Abstract
The perspectives and techniques that have been developed in the previous chapters will now be applied to calculation of wave propagation in solids. Their application to longitudinal and shear waves will be both familiar and simple. What you will find to be even more satisfying is the success of those same techniques for finding solutions for waves in a system that does not obey the wave equation and whose solutions are not functions of x ± ct.
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Notes
- 1.
Such gravitational wave detectors are called “Weber Bars” in honor of the first attempt to use a longitudinally resonant bar to detect gravitational waves that was made by J. Weber. Weber claimed to detect gravitational waves in an article entitled “Gravitational-Wave-Detector Events,” Phys. Rev. Lett. 20, 1307–1308 (1968), although his claim could not be substantiated.
- 2.
This is a slight underestimate because the crystalline structure of gold is face-centered cubic. The accepted atomic diameter for gold is 2.88 Å.
- 3.
An inexpensive digital wristwatch costing about $10 (including the strap) will have a frequency stability of better than ±30 s/month corresponding to a relative uncertainty of ±10 ppm. A laboratory-quality frequency counter can measure the frequency (or period ) of a 5 MHz oscillation with 8 digits in 1 s. In our example, such a counter would time the period of about 5,000,000 cycles to within ±½ cycle and invert the result to obtain frequency with an error of less than ±0.1 ppm.
- 4.
One horsepower [hp] is defined as 745.7 W.
- 5.
Using hydraulic fluid at 17 MPa, with an average flow rate of U = 4.5 L/s = 4.5 × 10−3 m3/s, the hydraulic power is 〈Π〉 t = (Δp) U ≅ 76 kW.
- 6.
This equation is only approximately correct since the moment is also opposed by the rotary inertia of the bar. The approximation is very good when λ ≫ a, which is our current focus. The complete equation for the dynamics, which includes rotary inertia and shear, is derived in several references, for example, D. Ross, The Mechanics of Underwater Noise (Pergamon, 1976); ISBN 0–08–021,182-8.
- 7.
As will be shown in Chap. 9 on dissipative hydrodynamics, this is not the case for equations that contain both odd- and even-order derivatives. The Navier–Stokes equation , describing viscous dissipation; the Fourier heat diffusion equation, describing thermal conduction losses; and Schrodinger equation of quantum mechanics all contain both odd- and even-order derivatives that require complex numbers to relate frequency and wavenumber.
- 8.
The laws of electromagnetism, known as Maxwell’s equations, which govern the propagation of light, are also linear differential equations.
- 9.
The full dispersion relation for surface waves on fluids produces the dispersion relation (below) that also includes the force of gravity g, which dominates the restoring force at long wavelengths, as well as surface tension (capillarity). When the wavelengths are long compared to the depth, h, so kh ≪ 1, tanh (kh) is proportional to kh and the speed is again dispersionless. On water, capillarity (surface tension, σ) dominates gravity for wavelengths less than about one-half centimeter.
$$ {c}_{\mathrm{ph}}^2=\frac{\omega^2}{k^2}=\left(\frac{g}{k}+\frac{\sigma}{\rho} k\right) \tan \mathrm{h}(kh) $$ - 10.
For ripple-tank demonstrations, the depth of the water is about 5 mm. At that depth, the restoring forces of gravity and surface tension balance to produce a frequency-independent surface-wave velocity, c ≅ 22 cm/s. [See M. J. Lighthill, Waves in Fluids (Cambridge, 1978); ISBN 0521 21,689 3. Sec. 1.8 (Ripple-tank simulations)]
- 11.
The glockenspiel has bars of uniform cross-section, so the ratio of their overtones to the fundamental is given in Table 5.3. The underside of the bars in a xylophone is thinned near their center to make their overtones the ratio of integers: f 2/f 1 = 3 and f 3/f 1 = 6.
Table 5.3 Clamped–clamped or free–free bar modal frequency ratios, f n /f 1, and normalized wavelengths, λ n /L - 12.
A screwdriver handle makes an ideal mandrel since the handle has grooves to improve grip that provide spaces to weave the last turn in and out of the grooves to hold the coil together when it is slipped off the screwdriver’s handle.
- 13.
Almost no current flows through the detection coil, so it could be much smaller gauge wire, but it is often convenient to make both coils from the same wire.
- 14.
If a low-noise voltage preamplifier is available (e.g., PAR 113, Ithaco 1201, or SRS 560), they usually also provide some adjustable low-pass filtering capabilities that can remove low-frequency seismic vibrations if the apparatus is on a table that is not rigid.
- 15.
Fitzgerald had thought he discovered a new attenuation mechanism but Leonard showed that the Fitzgerald’s “effect” was absent and Fitzgerald had just measured an artifact of the attachment of piezoelectric transducers to his sample.
- 16.
Although this sample is a composite, the glass fibers in the epoxy matrix are short (about 800 μm long) and randomly oriented. The sample behaves isotopically upon length scales that are on the order of the bar's diameter and the wavelengths.
- 17.
The stiffness contributions of the transducer coils have been calculated by Guo and Brown [25].
- 18.
Since the coil is actually on the surface of the bar, its diameter is slightly larger than d. We will neglect this difference by arguing that the part of the coil that crosses the bar’s end has a lower moment of inertia. It would be a small correction to an already small correction. (So works the rationalizations in the mind of an experimentalist.)
- 19.
Unlike the example in Sect. 2.5.3, which controlled a single degree-of-freedom simple harmonic oscillator, the bar has standing wave modes so the phase relation between force and velocity will alternate by 180° between adjacent modes.
- 20.
Large samples of plutonium self-heat by radioactive decay (or worse!).
- 21.
Morse’s solution for the frequency in his result for his ν n , equal to our f n , includes a mode-independent constant term. He is not able to produce the n 2 dependence without adding another term to his Taylor series expansion and his result is obviously incompatible with Young’s observations [38].
- 22.
Piano technicians compensate for this anharmonicity. Anharmonicity is present in different amounts in all of the ranges of the instrument but is especially prevalent in the bass and high treble registers. The result is that octaves are tuned slightly wider than the harmonic 2:1 ratio. The exact amount that octaves are “stretched” by a piano tuner, by tuning the octave to a match half the frequency of the second overtone instead of the first, varies from piano to piano and even from register to register within a single piano—depending on the exact anharmonicity of the strings involved. With small pianos, the anharmonicity is so significant that the tuning is stretched by matching the triple octave.
- 23.
The “speaking length ” of a piano string is the distance between the bridge, located on the sound board near the hitching pin, and the capo d’astro, near the tuning pin. It is the speaking length that determines the distance between the fixed–fixed boundaries .
- 24.
Recall from Sect. 3.3.3 that one cent is one-hundredth of an equal temperament semitone (in the logarithmic sense), or a frequency ratio of 21/1200 = 1.000578.
- 25.
Syntactic foam was developed in the 1960s to provide buoyancy for instruments deployed in the deep ocean. They are composite materials fabricated by filling a castable epoxy with hollow glass microspheres. Those microspheres (sometimes also called microballoons) are very rigid so the foam is not crushed when subjected to large hydrostatic pressures.
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Garrett, S.L. (2017). Modes of Bars. In: Understanding Acoustics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-49978-9_5
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