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Understanding Acoustics pp 423–493Cite as

Nondissipative Lumped Elements

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Part of the book series: Graduate Texts in Physics ((GTP))

Abstract

The goal of the chapter is to start applying the Laws of Hydrodynamics that were provided in Eqs. (7.32), (7.42), and (7.43). By applying these laws to some simple acoustical networks , we can begin to develop our understanding of their meaning and their broad utility in acoustics. We start by ignoring dissipation and by choosing acoustical elements that are small compared to the wavelength of sound. In this nondissipative lumped element approximation, the Continuity Equation (7.32) leads us to the definition of an acoustical compliance , C, that plays the same role as a capacitor in alternating current (ac) electrical circuit theory or a spring in the theory of mechanical vibrations.

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Notes

  1. 1.

    We can “ignore” dissipation by setting the shear viscosity equal to zero, μ = 0, in the Navier–Stokes Equation (7.34) and both μ and the thermal conductivity to zero, κ = 0, in the Entropy Conservation Equation (7.43).

  2. 2.

    In this case, “slowly” means that the relative change in mean pressure, δp m/p m ≪ 1 for distances on the order of the wavelength of sound.

  3. 3.

    The choice of “c” to represent the speed of sound (or the speed of light) evolved from the word “celerity ” meaning rapidity of motion or action.

  4. 4.

    The “acoustical impedance ,” Z ac ≡ p 1/U 1, which has just been introduced is the fourth “impedance” that has appeared in this textbook. We defined the “electrical impedance,” Z el = V/I, in Sect. 2.5.5, where it was applied to an electrodynamic loudspeaker. The “mechanical impedance ,” Z m = F/v, was introduced as the solution to the steady-state response for a damped-driven harmonic oscillator in Sect. 2.5.1. The limp string’s “characteristic impedance ,” Z m,0 = ρ L c, was introduced to calculate the steady-state response of a force-driven string in Sect. 3.7.

  5. 5.

    The design of springs for acoustical systems is a very important problem in areas like loudspeaker cone suspensions and vibration isolators for machinery. For mechanical springs, a crucial design consideration is fatigue failure of the spring material. Gas springs are attractive in some applications because the gas does not “wear out.” One of the most articulate engineers of acoustical systems is John Corey, the founder and owner of CFIC, Inc. and Q-Drive, in Troy, NY (now a wholly owned subsidiary of Chart Industries). In defense of gas springs, John likes to say, “Nobody has ever successfully measured the endurance limit of a helium atom.” On the other hand, in exchange for a spring that will not suffer fatigue failure, one needs to provide a clearance (piston) seal that is not too dissipative due to gas friction or due to fluid blow by, or flexure (diaphragm or bellows) seal that is not subject to fatigue failure.

  6. 6.

    The Santa Ana winds are katabatic winds, from the Greek, meaning “to flow downhill.”

  7. 7.

    A nice acoustic levitation chamber video demonstration is available on You Tube, http://www.youtube.com/watch?v=94KzmB2bI7s. In a truly bizarre application, Wenjun Xie, at a university in Xi’an, China, used acoustics to levitate live insects, spiders, and fish as shown in Fig. 15.20.

  8. 8.

    The “smallness” criterion for the amplitude of oscillation is somewhat arbitrary, depending upon the desired accuracy of any particular calculation, but a reasonable rule of thumb is to require that the peak-to-peak displacement of the gas in the neck, 2ξ 1, is at least ten times smaller than the length of the neck: 2ξ 1 = 2(U 1/A)/ω < (L/10).

  9. 9.

    As we will see, the standard “recommended effective length correction” improves the agreement between the measurement and the theory of Eq. (8.51), although it is not exact, since the flow transition between the neck and volume is somewhat shape dependent.

  10. 10.

    The current version of DeltaEC includes the thermophysical properties of the following gases and liquids: nitrogen, dry air, humid air, carbon dioxide, hydrogen, deuterium, helium, neon, He/Ar and He/Ne gas mixtures, natural gas combustion products, liquid sodium, and sodium/potassium eutectic (NaK). Solids include an “ideal” solid with infinite heat capacity and infinite thermal conductivity , copper, nickel, stainless steel, tungsten, molybdenum, kapton®, mylar®, and Celcor® (a porous ceramic matrix). It also allows the user to create their own *.tpf file to represent the temperature and pressure varying thermophysical properties of user-specified fluids and solids.

  11. 11.

    DeltaEC also supports some nonlinear (i.e., high-amplitude) acoustical effect such as boundary-layer turbulence, “minor loss,” and amplitude-dependent interfacial discontinuities.

  12. 12.

    The Los Alamos Thermoacoustics Group uses Rayleigh’s traditional notation for sound speed, a, instead of the more contemporary choice of c to represent sound speed. Perhaps this is because c is used quite frequently in thermoacoustics to designate specific heats.

  13. 13.

