There are a lot of factors to take into consideration when analyzing our headphone system: the periodicity of the sound, the frequency of the sound, the wavelength of the sound; also factors like the material of which the headphones are made and the capability of the op-amps we used.
In order for the sound-cancelling headphones to work, they must play the inverted sound into your ear exactly at the same time that the original sound reaches your ear. Because the speed of light is so much faster than the speed of sound, the signal will travel more quickly through the circuit than it will through the headphone to your ear, and so the inverted signal gets to your ear before the regular one does and it doesn't do much good.
When a signal is periodic, it is easier to find a place where the original signal and inverted-phase signal line up to produce a diminished sound. For our headphones, it seemed to be that a non-inverted signal did the best job on diminishing very low frequencies.
We also found that there was better attenuation the closer the mics were to the ear. The distance that the mic is from the ear could affect the device quite a bit because of a signal's wavelength. Wavelength is the distance between two points on a wave that have the same phase (or the length of the period of the signal). Wavelength is equal to the velocity of the signal divided by its frequency. Sound travels through air at 331 meters per second plus 0.6 * the number of degrees above freezing the air through which the sound travels is. Obviously, our sound was traveling through air and also plastic and foam, but just considering how quickly it travels through air, the following is a table of wavelengths of signals we used:
| Wavelength of a Sound Wave through air at 20 degrees celcius |
| frequency (hz) | wavelength (cm) |
| 100 | 343 |
| 250 | 137.2 |
| 500 | 68.6 |
| 750 | 45.73 |
| 1000 | 34.3 |
| 1500 | 22.87 |
| 2000 | 17.15 |