In noisy factory environments, it's possible to use a loudspeaker to cancel persistent...

Question:

In noisy factory environments, it's possible to use a loudspeaker to cancel persistent low-frequency machine noise at the position of one worker. The details of practical systems are complex, but we can present a simple example that gives you the idea. Suppose a machine 6.0 m away from a worker emits a persistent 90 Hz hum. To cancel the sound at the worker's location with a speaker that exactly duplicates the machine's hum, how far from the worker should the speaker be placed? Assume a sound speed of 340 m/s.

The Speed of Sound

Sound events comprise periodic compressions and rarefacations of the material that the sound event is traveling through. This implies that the density of this material (i.e. the coupling between neighboring atoms) plays a key role determining the speed of these sound events. For instance, sound travels much faster in solids than in liquids and faster in liquids than in gases. Examples for the speed of sound are:

• Granite: 5950 m/s
• Water: 1484 m/s
• Air: 343 m/s.

The given values are approximate and depend on temperature (strongly in gases) and (in fluids) on pressure.

At a frequency of 90 Hz, the humming sound has a wavelength of $$\lambda = \frac{c}{f} = \rm \frac{340~m/s}{90~Hz} = 3.8~m~.$$ To cancel the sound of the machine, the identical sound coming from the speaker has to have a phase shift of half a wavelength, which is 1.9 m. Therefore, two possible locations for the loudspeaker are:

• 6.0 m - 1.9 m = 4.1 m
• 6.0 m + 1.9 m = 7.9 m

The first option is better in so far as that the speaker has to be driven at a lower amplitude, since it is closer to the worker's ears.

Note that the speaker has to be placed on the line defined by the position of the worker and the position of the machine to yield the largest possible quiet zone. Another option would be to co-locate the speaker with the machine and drive the speaker with a signal, that is phase-shifted by 180{eq}^{\circ} {/eq}. 