5 13 j . The component of the listener's velocity towards the source is v_ L = v _L u = (26 j ) ( - 12 13 i + 5 13 j ) = 26 5 13 = 10 m s ⁻¹ . Us