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Research Papers

A Thévenin-Inspired Approach to Multiple Scattering in Acoustics

[+] Author and Article Information
Randall P. Williams

Department of Electrical and
Computer Engineering,
The University of Texas at Austin,
10100 Burnet Rd., Bldg. 160,
Austin, TX 78758
e-mail: randy.williams@utexas.edu

Neal A. Hall

Department of Electrical and
Computer Engineering,
The University of Texas at Austin,
10100 Burnet Rd., Bldg. 160,
Austin, TX 78758
e-mail: nahall@mail.utexas.edu

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 22, 2018; final manuscript received July 10, 2018; published online September 10, 2018. Assoc. Editor: Ronald N. Miles.

J. Vib. Acoust 141(1), 011016 (Sep 10, 2018) (12 pages) Paper No: VIB-18-1214; doi: 10.1115/1.4040927 History: Received May 22, 2018; Revised July 10, 2018

We have previously shown how Thévenin's theorem may be used to solve problems in linear acoustic scattering from a mobile body, by forming the solution as a superposition of the field scattered from the body when held immobile and the solution for radiation from the body in a quiescent field (Williams, R. P. and Hall, N. A., 2016, “Thévenin Acoustics” J. Acoust. Soc. Am., 140(6), pp. 4449–4455). For problems involving scattering from multiple mobile bodies, the approach can be extended by using multiport network formalism. The use of network formalism allows for the effects of multiple scattering to be treated using analogous circuit models, facilitating the integration of scattering effects into circuit-based models of acoustic transducers. In this paper, we first review Thévenin's theorem for electrical and linear acoustic systems, and discuss the Thévenin-inspired approach to scattering from one rigid, mobile cylinder. Two-port formalism is introduced as a way to address problems involving two scatterers. The method is illustrated using the problem of scattering from a pair of rigid, mobile cylinders in an ideal plane progressive wave. The velocities of the cylinders and the resultant pressure field in response to the incoming wave are found. Unique features of the method compared to more conventional approaches are discussed.

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References

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Figures

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Fig. 1

Thévenin equivalent representation of an electrical network, characterized by the parameters VTh and ZTh, connected to a load with impedance Zload

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Fig. 2

Problem geometry for scattering of a plane sound wave by a rigid, mobile cylinder

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Fig. 3

Thévenin equivalent system representation for a single cylinder in a sound wave

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Fig. 4

A plane wave with amplitude pinc propagates in the +x− direction and is scattered by a pair of rigid, mobile cylinders. The cylinders are separated along the direction of propagation by a distance b and react by moving with time harmonic velocities u1(t) and u2(t).

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Fig. 5

Schematic illustration of a general two-port linear electrical network containing passive elements without independent sources. Vj and Ij are the voltage and current on the jth port, respectively.

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Fig. 6

Illustration of cylinders with independent velocities uj and net forces due to radiation Frad,j (a), and equivalent two-port linear system representation (b)

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Fig. 7

Circuit representation of the two cylinder acoustic system. Each cylinder experiences an independent blocked force Fb,j(t) and has mass Mc,j. Motional acoustic coupling occurs through the two-port radiation impedance element.

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Fig. 8

Normalized force due to scattering on each cylinder when both are immobile and for a normalized separation distance of b/a = 8

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Fig. 9

Normalized radiation impedances Zrad,11 and Zrad,12 for the two cylinders, for a normalized separation distance of b/a = 8

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Fig. 10

Normalized velocities for the two cylinders as functions of frequency, for neutrally buoyant cylinders separated by a normalized distance of b/a= 8. The single-cylinder solution is shown for comparison.

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Fig. 11

Pressure magnitude distribution around the surface of cylinder 1 as a function of θ1, as a result of scattering from two neutrally buoyant cylinders separated by a distance b/a= 8 and ka= 0.1 (a), and for ka= 3 (b). The single-cylinder solution is also shown, which serves to highlight the impact of adding a second mobile cylinder on the total resultant pressure distribution.

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Fig. 12

Schematic illustration of acoustic scattering from a pair of rigid, mobile cylinders which are connected by a mechanical spring of stiffness k, and which are coupled with an electromechanical actuator which generates a signal Vout in proportion to their velocity difference

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Fig. 13

Equivalent circuit model using the mobility analogy for the two-cylinder system shown in Fig. 12. The elements at the top of the circuit capture the transducer and spring linking the cylinders.

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