Page 17 - Spring 2007

Fig. 8. Far-field sound intensity in air for various low-order simply supported 1 m square flat plate modes. Top–below coincidence, Middle–near coincidence, Bottom–above coincidence.
ated by a given plate mode, along with an integral for comput- ing the sound power radiation efficiency for each mode. We show the far-field intensity below, near, and above the plate coincidence frequency for the first three low-order modes of a square 1m x 1m plate in Fig. 8. Recall from the first article that we define mode orders as (m, n) pairs, where m and n corre- spond to the number of antinodes (regions of maximum defor- mation) in the plate’s x and y directions, respectively.
In the figure, the mode shapes of the plate are shown, along with the corresponding far-field intensity patterns. At low frequencies, the fundamental (1,1) mode radiates sound omnidirectionally, like a baffled circular piston. Also at low frequencies, the (1,2) mode radiates sound like a dipole, and the (2,2) mode radiates like a quadrupole.
The figure also shows radiation efficiencies as a function of acoustic wavenumber (frequency/acoustic wavespeed), with a line shown to indicate the frequencies of the directiv- ity plots. Notice how they resemble the normalized radiation resistance of the baffled circular piston (Fig. 4). Below coin- cidence, the efficiencies increase rapidly with increasing fre- quency. The efficiencies peak at coincidence (exceeding 1, showing that they are not true efficiencies!), and then asymp- tote to a value of one above coincidence.
Above coincidence, the far-field sound directivity changes, with ‘lobes’ of sound radiated from the structure at critical angles. These critical angles may be computed using the trace matching procedure described above for the infinite plate. The critical angles exist for all plate modes except the fundamental (1,1) mode, which radiates sound normal to the plate at all frequencies, but with a spatial ‘beamwidth’ that narrows with increasing frequency.
Let us re-examine the radiation efficiencies below coin- cidence for the different mode orders. We see that the (1,1) mode radiates sound most efficiently, followed by the (1,2) mode, and then the (2,2) mode, which radiates sound least efficiently. This trend shows an important result: modes with odd m and n orders radiate sound much better than those with mixed orders (odd-even or even-odd), which radiate better than those with purely even orders (even-even).
Using the analytical radiation efficiencies, we can com- pute how much sound a rectangular plate radiates when driv- en by a point force. To do so, we combine the mobility for- mula from Part 1 of this article with radiation efficiencies
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computed using the formulas in Wallace. Ignoring any fluid
loading effects on the structure (which we know will mass- load and radiation damp the structure from our exercises with the baffled circular piston), we show the mobility, radi- ation efficiency, and radiated sound power for a unit force drive in Fig. 9.
Starting with the mobility (top of the figure), we see how the contributions from the individual modes compare to the total mobility. The mobility is dominated by resonant response at the resonance frequencies, and a mix of non-res- onant responses away from resonance.
Next, examining the radiation efficiencies of the modes shows the same trends we observed in Fig. 8—that odd-odd modes radiate sound very well, but modes with mixed and even orders radiate sound poorly. The fundamental (1,1) mode radiates sound very much like a baffled piston, and the
radiated sound power transfer function (power normalized by the square of the input force) is dominated by the sound radiated by that mode. Peaks in the sound power occur for the other modes, primarily the other odd-odd (1,3) mode, but the non-resonant sound radiated by the (1,1) mode is nearly flat with increasing frequency.
This example brings up one of the key points of this arti-
Structural Acoustics Tutorial 2 15