Page 14 - Spring 2007

piston. In fact, power can be calculated knowing a structure’s radiation resistance and spatially- and time-averaged normal velocity:
. (4)
The radiation resistance and reactance are the real and imaginary parts of the fluid loading on the structure’s surface. Imagine once again the fluid as an encompassing elastic blob surrounding the structure. As the structure pushes against this blob, it encounters an impedance, which resists the structure’s motion. The impedance, Zο, is complex, and equal to Rο + iXο, where Xο is the reactance. The resistance and reactance are fluid-loading properties of the structure, and do not depend on how the structure vibrates. Equations for the resistance and reactance (shown without derivation) of a baf- fled circular piston are:
Popular approximations for the low-frequency resistance and reactance are:
(7)
At high frequencies, the resistance asymptotes to a con- stant value, while the reactance decreases inversely propor- tionally to frequency:
. (8)
Now, notice how Rο and Xο in the example are plotted in absolute terms, but also plotted normalized against ροcοA (see the axis on right side of the plot) and against dimensionless frequency ka (see the axis on the bottom of the plot). Recall that you can use ka to visualize how many acoustic wave- lengths fit over the characteristic dimension a. In our plot, the maximum ka is π, which corresponds to a half wavelength over the piston radius, or a full wavelength across the piston diam- eter. Ro /ροcοA converges to a value of 1 at high frequencies. In fact, Ro /ροcοA is called the normalized radiation resistance, or more commonly, the radiation efficiency:
. (9)
We will learn more about radiation efficiency when we dis- cuss how sound is radiated by plate modes.
At the bottom of Fig. 4, examples of how the fluid load- ing varies over the surface of a piston are shown at low, mid, and high frequencies. Below a ka of π/2, the fluid loading is primarily reactive, or mass-like, weighing down the piston. Above ka of π/2, the fluid loading becomes more resistive, absorbing energy in the form of sound from the vibrating structure.
Now, let us suppose that the piston is the mass element of a simple harmonic oscillator, where the mass (m) rests on a grounded spring (k) and dashpot (b). We now consider the effect of the complex fluid loading on the piston resonance, where the piston mobility in-vacuo is:
. (10)
The fluid loading (resistance Rο and reactance Xο) may be added to the mobility equation to produce:
(11)
For a 1 gram piston of 16.6 mm radius, with spring con- stant k=1x105 N/m, and a damping constant b of 1, we com- pute the drive point mobility v/F in air (ignoring fluid load-
, and
. (6)
Note once again that no piston structural properties appear in the above equations; they are solely determined by the geome- try of the piston and the acoustic properties of the fluid.
We plot the resistance and reactance for a 16.6 mm radius piston in water at the top of Fig. 4. At low frequencies, the resistance (in red) looks like a parabola, while the reac- tance (in green) looks like a line with a constant slope.
(5)
Fig. 4. Radiation impedance of baffled circular piston. Top–resistance and reac- tance as a function of frequency (and ka); Bottom–spatial variability of resistance and reactance at three discrete frequencies.
12 Acoustics Today, April 2007