Hi Marco, and Welcome to the forum!
As Bert already pointed out, predicting the isolation of a two-leaf wall is no simple matter. It's a lot more complex than for a single leaf wall, which is fairly straight forward. For a single-leaf wall, the Mass Law equations are pretty accurate. If you want to know about isolation at specific frequency bands and draw a graph of TL for the entire spectrum, then you can use the basic mass law equation:
TL(dB)= 20log(M) + 20log(F) -47.2
M is the surface density of the panel, and
F is the center frequency of the measurement band
If you just want an overall estimate of TL for the entire wall, without being too worried about individual frequencies, then you can use the simplified empirical mass-law equation:
TL = 14.5 log (Ms * 0.205) + 23 dB
Where: M = Surface Mass in kg/m2 )
That works for walls with low surface density. For walls with high surface density (above about 150 kg/m2), you need this version:
TL = 37.5 log (M) - 42 dB
That's all for single-leaf walls, and is rather simplistic (there are other factors involved too) but reasonably accurate for most studio-builder applications.
But when it comes to two-leaf walls, things get rather more complicated. Firstly, for any wall (no matter how many leaves) there are actually five different regions on the spectrum for, and each region is controlled by a different aspect of the wall. Like this:
five-regions-of-isolation-spectrum-2.jpg
At the very low end of the spectrum, isolation is controlled by stiffness (how rigid the wall is), then there is a region that is controlled by the resonant frequency of the wall system, followed by a region that is governed by mass law, then there's the "coincidence dip" where isolation is governed mostly by damping, and finally at the very high end of the spectrum, isolation is governed mostly by shear waves in the wall. This graph as shown applies to single leaf walls, and thus shows isolation as increasing by 6 dB per octave in general (mass law), but modified by the various types of resonance.
In the case of a two-leaf wall, isolation increases at a much higher rate, of 18 dB per octave (as you already discovered), but once again modified by the various resonances. With a two-leaf wall, the MSM resonance is usually higher up the curve than the simple resonance for a single leaf wall, which implies that two-leaf walls are much better at isolating higher frequencies, and not so good at lower frequencies.
In fact, at the resonant frequency of the wall, if there is no damping at all and the materials are not very stiff, then not only do you not get isolation, you can actually get amplification! Since the entire wall is resonating, the resulting sound transmission can be louder at that specific frequency range. Something like this:
MSM-isolation-graph--amplification--reduction-#2-negative-image-resonance-damping-GOOD!!!!.GIF
So you can actually get considerably LESS isolation at the resonant region than you would have gotten with just a mass-law governed single-leaf wall.
If you take a close look at the graph, you'll notice that there's a note saying that at 1.4 times the resonant frequency, the difference in isolation is down to zero, again, and at higher frequencies, the wall isolates well. Above that, you get your 18db per octave increase. (If you are interested in total accuracy, the zero point is actually at 1.414 times the resonant frequency, since the math related to resonance is also related to the square root of 2, which happens to be roughly 1.414).
Here's another version of that graph, showing how damping affects that peak:
MSM-AMPLIFICATION-graph---GOOD!!!Ifigtw-V2o.gif
That's why the general recommendation is to tune the resonant frequency of your wall to be at least one octave lower than the lowest frequency you need to isolate (one octave lower = half the frequency), and that you have good damping inside the wall cavity
So how do you tune your wall? Using the equation for 2-leaf MSM resonance, which goes like this:
msm-2-leaf-equation.jpg
Simplifying that, you get this:
Fc=c[(m1+m2)/(m1m2d)]^.5
where:
Fc=resonance frequency (Hz)
c=constant (60 for empty cavity, and 43 for with insulation)
m1=mass of first leaf (kg/m^2)
m2 mass of second leaf (kg/m^2)
d=depth of cavity between leaves (m)
So by adjusting the mass on each leaf of your wall, and the size of the cavity in between you can tune your wall to the resonant frequency you want, which should be no higher than half the lowest frequency that you need to isolate (in other words, it you want to isolate down to 50 Hz, then you tune the wall below 25 Hz, and if you want to isolate down to 40Hz then you tune the wall to below 20 Hz),
If you want an overall summary of figuring out isolation for a 2-leaf wall, then here it is:
Full-spectrum-TL-isoaltion-equations.jpg
You have three equations ther. The first one applies to frequencies below resonance, the second to frequencies around resonance, and the third to frequencies above resonance.
The above is all theoretical, of course, and since theory is no necessarily the same as reality, don't take it as being carved in stone. All of the above assume perfect materials that behave perfevtly under perfect conditions: there's no such thing as perfect materials, and you certainly can't build a perfect wall and keep conditions perfect... so you won't actually get that 18 dB / octave increase, MSM resonance will be higher than you expected, and the wall won't isolate as well as predicted. But the above does at least give you a reasonable basis for estimating overall isolation.
- Stuart -
can be this simple? TL of wall 1 + TL of wall 2 (for frequencies above 1.4 times the MSM resonance frequency of course) and that's it??
It's nice to see that I'm not the only one that thinks this.
If that were the case, then Eric and the guys at Galaxy Studios wasted a huge amount of time and money building enormously massive concrete bunkers, and floating them on complex spring systems in order to get their world-record 100 dB of isolation...