Page 20 - Spring 2007

Fig. 11. Typical transmission loss plot for variable angles of incidence. Grazing inci- dence refers to acoustic waves that are nearly in the plane of the plate.
computes averages of response over space, since <E>πλατε is averaged over the plate.
A full discussion of SEA is outside the scope of this tuto-
rial, but you can find more information on SEA in the review
paper by Burroughs10 and a more thorough treatment of how
structures respond to diffuse fields in the article by Shorter
11
need to know the incident pressure spectrum of the diffuse acoustic field, and the radiation and structural resistance. You also need an estimate of the plate’s modal density, which is
and Langley.
You can use Eq. 21 without using SEA, though. You just
,
(22)
Note that the plate’s modal density depends on frequency (as do all modal densities).
Examining the equation, we see that the higher the radi- ation resistance, the higher the plate energy (and therefore the higher the plate’s vibration). So, the better a plate radiates, the easier it is to excite with incident pressure fields!
Now, what happens when there is also fluid on the other side of the plate? How much of the incident sound gets through the plate to the other side? This is the classic sound transmission loss problem, and may be solved easily for an infinite plate, and not so easily for a finite one.
Fahy4 provides a derivation of the sound power transmission coefficient through an infinite plate in his text- book, and we repeat it here (assuming the fluids on both sides of the plate are the same):
stiffness is dominant, and the transmission loss increases with the 6th power of frequency, or 18 dB/octave. Figure 11 shows what most people already know from experience—it is hard to keep low-frequency sounds from propagating through barriers. Consider this the next time you close a door to block out sound from a hallway or another room. You stop hearing mid-high frequency sounds, but still hear ‘muf- fled’ low-frequency noise.
To visualize the sound field incident on and transmitted by an infinite plate, Fig. 12 compares pressure and displace- ment of a plate at two conditions: well below, and near coin- cidence. In the example, we have set the plate loss factor equal to 0. Try setting loss factor to zero and computing the transmission coefficient in Eq. 23 at coincidence (remember, this is where the acoustic wavenumber in the plane of the plate matches the free bending wavenumber in the plate, or kοsinφ = kβ). You should compute a transmission coefficient of 1, which is perfect sound transmission!
The strength, or depth of the coincidence dip depends strongly on the plate’s loss factor η. Designers of noise barri- ers (windows, doors) try hard to minimize the depth and breadth of the coincidence dips. The most common approach for mitigating coincidence dips is using constrained layer damping, or CLD (we learned about this in Part 1 of this arti- cle). Automotive glass in luxury vehicles, and glass in high- end office buildings usually have a thin layer of clear vinyl sandwiched between two panes of glass.
For zero, or normal angle of incidence (sound waves nor- mal to the plate’s surface), the transmission coefficient is not indeterminant (you might think it would be, since there are several terms in Eq. 23 that divide by sin(φ)). The transmis- sion coefficient actually simplifies to:
,
(24)
which corresponds to the well-known ‘mass law.’ The mass law transmission loss is shown in green in Fig. 11, and increases with the square of frequency over all frequencies,
The red, green, and blue terms in the equation represent the damping, mass, and stiffness of the plate, respectively. The amount of sound transmitted depends on the fluid proper- ties, the structural properties, frequency, and the angle of the incident pressure wave with respect to the plate.
Some typical transmission loss plots, computed as 10log10(1/τ) are shown in Fig. 11. For acoustic waves not nor- mally incident to the plate, sharp dips appear in the transmis- sion loss. These dips correspond to sharp peaks in the trans- mission coefficient (transmission loss is the inverse of the transmission coefficient), and act as strong pass-bands of inci- dent sound. The dips are at the coincidence frequencies of the plate. Recall that the coincidence frequencies depend not only on the plate, but on the angle of incidence of the sound waves. As the angle of incidence changes, the coincidence frequency and the frequency of the transmission loss dip changes as well.
At low frequencies, the mass term in Eq. 23 determines the transmission loss, which increases with the square of fre- quency (6 dB/octave, or 6 dB for each doubling of frequen- cy). At high frequencies, above the coincidence dip, plate
.
(23)
18 Acoustics Today, April 2007