Page 16 - Spring 2007

Fig. 6. Bending and acoustic wavenumber-frequency plot, with trace diagrams of bending and acoustic waves at and above coincidence. Below coincidence, an infi- nite bending wave radiates no sound!
waves now radiate sound, as shown in the image in the middle of Fig. 6. They do so in the plane of the plate, or grazing the plate. At frequencies above coincidence, the flexural waves continue to speed up—eventually becoming pure shear waves at very high frequencies (see Part 1 of this article). At these high frequencies, the sound radiated by the flexural waves propagates in a preferred direction, which is computed by trace matching the flexural wave to the shorter acoustic wave, as shown in the image on the bottom right of Fig. 6.
The angle of dominant radiation (taken from a vector normal to the structure’s surface) is computed as:
, (12)
where cΒ is the flexural wavespeed. You can compute a plate’s coincidence frequency by setting the flexural and acoustic wavespeeds (or wavenumbers) equal to each other. For bend- ing waves in thin plates at coincidence:
moduli (Young’s Modulus E and Shear Modulus G) speeds up flexural waves, and lowers the plate’s coincidence frequency. Increasing a plate’s density increases its mass, slowing down flexural waves, and raises the plate’s coincidence frequency. Increasing thickness increases both stiffness and mass, but increases the stiffness at a greater rate, so thickening a plate will lower its coincidence frequency.
Therefore, stiffening a plate lowers its coincidence fre- quency, allowing it to radiate sound at lower frequencies. Conversely, mass-loading a plate raises its coincidence fre- quency, so that the plate does not radiate sound at low fre- quencies. It would seem that the answer to most noise con- trol problems would be to simply add mass to a plate while reducing its stiffness! While this is a good way of reducing sound radiation, we have never had any of our sponsors accept it. As most of us have experienced, nearly all new structures are lightweight and stiff, like carbon-fiber com- posites reinforced with ribbing. These sorts of structures typ- ically have low coincidence frequencies, and therefore radiate sound very well. Also, as we will see later, lightweight stiff structures are very easy to excite by sound waves.
Since most practical structures are finite, we will now explain how well the mode shapes of a structure radiate (see Part 1 of this article for a discussion of structural resonances). The classic example studied by early structural acousticians is our old friend—the simply supported rectangular plate.
Before we proceed with modal sound radiation, remem- ber a key concept: the standing waves in a mode shape are comprised of multiple left and right (and forward and back) traveling waves, which propagate at the structure’s wave speed. It is these traveling waves that radiate, or do not radi- ate sound. If our plate were infinite, such that there were no reflections from any of the plate boundaries, subsonic flexur- al waves would radiate no sound. However, the discontinu- ities at the boundaries ‘scatter’ the energy in subsonic flexur- al waves into many wavenumbers, some of them supersonic, so that a finite plate radiates sound below its coincidence fre- quency. The amount of sound radiation depends on the radi- ation efficiency of each of the plate’s modes.
Maidanik7 and Wallace8 computed how much sound a finite rectangular simply supported plate’s modes radiate. In particular, Wallace provides formulas for the frequency- dependent far-field pressure and acoustic intensity fields radi-
Fig. 7. Effects of stiffening and mass-loading on plate coincidence frequencies. Stiffening a plate reduces the coincidence frequency, and adding mass increases it.
, so
(13)
(14)
How does coincidence frequency vary with plate parameters? To find out, let us examine Fig. 7. Increasing a plate’s elastic
14 Acoustics Today, April 2007
.