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Sound waves in a medium containing rigid spheres

IP.com Disclosure Number: IPCOM000128297D
Original Publication Date: 1986-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 9 page(s) / 24K

Publishing Venue

Software Patent Institute

Related People

Dov Bai: AUTHOR [+4]

Abstract

The effective speed of sound in a medium containing particles is different from that in the ambient medium. We calculate it for the idealized case of immovable rigid spheres with centers arranged in a simple cubic lattice. 'We assume that the wavelength in the ambient medium is large compared to the distance between particles. Then the wave equation for the pressure can be reduced to an ordinaxy differential equation with periodic coefficients. Floquet theory shows that there axe pass bands and stop bands along the frequency axis. We determine them, and the sound speed in the pass bands, by both analytical and numerical means.

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This is the abbreviated version, containing approximately 25% of the total text.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Sound waves in a medium containing rigid spheres*

Dov Bai Joseph B. Keller Depaxtment of Computer Science Depaxtments of Mathematics Stanford University and Mechanical Engineering Stanford California 94305 Stanford University Stanford, California 94305

Abstract. The effective speed of sound is calculated for a medium containing immovable rigid spheres arranged in a simple cubic lattice. Long waves propagating along a lattice axis axe treated. The wave equation for the pressure is reduced to an ordinary differential equation to which Floquet theory is applied. Both perturbation and numerical methods are use to find-the effective speed as a function of frequency, and to locate the pass and stop bands.

Keller's research was supported by , the Office of Naval Research, the Air Force-Office of Scientific Research, and the National Science Foundation. Bai's work was supported by the Center for Large Scale Scientific Computing at Stanford under the Office of Naval Research Contract N00014-82-K-0335.

1. Introduction

The effective speed of sound in a medium containing particles is different from that in the ambient medium. We calculate it for the idealized case of immovable rigid spheres with centers arranged in a simple cubic lattice. 'We assume that the wavelength in the ambient medium is large compared to the distance between particles. Then the wave equation for the pressure can be reduced to an ordinaxy differential equation with periodic coefficients. Floquet theory shows that there axe pass bands and stop bands along the frequency axis. We determine them, and the sound speed in the pass bands, by both analytical and numerical means.

2. Formulation

Let the x-axis.be a lattice axis with L the spacing between sphere centers along it, and let R be the radius of each sphere. A wave of frequency w propagates along this axis. By the periodicity of the lattice, the wave is symmetric about the planes

(Equation Omitted)

. Therefore it suffices to determine the wave within the region bounded by these four planes, which may be thou ght of a's a rigid-walled waveguide of square cross section with spheres placed periodically along its axis. When wLIc is small, where c is the sound speed in the ambient medium, the wavelength in this medium is large compaxed to L. Then the pressure p is practically constant over the cross-section of the waveguide, so we write it as p(x).

Under these conditions, p(x) satisfies the long wav~& equation [L1]

(Equation Omitted)

Here S(x) is the unobstructed,cross- sectional area of the guide, given by

Stanford University Page 1 Dec 31, 1986

Page 2 of 9

Sound waves in a medium containing rigid spheres

(Equation Omitted)

n view of (2.2), the ordinary differential equation (2.1) has a periodic coefficient. By Floquet's theorem it has a solution which satisfies the condition

(Equation Omitted)

for some constant K(k) which depends upon k. The complex conjugate p...