Page 12 - Spring 2007

Fig. 1. Low frequency vibration and acoustic fluid motion of a piston. Left – baf- fled piston; Right – unbaffled piston.
(FSI). FSI usually refers to how moving fluids interact with
solid objects, such as the turbulent flow over a structure. To
learn more about FSI, we recommend the references by
23 Blevins and Naudascher and Rockwell.
Piston vibrating against an acoustic fluid
Perhaps no other structural-acoustic system has been studied more than a circular baffled piston vibrating against an acoustic fluid (see, for example, Fahy,4 pages 58–60 and 118–121; Pierce,5 pages 220–225; and Junger and Feit,6 pages 95–100 and 105–109). We will describe how a piston interacts with a surrounding acoustic fluid, and how the acoustic fluid affects the piston’s vibrations. This simple problem allows us to introduce many structural-acoustic quantities of interest.
Let us start by considering a very slowly oscillating pis- ton in a rigid baffle, as shown in Fig. 1. As the piston moves outward, fluid flows from the area of high pressure near the center of the piston to its outer edge, which is nearly at ambi- ent pressure. Conversely, as the piston moves inward, fluid flows toward the center of the piston, where the pressure is less than the ambient pressure. The level of sound radiation is a function of frequency because as the speed of the oscilla- tion of the piston increases, there is less and less time for the pressure to equalize, and eventually the surface vibrations become much more efficient in compressing the adjacent fluid and causing sound radiation. Since the piston’s size con- trols how far the pressure pulses have to travel before they can equalize with the surrounding fluid, it is important in determining when it begins to radiate sound efficiently.
Thus, we can conclude purely from physical arguments that the main parameters which control the acoustic radiation from a vibrating structure are its speed of vibration (or fre- quency) and its size. It is common in acoustics to nondimen- sionalize the problem in terms of the quantity ka, where k is the acoustic wavenumber and a is the characteristic dimension of the vibrating structure. For a piston source, the transition from an inefficient, low frequency oscillation to an efficient, high fre- quency oscillation occurs at approximately ka = 1, where a is the radius of the piston. At this frequency, the acoustic wave- length is of the same order as the size of the structure.
A source with zero average displacement is an even more inefficient radiator of sound at low frequencies, like an unbaffled piston, i.e., a loudspeaker without a cabinet. When the diaphragm of the speaker moves, the fluid on one side of the speaker is compressed while the fluid on the other side is expanded (see Fig. 1). There is a natural flow of fluid from high pressure to low pressure, and thus the fluid flows around the edges of the speaker. The pressure equalization is very nearly complete because the average displacement of the sur- face is zero when both sides are taken into account, such that there is very little residual compression of the acoustic medi- um and sound radiation. As the frequency of vibration increases, the sound radiation also increases because there is not as much time for the fluid to flow around the edges of the speaker before it is compressed. At approximately ka = 1, a source with zero average displacement becomes as efficient a radiator of sound as a baffled piston.
Let us consider the pressure field radiated by an oscillat- ing piston at discrete frequencies. We assume the piston’s
vibrations are time harmonic (eiwt), and consider pressures in the far-field, or far away from the structure. The far-field pressure radiated by a baffled circular piston as a function of angle θ and distance r is:
, ( 1)
where ρ0 is the fluid density, a is the piston radius, vν is the piston velocity (assumed to be constant over the surface of the piston), and kο is the radial frequency ω divided by the acoustic sound speed cο. We do not include the time-har- monic dependence in this, or in any future equations.
Figure 2 shows the far-field pressure at two frequencies, normalized to kοa, where kοa=π and 3π. For kοa=π (a full acoustic wavelength spanning the piston diameter), the pres- sure is in phase at all directivity angles (note that the direc- tivity angle is taken from a vector pointing normal to the pis-
10 Acoustics Today, April 2007
Fig. 2. Pressure distribution in the far-field of a radiating baffled piston. Top – ka=π; Bottom – ka=3π.