Abstract
Sound and light propagate through space from their sources and are ultimately detected by a receiver such as a person or a device. The subject of the principle of superposition in Chap. 7 dealt with the waves produced by more than one source. In this chapter we will deal with effects on waves when their propagation is complex. The phenomena to be studied are:
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Diffraction, which refers to the way waves bend around obstacles. In particular, we will deal with how it is that we can hear sound from a loudspeaker without being able to see the loudspeaker. We cannot see something that is blocked by an opaque object. There is a great difference between sound and light. Why is this so?
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Reflection of waves off interfaces between two media (such as sound off a wall or light off a mirror or rough surface). What are the conditions for a surface to be shiny? Why does polishing a surface make it shiny?
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Refraction, which refers to the way waves behave when they are transmitted (pass on) from one medium to another (such as light from air to glass or sound from air to water). The operation of lenses, which are used in eyeglasses, microscopes, and telescopes, relies on the phenomenon of refraction.
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Scattering of waves by a tenuous distribution of obstacles, such as light off air molecules. In particular, we will learn why the sky is blue in color.
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The Doppler Effect, which characterizes the effect on the frequency of a sine wave that is observed by a receiver that is moving with respect to the source of the wave. We will understand why the frequency of sound emitted by the horn of an approaching train starts off at a large value and gradually decreases as the train moves away from us. This phenomenon allows an astrophysicist to determine the speed of distant star towards or away from us.
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Notes
- 1.
It is interesting to note that the above relation is similar to the expression for the angle for first order constructive interference for two sources of waves. [See Eq. (7.6).] As long as the angle θ is much less than one radian,
$$\displaystyle \begin{aligned} \begin{aligned} \sin \theta &\sim \theta \end{aligned} \end{aligned} $$(9.2)where θ on the right-hand side is expressed in radians. Then the equation for the angle for first order interference becomes
$$\displaystyle \begin{aligned} \begin{aligned} \frac{\lambda}{d}=\sin \theta_{1} &\sim \theta_{1} \end{aligned} \end{aligned} $$(9.3)In this last equation, the parameter d refers to the distance between two sources, whereas in this chapter d refers to the diameter of a hole.
- 2.
The figures were produced with applet on the website (2-11-2011): http://www.falstad.com/ripple/.
- 3.
The figure was made from the applet on the website (2-5-2011): http://www.falstad.com/ripple/.
- 4.
Interestingly, one can show that if the density of air molecules were to be perfectly uniform, this sum would result in no net scattering; the sky would be perfectly transparent! It is the modest degree of non-uniformity of the density that is responsible for the scattering. In fiber optics communication, the glass is so pure, that is, free from impurities and inhomogeneities, that it is the small degree of non-uniformity in the molecular density associated with the random thermal motion of the molecules, that is responsible for the small attenuation in the fibers.
- 5.
We are used to referring to the interface between air and water as the surface of water. However, how should we refer to the boundary between water and oil? The word interface is a neutral term, clearly superior to the term surface .
- 6.
Not to be confused with the angle of reflection θ rfl.
- 7.
To see this, suppose that we have observers at points P and Q, in the two respective media. The wave proceeds in a continuous manner. Thus, the rate f1 at which crests pass point P must equal the rate f2 at which crests pass point Q. Thus, we will replace the two symbols f1 and f2 by the common symbol f.
- 8.
It is often thought that the ray described above originates from the sun. A bit of analysis shows that this is impossible: In outer space n = 1, so that \(n \cdot \sin \theta = \sin \theta \leq 1\). At the turning point, θ = 90∘, so that \(n \cdot \sin \theta > 1\). The only way we can have total internal reflection is for the ray to originate from light scattered by the atmosphere.
- 9.
- 10.
Reference: Handbook of Chemistry and Physics, 65th edition, (Chemical Rubber Comp., Boca Raton, FL, 1984).
- 11.
