The acoustic ruler

Question 1. What SPL would be needed at loudspeaker B to achieve an SPL of 80 dB for listener 1?

104 dB SPL. Level drop from loudspeaker B to listener 1 is 24 dB, add this to the 80 dB SPL required and you get 104 dB SPL.

Question 2. What delay must be applied to loudspeaker C relative to loudspeaker A for listener 3 to hear the same arrival time?

46 milliseconds. The pathlength from loudspeaker A to listener 3 is 73 milliseconds, while the pathlength from loudspeaker C to listener 3 is 27 milliseconds. Subtract 27 from 73 to get the difference. Since the speed of sound is constant, the pathlength scale is linear, so you could also measure the pathlength between loudspeaker A to loudspeaker C but this will only work if the two loudspeakers and the listener are in line.

Question 3. What is the direct sound level variation between listener 1 and listener 2 with only loudspeaker B switched on? Assume no off axis level drop.

4 dB. Level drop from loudspeaker B to listener 1 is 24 dB. Level drop from loudspeaker B to listener 2 is 28 dB. Subtract one from the other to get the answer.

Question 4. At what level would loudspeaker D need to be set to be the same level as loudspeaker B at listener 2?

-10 dB below the level of loudspeaker B. Level drop from loudspeaker B to listener 2 is 28 dB. Level drop from loudspeaker D to listener 2 is only 18 dB, so you'd need to turn loudspeaker D down by 10 dB for it to be the same level as loudspeaker B.

Question 5. If the SPL at listener 3 is 85 dB SPL, what will the level at listener 5 be?

87 dB SPL. Level drop from loudspeaker A to listener 3 is 28 dB, so the level at loudspeaker A will be 85 dB SPL + 28 dB = 113 dB SPL. Level drop over distance from loudspeaker A to listener 5 is 14 dB level drop over distance, plus 12 dB off axis level drop, a total of 26 dB. 113 dB SPL – 26 dB = 87 dB SPL.

Question 6. What is the level variation between listener 3 and listener 4 with only loudspeaker A on?

4 dB. Level drop from loudspeaker A to listener 3 is 28 dB. Level drop over distance from loudspeaker A to listener 4 is 18 dB plus a further off axis level drop of 6 dB, making a total of 24 dB. The difference between the two is 4 dB.

Question 7. At what frequency will there be a boost as a result of the boundary effect of the wall adjacent to loudspeaker A?

250 Hz. If the front of the loudspeaker is half a wavelength from the wall, the reflection off the wall will be a whole wavelength or 360 degrees out of phase and therefore be back in phase again, so it will add constructively with the direct sound from the loudspeaker. Using the Wavelength scale, you can see that the front of the loudspeaker is the same distance from the wall as a 500 Hz wavelength. This distance represents half a wavelength at 250 Hz. This will be true for listener 3 who is perpendicular to the wall and in the far field where the extra distance to the wall and back is a minimal increase. If the reflection was perfect, the addition would be 6 dB, in reality, it will be between 3 and 6 dB. For listener 5, the pathlength of the reflection will be different so the frequency will move slightly.

Question 8. At what frequency will there be a trough as a result of the boundary effect of the wall adjacent to loudspeaker A?

125 Hz. If the front of the loudspeaker is a quarter of a wavelength from the wall, the reflection off the wall will be half a wavelength or 180 degrees out of phase with the direct sound from the loudspeaker, so it will cancel out the sound at that frequency, causing a trough. The front of the loudspeaker measures the same distance from the wall as a 500 Hz wavelength, which is quarter of a wavelength at 125 Hz. Again, this is most pronounced for listener 3. Most loudspeakers will be omnidirectional at this frequency, so the energy going backwards will be virtually the same as the energy going forwards. If the reflection is strong, the trough can be very deep.