Abstract
Having already invested in understanding both the equation of state in Chap. 7 and in the hydrodynamic equations in Chap. 8, only straightforward algebraic manipulations will be required to derive the wave equation , justify its solutions, calculate the speed of sound in fluids, and derive the expressions for acoustic intensity and the acoustic kinetic and potential energy densities. The “machinery” developed to describe waves on strings will be sufficient to describe one-dimensional sound propagation in fluids, even though the waves on the string were transverse and the one-dimensional waves in fluids are longitudinal.
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Notes
- 1.
Prof. Polack was on the faculty at the Danish Technical University (less than a 10 min drive from Lyngby to the Brüel & Kjær Headquarters in Nærum, Denmark) when he made this design. He is currently a professor at Université Pierre et Marie Curie and the Head of Doctoral School of Mechanics, Acoustics, Electronics and Robotics (SMAER, ED 391).
- 2.
The resultant resonator was conical. It can be modeled easily in DeltaEC as a single CONE element (plus an electrodynamic driver VESPEAKER at one end and an OPENBRANCH radiation condition at the other end). This is a lot easier than 28 lumped elements once you accept that a cone will solve the problem.
- 3.
Some say thermodynamics is the field where every partial derivative has its own name.
- 4.
Analog Devices, Inc. sells a wonderful temperature sensing integrated circuit (AD 592) that sources one microampere of current for each degree of absolute temperature making electronic temperature compensation nearly trivial.
- 5.
In acoustics, the equivalent of “optical trapping” is acoustical levitation superstability (see Sect. 15.4.7): M. Barmatz and S. L. Garrett, “Stabilization and oscillation of an acoustically levitated object,” U. S. Pat. No. 4,773,266 (Sept. 27, 1988).
- 6.
In my estimation, one of the most significant engineering breakthroughs of the twentieth century was the invention, by Harold S. Black, of the negative feedback amplifier in 1928, now known as the “operational amplifier” (op-amp). In Black’s own words, “Our patent application was treated in the same manner as one for a perpetual-motion machine. In a climate where more gain was better, the concept that one would throw away gain to improve stability, bandwidth, etc., was inconceivable before that time.”
- 7.
The “true” definition of an r.m.s. amplitude is always tied to the power associated with that amplitude. To account for nonsinusoidal waveforms, engineers introduce a dimensionless “crest factor,” CF, that is defined as the ratio of the peak value of a waveform to its rms value. For a sine waveform, CF (sine) = √2, for a triangular waveform, CF (triangle) = √3, and for a half-wave rectified sinewave, CF (half-wave rectified) = 2. Most instruments that measure the “true rms” value of a parameter (e.g., voltage) will exhibit an accuracy that decreases with increasing crest factor. For measurement of Gaussian noise, instruments that tolerate 3 < CF < 5 are usually adequate.
- 8.
The weighting of A, B, C, and D (no weighting) levels are specified in several international standards, for example, the International Standards Organization ISO 3746:2010.
- 9.
The transfer function for sound pressure at the eardrum to the sound pressure presented to the outer ear is above 14 dB from 2.5 to 3.2 kHz as shown in Table 2 of the ANSI-ASA S3.4 Standard.
- 10.
When we include thermoviscous dissipation on the walls of the tube, the velocity will not be constant throughout the tube’s cross-section, since the no-slip boundary condition for a viscous fluid requires that the longitudinal velocity vanish at the tube’s walls. In many cases, the ratio of the viscous penetration depth, δ ν, to the tube’s radius, a, is small, δ ν ≪ a, so the flow velocity is nearly uniform throughout most of the tube’s cross-section.
- 11.
The Reciprocity Theorem also applies in vector form. If we applied a vector force at some location, ①, on a flexible structure, and the vector displacement is measured at some other location, ②, we would observe the same vector displacement at ① if the same vector force were applied at ②.
- 12.
The primary calibration of voltage is simpler (in principle, although it requires temperatures near absolute zero for the junction) since it is possible to relate the voltage across a superconducting Josephson junction to the resulting oscillation frequency , f, since their ratio is determined by Planck’s constant , h, and the charge on an electron, e: f = nhf/2e, where n is an integer. 2e/h = (483.5978525 ± 0.000003) MHz/μV.
- 13.
This is equivalent to saying that the transducers are noncompliant in much the same way that the “constant displacement drive” for a sting does not “feel” the load of the string and preserves the “fixed” boundary condition. For reciprocity calibration of transducers that behave as an ultra-compliant driver (i.e., a constant force drive equivalent), see [36].
- 14.
The acoustic center of a reversible microphone or a sound source under free-field conditions is defined as the extrapolated center of the spherically diverging wave field.
- 15.
Within the cone, the pressure amplitudes would be larger than those of a spherically spreading wave if the source’s volume velocity was the same in both cases. If the cone subtends a solid angle , Ω, then the pressures would be enhanced by a factor of 4π/Ω, assuming the additional load would not reduce the source’s volume velocity.
- 16.
If the constant is negative, then there is a family of sinusoidal horn shapes (e.g., a globe terminated in a cusp) that describe the shape of the bell of the flute commonly associated with Indian snake charmers or the English horn, first used by Rossini, in 1829, in the opera, William Tell. [See B. N. Nagarkar and R. D. Finch, “Sinusoidal horns,” J. Acoust. Soc. Am. 50(1) 23–31 (1971).]
- 17.
A nonuniform power transmission coefficient does not necessarily reduce the value of guided-wave enhancement of the coupling to an electrodynamic transducer, since human perception of low-frequency musical content is not particularly sensitive to such nonuniform response. The best example of such a psychoacoustic tolerance may be the success of the Bose Wave™ radio that employs a long serpentine duct that is driven by the rear of the forward-radiating speakers as shown in US Pat. No. 6,278,789 (Aug. 21, 2001).
- 18.
Fritz Haber was also one of the world’s most infamous chemists, having developed the gas that was used to murder prisoners in the Nazi death camps.
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Garrett, S.L. (2017). One-Dimensional Propagation. In: Understanding Acoustics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-49978-9_10
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