Abstract
A defining property of waves is that their displacements add together, which is called superposition. This is a simple process but leads to a wide range of interesting behaviors. It means that waves traveling in different directions add as they cross each other and then separate back into separate waves afterward. Waves can also add to create larger waves, smaller waves, or beating patterns. Additionally, reflected waves create standing waves in which they just go up and down without appearing to move. If waves reflect off both boundaries of their region, then that limits the possible standing waves to what are called the normal modes and harmonics of the system, which are central aspects of musical instruments. Wave superposition also causes thin films to create rainbow patterns, and produces the vibrant colors of many insects, birds, and other animals.
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Notes
- 1.
Despite the widespread validity of the superposition principle, which is assumed in almost all parts of this book, there are some exceptions. These arise from non-linear effects, which are simply defined as wave behaviors that violate the superposition principle. Nonlinear effects are typically negligible for low amplitude waves, but can become important when waves have large amplitudes. They are studied in the fields of nonlinear optics and nonlinear acoustics. The primary situation in which nonlinear effects are commonly observed is for water waves, where they give rise to phenomena such as wave breaking (note that some high amplitude waves are breaking in Figure 3.4) and wave development over time.
- 2.
Rogue waves also arise partly from non-linear effects.
- 3.
For a rope, “hard” boundaries, such as a rigid doorknob, produce the opposite polarity; on the other hand, “soft” boundaries, such as a ring looped around a pole, produce the same polarity. These are shown in Figure 3.10. Boundaries also occur where the wave speed changes, such as where a heavy rope (slow) is tied to a light rope (fast), or where light goes from air (fast) into water (slow). In these cases, the reflected wave has the same polarity if the new medium has a faster wave speed and the opposite polarity if the new medium has a slower wave speed.
- 4.
The term “standing waves” can also refer to waves that propagate at the same speed as a moving medium but in the opposite direction, causing the waves to appear stationary. Waves in river rapids are common examples. They are quite different from the standing waves that this section focuses on.
- 5.
Boundaries that reverse wave polarity have nodes and those that preserve wave polarity have antinodes.
- 6.
Both this frequency result and Eq. 3.2 assume that the wave velocity is independent of wavelength, which is called nondispersive. This is true for string waves, sound waves, light waves in vacuum, and many other types of waves, but is not always true. Water waves are a notable exception.
- 7.
Newton, Opticks, p. 202.
- 8.
Newton, Opticks, p. 281.
- 9.
Actually, they only radiate in all “forward” directions, which is not explained well by the theory
- 10.
Arago had a remarkably adventurous early career in science, including being detained by Spain as a suspected French spy while he was actually conducting surveying work. Later on, he became the Prime Minister of France. See http://lousodrome.net/blog/light/2015/07/07/fresnel-and-the-poisson-spot/.
- 11.
Each strut creates a line that is perpendicular to itself in the image, on both sides of the star. Thus, a vertical strut creates a horizontal line and vice-versa. Telescopes that have “Y”-shaped struts, such as the James Webb Space Telescope, also have a full line from each strut, this time creating 6-pointed diffraction spikes.
- 12.
Named for the English astronomer Sir George Airy, who first described it mathematically.
- 13.
A good reference is: Sun, Jiyu, Bharat Bhushan, and ** Tong (2013) “Structural coloration in nature” Royal Society of Chemistry Advances 3:14862-14889.
- 14.
Vinther, Jakob, et al. “Structural coloration in a fossil feather” Biology Letters 6:128-131 (2010).
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Andrews, S.S. (2023). Superposition. In: Light and Waves. Springer, Cham. https://doi.org/10.1007/978-3-031-24097-3_3
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