To start with I'll show you my double leaf wall calculations and then I'll move on to the triple leaf questions.

(I'm going to show my working, so you can double check the method is correct)

**For my double leaf walls:**

First calculate MSM wall resonant frequency:

Fx = c[(m1+m2)/(m1m2d)]^0.5

18mm OSB3 and 15mm Plasterboard on each decoupled wall frame. Insulated cavity is 225mm panel to panel.

18mm OSB3 at 11.8kg/m2.

15mm Plasterboard at 10.25kg/m2.

Total: 22.05kg/m2

Giving:

Fx = 43 * [(22.05 + 22.05) / (22.05 * 22.05 * 0.225)] ^ 0.5

Fx = 43 * [ 44.1 / 109.4 ] ^ 0.5

Fx = 43 * 0.4 ^ 0.5

Fx = 43 * 0.63

**Fx = 27.09Hz**

Single leaf transmission loss for each leaf:

TL = 14.5 log(M * 0.205) + 23 dB

TL = 14.5 log(22.05 * 0.205) + 23

TL = 14.5 log(4.52) + 23

TL = 14.5 * 0.66 + 23

TL = 9.57 + 23

**TL = 32.57dB**

Now move on to transmission loss formulas:

R1 and R2 are using my single leaf transmission loss calculated above (32.57dB).

f0 is my wall resonant frequency.

f1 is 55/d Hz = 55 / 0.225 = 244.45Hz.

R = 20log(f (m1 + m2)) - 47 ...[for the region where f < f0]

R = R1 + R2 + 20log(f * d) - 29 ...[for the region where f0 < f < f1]

R = R1 + R2 + 6 ...[for the region where f > f1]

The lowest frequency I'm realistically aiming to isolate is the low E of a 4 string Bass guitar so I'm looking at 40Hz upwards.

So as 40Hz is above my resonant wall frequency and less than 244.45Hz I use formula two above.

R = R1 + R2 + 20log(f * d) - 29

R = 32.57 + 32.57 + 20 log( 40 * 0.225) - 29

R = 65.14 + 20 log( 9 ) - 29

R = 65.14 + 19.08 - 29

**R = 55.22dB**

I am happy with this level of isolation if I can achieve it.

Now moving onto the the triple leaf, this is my roof. I need a vented roof space so will have to seal the underside of the roof joists, this forces me to have a triple leaf ceiling.

How do you calculate the resonant frequencies is this system? As I have read this is not just a case of calculating them independently?

I've seen these equations listed on another thread here: The author of the thread listed the variables as:

m1,m2,m3 - mass per unit area (kg/m2)

c0 - speed of sound 343 m/s

ro0 - air density 1,18 kg/m3

These are more complicated formulas and I'm struggling to understand the triple leaf one especially. This is my working for the double leaf partition:

m0 = (2 * m1 * m2) / (m1 + m2)

m0 = (2 * 22.05 * 22.05) / (22.05 + 22.05)

m0 = 972.4 / 44.1

m0 = 22.05

f0 = 1 / (2π) * [(3.6 * ro0 * c0^2) / (m0 * d)] ^ 0.5

f0 = 0.16 * [(3.6 * 1.18 * 343^2) / ( 22.05 * 0.225)] ^ 0.5

f0 = 0.16 * [499772.95 / 21.07] ^ 0.5

f0 = 0.16 * 23719.65 ^ 0.5

f0 = 0.16 * 154.01

**f0 = 24.64Hz**

This is reasonably close to the simplified formula I've used before (although I'm not sure if insulation is figured in this formula at all??). Have I calculated this correctly?

Then can someone help me with the second formula for triple leaf partitions please?

This is my working so far on the triple leaf calcs:

m1 = 22.05

m2 = 22.05

m3 = 11.8 (single OSB3 18mm)

d1 and d2 = 0.2 (200mm cavities)

a = 1 / (2 * m2) * ( [(m1 + m2) / (m1 * d1)] + [(m2 + m3) / (m3 * d2)] )

a = 1 / ( 2 * 22.05) * ( [(22.05 + 22.05) / (22.05 * 0.2)] + [(22.05 + 11.8 ) / (11.8 * 0.2)] )

a = 0.023 * ( [44.1 / 4.41] + [33.85 / 2.36] )

a = 0.023 * ( 10 + 14.34)

**a = 0.56**

b = M / (m1 * m2 * m3 * d1 * d2)

I'm assuming capitalised M means the combined mass of the all of the layers on the leaves (as in the single leaf equation). So:

b = (22.05 + 22.05 + 11.8 ) / (22.05 * 22.05 * 11.8 * 0.2 * 0.2)

b = 55.9 / 229.49

**b = 0.24**

**Now the actual resonant frequency formula:**

For the first frequency:

fx = 1 / (2π) * [(3.6 * ro0 * c0 ^ 2) ^ 0.5] * [ ( a + [ (a^2 - b) ^ 0.5 ] ) ^ 0.5 ]

fx = 1 / (2π) * [(3.6 * 1.18 * 343^2) ^ 0.5] * [ ( 0.56 + [ (0.56^2 - 0.24) ^ 0.5 ] ) ^ 0.5 ]

fx = 0.16 * [499772.95 ^ 0.5] * [(0.56 + 0.27) ^ 0.5]

fx = 0.16 * 706.94 * 0.91

**fx = 102.93Hz**

For the second frequency:

fy = 1 / (2π) * [(3.6 * ro0 * c0 ^ 2) ^ 0.5] * [ ( a - [ (a^2 - b) ^ 0.5 ] ) ^ 0.5 ]

fy = 1 / (2π) * [(3.6 * 1.18 * 343^2) ^ 0.5] * [ ( 0.56 - [ (0.56^2 - 0.24) ^ 0.5 ] ) ^ 0.5 ]

fy = 0.16 * [499772.95 ^ 0.5] * [(0.56 - 0.27) ^ 0.5]

fy = 0.16 * 706.94 * 0.54

**fy = 61Hz**

These don't quite look right as the lowest frequency is almost 3 times what I calculated for the double leaf.

Thanks (I feel like I'm back at school)

Dan