Abstract
As discussed in Chap. 1 in the case of vibrating strings and organ pipes, there are generally many different modes in which a resonant system may vibrate. (See Fig. 1.6 and related discussion.) Generally, more than one of these modes are excited simultaneously in the sounding of a musical instrument. Indeed, their presence or absence is what determines the beauty of a particular tone as well as the difference in sound from one instrument to another. Which modes are excited is not only determined by the characteristics of the resonant system but also by the way in which it is excited. For example, the slipping of the violin string on the bow, or the vibration of the reed in an oboe or a bassoon excites a particular set of modes in those instruments.
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Change history
18 July 2020
The inadvertently published equations have been corrected as mentioned below.
Notes
- 1.
Note by definition,
a =log10b means that b = 10a.
Therefore,
log10b 2 =log10102a = 2a = 2log10b
- 2.
Baron Jean-Baptiste-Joseph von Fourier (1768–1830) accompanied Napoleon in 1798 on his expedition to Egypt, where he served as Secretary for Napoleon’s newly formed Institut d’Egypt. In Cairo, he did extensive research on Egyptian antiquities and gave advice on engineering matters. He returned to France in 1801, about the same time that the Rosetta Stone and other major ancient Egyptian relics were surrendered to the British. Back in France, he was charged with the publication of an enormous mass of Egyptian material which became known as Description de l’Egypt (in 21 volumes from 1808 to 1825). He was also the first to describe the atmosphere’s trapping of heat as “The Greenhouse Effect” in the 1820s. (See Segré 2002, p. 119.)
- 3.
The notation \(\sum _{n=1}^\infty (A_n \sin {}(n\pi x / L) \cos (n\pi ct / L))\) means that you sum the expression for the values of n = 1, 2, 3, … to infinity.
- 4.
Ramanujan (1914) gave the most rapidly convergent series for 1∕π ever discovered: \( \frac {1}{\pi } = \frac {1}{4}\left [\frac {1123}{882}-\frac {22{,}583}{882^3}\cdot \frac {1}{2}\cdot \frac {1\cdot 3}{4^2}+\frac {44{,}043}{882^5}\cdot \frac {1\cdot 3}{2\cdot 4}\cdot \frac {1\cdot 3\cdot 5\cdot 7}{4^2\cdot 8^2}- \cdots \right ]\). Amazingly, the first term by itself gives π = 3.141585041… (Ramanujan liked to entertain his friends by reciting the endless digits of π at parties.
- 5.
These results are obtained by applying the trigonometry identities \(C_n \sin {}(n\theta +\phi _n ) = C_n \sin n\theta \cos \phi _n + Cn \cos n\theta \sin \phi _n\)
and
\(\tan \phi _n = \frac {\sin \phi _n}{\cos \phi _n}\) together with \(\cos ^2 \phi _n + \sin ^2\phi _n = 1\).
- 6.
In Heisenberg’s formulation, the energy of the electron (or other particle) is given by \(\mathcal {E} = h\nu \) where ν is a frequency corresponding to the de Broglie wavelength and h is Planck’s constant. Hence, in Heisenberg’s formulation of the Uncertainty Principle, \(\varDelta \mathcal {E} \varDelta t \approx h/2\pi \).
- 7.
My personal suspicion is that the peculiar nomenclature arose as a typographical error, compounded by the fact that there actually was an electrical engineer at the Bell Laboratories named Richard W. Hamming who developed his own window function for numerical analysis that is called the “Hamming Window” in the literature.
- 8.
Mitral prolapse (verified in the present case by an echocardiogram and open-heart surgery) is a common condition in which the mitral valve (so-called because it is shaped like a Bishop’s mitre) is pushed backward toward the left atrium when the left ventricle contracts. Some blood from the ventricle is then forced back through the leaky mitral valve into the atrium in turbulent flow, instead of going out through the aortic valve in laminar flow, as in the normal case.
- 9.
The assumption of an inverse square law is not terribly good here because the source of noise extended over a distance larger than the island of Verlaten. Also, the radiation pattern appears to have a strong dipole component. (See Winchester 2003, p. 271.)
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Bennett Jr., W.R. (2018). Spectral Analysis and Fourier Series. In: Morrison, A. (eds) The Science of Musical Sound. Springer, Cham. https://doi.org/10.1007/978-3-319-92796-1_2
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