The formula f = 170/√(m x d) is often quoted as applying to limp mass membrane bass traps. The m term is surface weight (lb/sqft) of the membrane and d is the depth of the air cavity (inches) behind the membrane. This formula is often recommended to people who are designing membrane bass traps. A membrane bass trap is usually a wide, shallow box mounted on the wall with the front side of the box covered by a thin sheet, an airtight membrane of material.
See Home Recording Studio, Gervais, Fig 9.10 (Panel Trap Formula). This type of limp mass membrane bass trap is essentially a one sided, wall mounted bass trap.
A) Limp Mass Membrane
There are two kinds of membranes, one being a true limp mass such as MLV and the other being a thin sheet of wood. The original formula is actually only appropriate for true limp mass membranes, f = 170/√(m x d). The m term is surface weight (lb/sqft) of the membrane that covers the air cavity and d is how deep the air cavity is (inches).
We have a 3’ by 6’ wall mounted membrane bass trap with a 10″ deep cavity. Using 1/16th inch MLV (1 lb per sqft) for the limp mass the resonant frequency is calculated by using the basic formula: f = 170/√(m x d) = 170 /√(1.0 x 10) = 54 Hz.
This formula is only useful for large surface limp mass membrane bass traps.
B) Thin Panel Membrane
Experimenters (Engineering Noise Control, Bies and Hansen, 4th Ed, 2009,CRC Press, Equation 8.55) with panel style membrane bass traps have had to make an adjustment to the traditional limp mass formula because they keep measuring resonate frequencies that are higher than the formula predicts. The empirically adjusted formula raises the resonant frequency by about 1/3rd.
fo = 1.34 x 170/√(m x d) = 228/√(m x d)
Again, m is thin panel surface weight (lb/sqft) and d is air cavity depth (inches).
Why the change? The original formula is theoretical, based on an infinitely wide limp mass over an infinitely wide air space. In other words there were no edge effects. But in the real world there are edge effects. The panel or membrane is rigidly attached air tight to a frame around the edge, which means the section of the panel near the edge is not as free to move under pressure as the more central area. Net result is that we have less weight moving on the same air spring and that means a higher resonant frequency.
If we have a panel, the light-mass edge effect is stronger than if we have MLV type limp mass. A panel is stiffer around the nailed down edge than a limp mass sheet, which accounts for the panels having a stronger edge effect, a lower effective mass compared to equivalent weight limp mass.
For a thin ¼” plywood (0.7 lb/sqft) panel trap over a 10″ air space we have to use the thin panel version of the membrane bass trap formula: f0 = 228/√(m x d).
fo = 228/√(0.7 x 10) = 86 Hz
Had we used the limp mass formula fo = 170/√(m x d) we would have predicted:
fo = 170/√(0.7 x 10)= 63 Hz, which would have been too low.
The predicted frequency for a limp mass membrane bass trap should use the original membrane formula fo = 170/√(m x d) while a thin panel membrane bass trap should use the modified formula f0 = 228/√(m x d).
C) Isothermal Improvement
If the air cavity is stuffed with batt insulation, the lightweight fiberglass material that usually comes in a roll and when expanded weighs about 0.4lb/cuft, the air in the cavity is changed from adiabatic to isothermal air. Isothermal air cavities are softer than adiabatic air cavities.
Filling the cavity with batt insulation distributes heat sinks throughout the air cavity which eliminates the heating of the air when compressed or cooling when expanded. Isothermal air stays at a constant temperature and does not develop the thermal back pressures due to warming and cooling that regular air experiences with the alternating pressures of sound waves.
When the limp mass membrane bass trap formula is converted to an isothermal air cavity,
the frequency is lowered to 84% of the adiabatic formula: fo = 143/√(m x d).
When the thin panel membrane formula is converted to an isothermal air cavity,
the frequency also lowers to 84% of the adiabatic formula: fo = 192/√(m x d).
Add batt insulation to our limp mass membrane bass trap, 1 lb/sqft sheet over 10″ air cavity lowers resonant frequency: fo = 143/√(1.0 x 10) = 45 Hz
We stuff the 10″ air cavity of a ¼” thin panel (0.7 lb/sqft) membrane trap with batt insulation to get a lower frequency for the same sized unit: fo = 192/√(0.7 x 10) = 73 Hz.
D) Double Sided Bass Trap
Sometimes it is of interest to make double sided limp mass bass traps. It’s a lot lighter and absorbs twice as much bass power. Instead of hanging against the wall it is mounted perpendicular to the wall so both sides can react to bass pressure, or it is mounted on the diagonal in the corner. It can also be flown overhead in the open space of larger rooms. The only difference between the limp mass membrane bass trap and the double membrane space trap is that the space trap is built twice as deep.
These double sided membrane bass traps do not push against the wall when the bass wave is being absorbed. Bass pressure pushes equal and opposite on both sides of the double membrane trap at the same time, which is why it does not transfer momentum to the wall it is attached to.