Another term for a converging lens is a convex lens , since both sides of the lens are convex. The diverging lens, discussed later, is also called a concave lens . There also exist lenses that are concave on one side and convex on the other. If these are possibilities for consideration, one must remove any unambiguity by referring to a biconvex lens , or a biconcave lens , or a convex-concave lens .
- 12.
Polycarbonate is a material often used for eye lenses because of its strong shatter resistance and light weight. It has a drawback in having stronger chromatic aberration than glass. The so-called \(Abb\acute {e}\) number is used as a material’s level of dispersion. The larger the number, the lower the level is dispersion. Thus, while crown glass has an \(Abb\acute {e}\) number of about 55 while polycarbonate has a value of about 32. See Wikipedia (1-6-2011): http://en.wikipedia.org/wiki/Abbe_number.
- 13.
In Appendix H we discuss magnifying power, which is a property of a lens or instruments such as a telescope or a microscope that consist of a series of lenses - referred to as a compound lens. Magnifying power represents the ability of an optical instrument to increase the image size on the retina that is produced by an object. In order to appreciate this material, it is necessary to understand how the eye works, as discussed in Chap. 13.
- 14.
To determine the ultimate position of the image produced by a compound lens, one must apply the thin lens equation sequentially. For the effect of each lens, one must make sure to use the distance from that lens of the image produced by the previous lens as the object distance of that current lens. In the case of eyeglasses, the distance of the eyeglasses from the eyes is so small that one usually can assume that the eyeglasses are coincident with the center of the compound lenses of the eyes.
- 15.
Reversibility is manifest in the orbit of a planet about the Sun: If a planet were stopped dead in its tracks and its path reversed so that at that point the original direction is reversed while the speed is the same, the planet would retrace its path into the past, where it came from. What we would observe could be seen by taking a movie of the planet’s motion and then running the movie backwards. Reversibility is manifest in the basic laws of physics. The consequence is that every sequence of events has a possibility of occurring. Yet, there are movie scenes that are hilarious if they are run backwards. Why? Because the reversed sequence is regarded as impossible. [Imagine someone shown jumping off a ladder onto the ground…. Now reverse the sequence.] Such sequences are referred to as being irreversible . One of the challenges of physics is to understand how such extremely unlikely, irreversible sequences are never seen and yet have a possibility of occurring, in principle.
- 16.
See the Appendix A for the definition of the relative change of a parameter .
- 17.
Since there is no medium for the propagation of light, the formulas below are not relevant for light. This case will be discussed later.
- 18.
We add the two speeds: For example, if you are running towards me at a speed of 10ft/s with respect to the ground and I am moving towards you at a speed of 1ft/sec with respect to the ground, you would be moving towards me at a speed of (10+1)=11-ft/s.
- 19.
See http://www.hypography.com/article.cfm?id=32479 for a summary of recent research on the cracking of a whip. Also, see the following website for a discussion of how shock waves are the clue behind the trick for cracking a piece of wood with one’s bare hand: http://www.worldkungfu.com/whip.html#WHIPS.
- 20.
The letter Č is pronounced like “ch” in “cheer.”
- 21.
This quasar is estimated as having a mass equal to about one-billion solar masses.
- 22.
See Wikipedia (1-11-2011): http://en.wikipedia.org/wiki/Bat.
- 23.
See https://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiment for a fascinating description of the application.
- 24.
As discussed in Chap. 3, the speed of sound in a bulk sample of a medium, that is having no boundaries, is given by \(\sqrt {B/\rho }\), where B is the bulk modulus. When sound travels down a solid pipe of material, the walls of the pipe are free to expand outwards without any material. As a result, one has to use a difficult modulus, called the Young’s modulus. For aluminum, this modulus differs less than 10% from the bulk modulus.
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Gunther, L. (2019). Propagation Phenomena. In: The Physics of Music and Color. Springer, Cham. https://doi.org/10.1007/978-3-030-19219-8_9
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