In a school auditorium the stage is a horizontal floor. The last row of seats is 20m from the centre of the stage and 1.0m above the stage floor. A rectangular wooden reflector panel is to be suspended such that its centre is 4.0m above the stage floor and directly above a point that is 6.0m in front of the stage centre. A singer standing at the centre of the stage, with her mouth 1.6m above the floor, must be clearly heard in the last row by both the direct sound and the single reflected sound from the panel.
(i) Treating the sound as travelling along straight rays, derive the angle which the reflector panel must make with the horizontal so that a sound ray from the singer is reflected toward the ear level (1.0m above the stage floor) of a listener in the last row.
(ii) Show that the point on the panel at which the sound is incident lies within ±0.5m of its centre.
(iii) Calculate the path difference between the direct sound and this reflected sound. Comment on whether the reflected sound will reach the listener distinctly after the direct sound, given that echoes in an auditorium become distracting when the time gap exceeds 0.050s. Take the speed of sound to be 340ms−1.