    The ability of a solid to hold the temperature of the gas constant at the solid–gas interface is quantified by the ε s parameter discussed in G. W. Swift, “Thermoacoustic engines,” J. Acoust. Soc. Am. 84(4), 1145–1180 (1988), Eq. (59). For most solids in contact with ideal gasses at “ordinary” pressures, ε s ≅ 0, although if a sound wave is propagating through a liquid metal it is difficult to find any material with sufficient heat capacity and thermal conductivity to hold the liquid isothermal at the liquid–solid interface.

  14. 14.

    In using DeltaEC, choosing the guesses and targets will require that the user has a reasonable understanding of the network that is being modeled. For example, targeting the frequency while guessing the pressure would make no sense, since the sound speed of an ideal gas is pressure independent as demonstrated in Sect. 10.3.2.

  15. 15.

    After having run successfully, planewav.in has converted itself to planewav.out.

  16. 16.

    Since this is a linear system, the frequency will be amplitude independent. By choosing the pressure amplitude to be unity at the entrance to the neck (0d), the numerical value of the pressure in the Helmholtz resonator ’s volume will correspond to the quality factor of the resonance as expressed in Eq. (C.1).

  17. 17.

    The fact that the resonance frequency found by DeltaEC is lower than the calculation based on Eq. (8.51) reflects the fact that DeltaEC includes the additional inertance of the fluid in the viscous boundary layer “attached” to the surface of the resonator’s neck and the isothermal compressibility of the gas in the thermal boundary layer on the surface of the cavity. These dissipative boundary layer effects will be the focus of Chap. 9.

  18. 18.

    Many mature users of DeltaEC (i.e., old guys like me) call the RPN segment an “RPN Target.” This is because DeltaEC can “target” an RPN variable.

  19. 19.

    Reverse Polish notation (RPN) is a system where the “operator” follows the variable(s). The “Polish” designation is in honor of its inventor, Jan. Łukasiewicz (1878–1956). That form of data entry and calculation was used in the scientific calculators made by Hewlett-Packard since the introduction the HP-35 in 1972, the first handheld scientific calculator. RPN is used in HP calculators to the present day. It is preferred by most scientist and engineers of my generation because it takes less key strokes and operations are unambiguous without requiring parentheses.

  20. 20.

    See G. W. Swift, Thermoacoustics: a unifying perspective for some engines and refrigerators (Acoust. Soc. Am., 2002), ISBN 0-7353-0065-2, Chap. 5.2, for a discussion of the difference between total power and acoustic power.

  21. 21.

    The radiation efficiency will be calculated later in this textbook (see Sect. 12.8.3). For those who cannot wait, Πrad = (ρf 2/2c)|U|2. DeltaEC could have calculated those automatically, as well, if a PISTBRANCH or OPENBRANCH segment was placed before the neck that models a flanged open end or unflanged open end.

  22. 22.

    As with many items in this section, DeltaEC supports a lot more capabilities that we have space to explore in an introduction. If you want to know more about Master–Slave links, or any other feature, you are referred to the User’s Guide.

  23. 23.

    There is also some dissipation due to the fluid shear which accompanies the divergence of the streamlines at both ends of the neck. A detailed analysis of this dissipation mechanism and the effective length correction is provided in K. A. Gillis, J. B. Mehl and M. R. Moldover, “Theory of the Greenspan viscometer,” J. Acoust. Soc. Am. 114(1), 166–173 (2003).

  24. 24.

    Using these box parameters and neglecting damping, the Helmholtz frequency for the box containing air with a sound speed of 347 m/s is 43.6 Hz, using Eq. (8.51) and adding one flanged end correction to create an effective length for the port, L eff = 28.4 cm.

  25. 25.

    The particular choice of parameters used in the DeltaEC electrodynamic speaker specification is not unique. Within the loudspeaker design community, the Thiele–Small parameters are far more common, especially in catalog descriptions of commercial drivers (see Fig. 2.42), although the DeltaEC parameter choice is more general, since DeltaEC must accommodate a variety of gases, pressures, and temperatures.

    Of course, there is a one-to-one correspondence between the parameters required by DeltaEC and the Thiele–Small parameters. For example, instead of specifying k, m, and R m, the stiffness, k, will be expressed as the equivalent volume stiffness of air, V AS [m3], if the speaker’s radiating area, S D [m2], is known (see Fig. 7.5). The moving mass, m, can be extracted from the free-cone resonance frequency, f s [Hz], and the mechanical damping, R m [kg/s], will be related to the dimensionless mechanical quality factor , Q MS.

    $$ k={gp}_{\mathrm{m}}{S}_{\mathrm{D}}^2/{V}_{\mathrm{AS}} $$
    $$ m= k/4{p}^2{f}_{\mathrm{s}}^2 $$
    $$ {R}_{\mathrm{m}}=2{pf}_{\mathrm{s}} m/{Q}_{\mathrm{MS}} $$

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  1. Pine Grove Mills, PA, USA

    Steven L. Garrett

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Garrett, S.L. (2017). Nondissipative Lumped Elements. In: Understanding Acoustics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-49978-9_8

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