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Institute of Sci. & Tech.

Corrections to be noted in Volume 65 of the JOURNAL OF RESEARCH of the
National Bureau of Standards—B. Mathematics and Mathematical Physics

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CONTENTS OF VOLUME 65B

61

Page

99

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III

JOURNAL OF RESEARCH of the National Bureau of Standards-B. Mathematics and Mathematical Physics Vol. 65B, No. 1, January-March 1961

On Transient Solutions of the "Baffled Piston"

Problem

F. Oberhettinger 1

(September 14, 1960)

The acoustic field produced by the movement of a piston in an infinite rigid wall for arbitrary time dependency of the motion is given.

1. Introduction

The case of the time harmonic movement of a piston membrane in an infinite rigid wall ("baffle") can easily be generalized to the case when the motion of the piston is not periodic but an arbitrary function of time. Such transient solutions have become of considerable interest in recent times (for a detailed treatment of the propagation of such sound pulses, see [5]2). The procedure for the case treated here is the same as used elsewhere [7], i.e., the Green's function for the exponential decay case (modified wave equation AU-2U=0, y=ik) is used to obtain the solution for the pulse problem. The accustic field (velocity potential) for the time. harmonic movement of the piston include representations given by Bouwkamp [1], King [6], and Wells and Leitner [9]. The first of these contributions gives the solution in the form of a series expression while the second and third involve integral representations that are obtained using integral transform methods (Hankel transform [4, p. 73] and Lebedev transform [4, p. 75] respectively). These representations can be used to treat the general case of an arbitrary movement of the piston. In view of the method to be employed here, such representations should be used for which the inverse Laplace transform of the velocity potential with respect to the purely imaginary wave parameter y=ik can be given. Such an expression can be obtained in a direct way by regarding each point of the moving disk as an accustic point source and integrating over all points of the disk.

2. Exponential Decay and Time Harmonic Solution

The piston is represented by an infinitely thin circular disk of radius a located in the x,yplane with its center at the origin of a three-dimensional Cartesian system of coordinates. The remaining part of the x,y-plane consists of a rigid wall. The movement of the disk is time harmonic of the form

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Each point Q of the disk can be considered as a point source with the velocity potential

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(1)

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(w kc, c is the velocity of sound), which represents Green's function of the wave equation in free space (see fig. 1).

1 Oregon State College, Department of Mathematics (invited paper).

2 Figures in brackets indicate the literature references at the end of this paper.

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