17.11:
Interference: Path Lengths
Imagine two speakers being fed by the same sound source of a single frequency. In this case, the two waves have no phase difference at the speakers.
Recall that sound waves are traveling waves involving both space and time coordinates. So, at a point P, the distance traveled by the two waves differs.
The waves are expressed in terms of the traveled paths. A phase difference has been introduced simply because of the difference in path lengths, called the path difference.
If the path difference is an integral multiple of the wavelength, the waves interfere constructively. If it is an integral multiple of the half-wavelength, they interfere destructively. For this reason, when two crests or two troughs meet, they interfere constructively, while the crests and troughs interfere destructively.
At points other than the crests and troughs, the amplitude is intermediate.
If the speakers were fed by different sources with a fluctuating phase difference, the interference pattern would change rapidly and be washed out.
17.11:
Interference: Path Lengths
Consider two sources of sound, that may or may not be in phase, emitting waves at a single frequency, and consider the frequencies to be the same.
Two special sources may be considered when they are in phase. This can be easily achieved by feeding the two sources from the same source. An example would be synchronizing the two speakers by feeding them with the same source, such as the sound waves produced by a tuning fork. This setup ensures that the two sources have the same frequency and are in phase at the start. They then cause interference in the surrounding region.
However, the waves travel different paths to different points, and in general, the path lengths are different. Recalling that sound waves, or any traveling waves, depend both on space and time coordinates, it is easy to see that the difference in path lengths gives rise to a phase difference in the waves at different points. The interference pattern is affected by the path difference.
This setup explicitly explains that interference is not only caused by phase differences due to the initial phase difference or the difference in frequencies, but also due to the difference in path lengths. It has important implications for light waves.
Note, however, that the path difference would affect the interference pattern even if the two sources have a constant starting phase difference. If the two sources do not have a constant starting phase difference, but the phase difference fluctuates rapidly, then the interference would rapidly fluctuate between constructive and destructive at all points. The intensity would also change rapidly between the maximum and minimum at all points. Human ears are not perceptive to rapid changes in intensity and hear the average effect, which is simply the sum of the intensities of the two sources.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson: section 16.6; pages 529-530.
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-introduction: section 17.4; pages 869-871.
- Verma, H.C. (2008). Concepts of Physics. Volume 1, Bharati Bhawan Pub. & Dis.; section 16.8; pages 335-336.