Abstract
In Chap. 7 we studied the wave that results from adding two sine waves having the same frequency, typically produced by having a single sine wave incident upon a set of slits or a grating. In this chapter we will study the nature of more complex waves: Topics included are:
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beats that result from having two waves with nearly equal frequencies
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modulated waves, such as AM and FM waves, that are central for radio communication
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spectrograms that provide us with a visual representation of a complex wave such as is produced by the human voice, a bird or a complex electronic signal
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polarized light that is associated with the fact that the electric field, which characterizes a light wave, oscillates in a direction that is transverse to the direction of propagation. This characteristic of light enables bees to navigate, enables us: to manipulate the path of a light beam, to cut down glare from sunlight, and to study the Universe soon after the Big Bang.
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Change history
15 October 2022
The original versions of chapters 8 and 11 were revised and the corrections for these chapters have been updated in the book.
Notes
- 1.$$\displaystyle \begin{aligned} \cos{}(a+b)=\cos~a~\cos~b~-\sin~a~\sin~b \end{aligned} $$(8.2)
Now let \(\overline {f}=(f_1+f_2)/2\). This is the average of the two frequencies. Assume also that f 2 > f 1. The result doesn’t depend upon which frequency is greater. Notice that
$$\displaystyle \begin{aligned} f_1=\overline{f}-f_{B}/2 \end{aligned} $$(8.3)and
$$\displaystyle \begin{aligned} f_2=\overline{f}+f_{B}/2 \end{aligned} $$(8.4)We can then show that
$$\displaystyle \begin{aligned} \cos{}(2\pi f_1~t)~ + ~\cos{}(2\pi f_1~t) = 2 \cos{}(\pi f_B~t)~\cos{}(2\pi \overline{f}~t) \end{aligned} $$(8.5)The significance of this equation is that the resulting wave is a sine wave with frequency \(\overline {f}\) modulated by a sine wave having the beat frequency.
- 2.
Regarding perfection: The Enso (https://en.wikipedia.org/wiki/Ensō) is a Zen Buddhist circle that is hand drawn and meant to help a person achieve a high level of enlightenment. There is a Zen tale that focuses on the circle: In a version I read years ago, a novice sees a monk seated next to a piece of paper on which the monk has drawn a circle. The novice asks, “Dear Master, the figure you have drawn does not appear to be a perfect circle.” To which the master replies, “What I have drawn is a perfect whatever it is.”
- 3.
The physical basis for the beats is attributed to Lord Rayleigh, who discussed it in Volume 1 of his Theory of Sound, second edition, [MacMillan and Co. London, 1894]. Rayleigh is most famous for his explanation for why the sky is blue, a subject that is discussed in Chap. 9.
- 4.
We can see the required bandwidth of about 20-kHz reflected in the list of radio stations that were less than 45 miles away from Boston on December 12, 2018: in kHz: 590, 630, 650, 680, 740, 800, 830, 850, 890, 920, 950, 980, 1030, 1060, 1090, etc.
- 5.
From trigonometry: \(\sin ^2(\theta ) = (1/2)[1-\cos {}(2 \theta )]\).
- 6.
The figure was produced from the Mathematica website: http://demonstrations.wolfram.com/PowerContentOfFrequencyModulationAndPhaseModulation/.
- 7.
A high pass filter allows only frequency components above a certain value to pass through, while a band pass filter allows only frequency components lying between two frequencies to pass through.
- 8.
The software Raven Lite is available free of charge from the website http://ravensoundsoftware.com/raven-downloads/.
- 9.
Recall that our musical perceptive and esthetic experience depend quite a bit upon the processing that occurs in our brains.
- 10.
In Sect. 4.7 we mentioned in a footnote the “Fourier transform.” This transform is the spectrum obtained by the mathematical method referred to as “Fourier transformation.” The technical term for the short time Fourier spectrum is that which is obtained by the Short Time Fourier Transform, of STFT for short. For more details see the Wikipedia article on the STFT: https://commons.wikimedia.org/wiki/File:STFT_colored_spectrogram_1000ms.png Note that this article is mathematically advanced.
- 11.
The drop in frequency is close to 7% and corresponds to a drop in the radius of the gear of the same percentage.
- 12.
- 13.
The source is a Wikipedia site https://en.wikipedia.org/wiki/Short-time_Fourier_transform. According to the website, the spectrogram was produced by Alessio Damato.
- 14.
Here are a few apps that you can use to obtain a spectrogram of a sound input using a mobile device: For an Android mobile device you can download SpectralPro Analyzer from the website https://play.google.com/store/apps/details?id=radonsoft.net.spectralviewpro&hl=en_US . For an iOS mobile device you can download SpectrogramProSpectrum Analyzer RTA by Onyx or SpectrumView from the App Store.
- 15.