If the double sided bass trap is sheeted with limp mass on both sides and has an air cavity D inches deep, then the limp mass membrane formula fo = 170/√(m x d) can be used as long as d equals half the depth D of the double walled bass trap, d = D/2.
If the double sided membrane trap is made out of thin panels, then the thin panel formula is used: f0 = 228/√(m x d), where d = D/2, half the spacing between the panels.
Isothermal cavity will use the isothermal trap formulas, where d = D/2.
This formula is for a bass trap with a sound panel mounted close to but not touching the back side of the limp mass membrane bass trap. This panel has to be rigidly held in position or else the bass pressures will cause it to vibrate, and then it’s not absorbing much sound.
If the remaining air cavity of either trap is backfilled or stuffed with loose batt insulation, the air cavity is changed from an adiabatic cavity to a much softer isothermal air cavity. Because the air spring inside the trap is softer, the resonant frequency formula is lower and the formula is fo (isothermal) = 143/√(m x d), where d = D/2, half the space between the two identical weight sheets.
E) Changing Air Volume
Let’s try something else. When we are working with bass pressures, as long as the dimension of the pressure zone is 1/10 wavelength in size, its shape doesn’t much matter. If we are trying to hit 60 Hz, the wavelength is about 20’ and the 1/10 wavelength is 2’. This means we can make minor variations in geometry of the air cavity, within the range of 2’ that result in overall volume changes while retaining the same membrane and we can change the resonant frequency.
If we could add a 6″ wide cavity around the edge of the panel, but otherwise keeping the panel the same size, we are adding ½’ x (6 + 3) x 2 = 9 square feet to the volume of air behind the panel, which was already 18 square feet. This is the same as changing the depth of the panel from 10″ to 15″.
Now we add isothermal batt into the enlarged air volume to soften the air spring, we have fo = 192/√(0.7 x 15) = 59 Hz, pretty close to the 60 Hz we were trying to hit.
F) The Air Cavity
All of these limp mass membrane bass traps are in effect a mass on top of a spring with some resistance added, essentially a series LRC circuit. Sound pressure pushes the membrane which moves. As it moves it pushes air through the flow resistive pad that is located just below the membrane surface. As air flows through the pad, friction occurs and energy is lost. Then the air flowing through the pad pushes onto the internal air cavity and compresses the air of the cavity. The moving membrane (mass) and the air cavity (spring) are separated by an air flow resistor that absorbs energy out of the movement of air between the membrane and the cavity.
Adding batt insulation into the cavity is primarily done to change the cavity from an ordinary adiabatic cavity into a softer isothermal air cavity, which has the same effect as if the cavity was enlarged. It is often said that adding batt insulation to the air cavity is done in order to increase the sound absorbing effect. Batt insulation in the cavity lowers the spring constant, makes it softer, which lets more air flow through the resistor for the same given air pressure outside the membrane trap. In that sense, adding batt insulation enables the trap to absorb more sound.
And in a second sense, adding batt to the cavity does add sound absorption to the cavity but only in the mid bass frequency range where the batt is no longer providing the isothermal condition to the air in the cavity. Usually this mid bass absorption quality is not useful because it is the lower bass that needs absorption and secondly, because of the limp mass law, less and less bass energy passes through the membrane in the upper bass registers.
G) Adding the Damping Factor
What is the right damping factor for a bass trap to have? What is the damping factor and how do we measure it? Essentially the damping factor is the % or fraction of energy lost per cycle. If we have a light damping factor the membrane moves easily and requires many cycles to absorb some amount of energy. If we have a strong damping factor, the membrane barely moves because of the friction.
This is a time old question. In general, the rule is to not use a light damping factor. The higher the resistance, the more power is transferred into the resistor. But if it is too high, no power is transferred. The resistance offered by a sound panel located directly under the membrane is a velocity damper, in that the membrane pushes air back and forth in the wall of the sound panel. Most, but not all damping, is velocity damping. Some damping is displacement dependent such as with constrained layer damping.
Critical damping factor is achieved when the energy of the system goes to zero as quickly as possible but without any overshoot or vibration. If the damping factor is set at ½ then when the system is thumped, there is just the initial overshoot with a very small and slow spring back and then the energy is gone. This ½ damping factor setting allows the membrane to quickly respond to transients as well and frequencies.
So, put the membrane bass trap together and test it. Thump it. If it doesn’t spring back at all, it’s over damped and needs more freedom of movement. Lighten the friction load and thump it again. If it twangs at all, it is under damped and needs more friction added back in. Thump it again and it sounds dead and does only one overshoot followed by a slow return to center, that’s the desirable damping factor.
LMV or limp mass membranes behave nicely in this manner. But light panel membrane traps do not. If you thump them, they twang like a board will twang. But that’s in a much higher register. In this case we are looking for very low frequencies, in the 40 to 60 Hz range. A softer thump will better stimulate the lower registers only. Use the heel of your fist instead of your knuckle to deliver the thump on your limp mass membrane bass traps.