Mass Law, Resonance and coincidence

The Mass Law of sound insulation provides a guide by which the level of transmission loss (TL) of a structure (how much sound energy it inhibits) can be predicted. It tells us that, in a single leaf wall or partition, TL increases by 6dB with every doubling of mass at a given frequency (this is a theoretical figure – in the real world it’s nearer to 5dB) and with every doubling of sound wave frequency, TL also increases by 6dB. This illustrates two important facts that we need to bear in mind when designing a studio. Firstly, low frequencies are harder to control than higher frequencies and if you intend to control low end frequencies with a single leaf construction, it will need to be dense and very, very heavy. That also means it will be extremely expensive to build and, given that the ceiling needs to be the same density as the walls, incredibly difficult to construct.

Fortunately, there’s a way to dramatically improve TL without hugely increasing cost, time spent and effort. This requires building a ‘room within a room’ structure, sometimes referred to as a MAM (mass air mass) or MSM (mass spring mass) system. It’s important to know that in order for a MSM partition to work effectively, the two leaves must be completely separate from each other structurally i.e. they cannot connect at any point. If they do, this will form a ‘flanking path’ which will turn the structure (at least partially) into a resonator rather than an isolator. Look at the cross-sectional wall diagram below. This represents a top down view of wooden framed, stud walls of varying construction, clad with plasterboard and shows the rather contra-intuitive way in which MSM isolation works. The STC (Sound Transmission Class) measurement is used in the building industry to quantify isolation in houses and offices etc. It’s not very useful in music applications because it only considers a limited range of frequencies but is illustrative in this instance.

The addition of insulation in the example second from left (rock wool or fibre glass flock) in the cavity of a basic wall (far left) improves the sound isolation by providing ‘damping’ of the plasterboard cladding (i.e. it reduces vibration). Notice how doubling the mass of the wall in the same configuration (third from the left) produces only a modest improvement (STC 40) and removing one inner layer of plasterboard increases this to STC 50. Better still, removing both inner layers increases this to STC 57 (second from right) and doubling up the plasterboard on the outside faces of the wall (far right) gives a rating of STC 63. The materials used are identical to those in the STC 40 wall but by arranging them differently in the STC 63 wall, insulation is massively improved at exactly the same cost. The take away from all this is that if you want a reasonable level of sound insulation, build an air-tight, two leaf MSM system of appropriate mass with loose cavity insulation for the best, most cost effective result.

So far so good! But there other factors to consider (this is often the case in acoustics). Strictly speaking, mass law applies directly to non-rigid partitions or ‘limp mass’ (insert your own single entendre here). Building materials invariably have an inherent rigidity, so mass law is only really an indicator of possible T/L. The actual sound insulation of a given structure depends on the interaction of the mass involved, the stiffness of the materials and how effectively they are damped. As well as this, mass law is subject to resonance at lower frequencies (we already know that low frequencies are harder to contain) and losses at higher frequencies due to the coincidence effect or coincidence ‘dip’.

Every panel you build will have a natural range of of frequencies at which they will vibrate more easily than others. These are called resonant frequencies and consist of a fundamental frequency and multiples of this known as harmonics. The critical frequency of a wall is that at which the bending waves within the wall match the frequency of a sound wave impacting on it. When this coincidence happens it enables more efficient transfer of sound wave energy across the partition. The effect of resonance and coincidence dip can’t be eliminated but you can engineer your design to have the lowest possible fundamental frequency and highest possible critical frequencies. The algebraic formulae for calculating these effects on transmission loss are listed on the table below this paragraph if you want to do your own calculations. The link below that is for an online transmission loss calculator. It was built by ‘Gregwor’ who is a moderator of The John Sayer’s Recording Design Forum and I recommend that you subscribe to it if you have any interest in building your own studio. The moderators and senior members of the forum have a great deal of sage advice on this and other related topics and I’m very grateful to Greg for his permission to use the link.

TL = 14.5*LOG(M*0.205)+23dB
where M = surface density of leaf
F0 = C((m1+m2)/(m1*m2*d))^0.5
where C = constant for empty or insulated cavity
where m1 and m2 are mass of each leaf (kg/m^2)
where d = cavity depth in metres
F<F0 = 20LOG(f*(m1+m2))-47
F0<F<F1 = R1+R2+20LOG(f*d)-29
F>F1 = R1+R2+6
where f = frequency you want to check TL of
where m1 and m2 are mass of each leaf (kg/m^2)
where d = depth of cavity
where F1 = 55/depth of cavity
MASS LAW = 20LOG(f*(m1+m2))-47.2

© John Steel 2020

Published by johnmsteel

Musician, editor and now studio builder.

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