Halliday, Resnick & Walker — Waves–II: Problem 10

Underwater illusion. One clue used by your brain to determine the direction of a source of sound is the time delay \(\Delta t\) between the arrival of the sound at the ear closer to the source and the arrival at the farther ear. Assume that the source is distant so that a wavefront from it is approximately planar when it reaches you, and let \(D\) represent the separation between your ears. (a) If the source is located at angle \(\theta\) in front of you (Fig. 17.7), what is \(\Delta t\) in terms of \(D\) and the speed of sound \(v\) in air? (b) If you are submerged in water and the sound source is directly to your right, what is \(\Delta t\) in terms of \(D\) and the speed of sound \(v_w\) in water? (c) Based on the time-delay clue, your brain interprets the submerged sound to arrive at an angle \(\theta\) from the forward direction. Evaluate \(\theta\) for fresh water at \(20^\circ\text{C}\).