An endoscope is a long flexible tube that is introduced down the digestive tract, with a light and camera at the far end.
- 16.
Known by the layperson as vocal cords .
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I am deeply grateful to Barbara Arboleda, for not only sharing the video with me, but also for revealing to me the reality and wonder of vocal folds.
- 19.
See website of Rob Hagiwara— https://home.cc.umanitoba.ca/~robh/howto.html , which describes how the information contained in a voice spectrogram can be extracted.
- 20.
Interestingly, this relation is analogous to what is called the Heisenberg Uncertainty Principle in Quantum Theory. In this case, we cannot pin down the details of a wave at a specific time without losing information about a frequency. At the level of quantum systems, such as electrons, we cannot pin down an electron’s position without losing information about its momentum or velocity. Specifically, the uncertainty of the position Δx is related to the uncertainty of the velocity Δv via the relation Δx ∼ħ∕mv, where m is the mass.
- 21.
Source NASA: See https://svs.gsfc.nasa.gov/vis/a010000/a010500/a010543/ . You can see a simulation of the collapse in this link.
- 22.
- 23.
Source: https://www.ligo.caltech.edu/system/avm_image_sqls/binaries/98/original/gw170817clean.jpg?1508108648. You can hear an excellent audio replication on this website: https://www.youtube.com/watch?v=X6dJEAs0-Gk.
- 24.
More precisely, the semi-major axis of its elliptical orbit . The orbits are very close to being circular and I will refer to the radius of a nearly circular orbit for simplicity and therefore without significant error.
- 25.
In the case of the earth and Sun, the Sun is very nearly stationary because its mass is so much greater than the earth’s mass. On the other hand, in the case of the binary neutron stars, the masses have equal orders of magnitude. The two stars spiral about each other. If they have equal masses, Kepler’s expression is missing a factor of two and our value of 60-Hz should be replaced by 30-Hz.
- 26.
You might find a different choice for the angle of a polarization axis—for example, the angle with respect to the vertical.
- 27.
Refraction is discussed in detail in Sect. 9.3.
- 28.
See Hecht, E., Optics, 5th ed.
- 29.
The angle θ′ is related to the input angle θ through the equation tan(θ′)=\( \sqrt {\mbox{T}_v/\mbox{T}_h }\) tan(θ).
- 30.
The figure is based upon the discussion of Karl von Frisch in his book, The Dance Language and Orientation of Bees, Belknap Press of Harvard University Press, Cambridge, MA, 1967.
- 31.
The strikingly different chemical properties of enantiomorphs have played an extremely important role in pharmaceuticals. I will present two different scenarios, one which led to wonderful pharmaceutical benefits and the other to more expensive and probably unnecessary medications. (1) The first scenario is represented by thalidomide . Among its early uses was its treatment of morning sickness in pregnant women. Unfortunately, the drug was prone to producing severe birth defects and was withdrawn from the market. Subsequently, it was found that one of its enantiomers is responsible for the birth defects while the other provides the desired pharmaceutical effects. Separating out the desired enantiomer made the drug available for numerous diseases and medical issues. (2) The second scenario is the role of patents in the pharmaceutical industry. Consider, Prilosec, which is a drug used to treat heartburn. When the patent owned by the pharmaceutical company AstraZeneca ran out, the company produced a form that had the pure drug-effective enantiomer and called this new drug Nexium . The company was able to obtain a new patent on the drug and sell it at a much higher price than Prilosec. I have researched the web for many studies that compared the two drugs and have yet (7-1-2019) to find one that reported a significant difference in their effectiveness—that is, more than a few percent in whatever way improvement can be measured. You can read information on this subject on the following websites (1-29-2011): http://en.wikipedia.org/wiki/AstraZeneca#Nexium; http://en.wikipedia.org/wiki/Esomeprazole#cite_note-12; http://www.medscape.com/viewarticle/481198_8. In spite of the negative responses towards AstraZeneca’s actions, you should always be on the lookout for contrary opinions about the Nexium-Prilosec controversy. Beware about the significance of a claim that drug A is more effective than drug B. The comparative effectiveness might mean, in simple terms, that A is 5% more effective than B. If so, A might be 90% effective while B might be 85% effective. The ratio is a mere 1.05. On the other hand, the respective effectivenesses might be 10% and 5%, in which case A is twice as effective as B!
- 32.
See the two articles in the June 22, 2000 issue of the science journal NATURE.
- 33.
The technique used in this problem is a simple application of the technique that Galileo is conjectured to have used to study the motion of a ball down an inclined plane. See the applet on the website (2-11-2011): http://www.joakimlinde.se/java/galileo/.
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Gunther, L. (2019). Complex Waves. In: The Physics of Music and Color. Springer, Cham. https://doi.org/10.1007/978-3-030-19219-8_8
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