# Why isolation increases with lower frequency in stiffness region?

Acoustic isolation of wall depends on frequency.When the wall doesnt have porous material or some kind of absorber,we can ignore sound absorbtion and think entirelity in terms of reflection and transmission,like a impedance mismatch in RF circuit,some energy gets through,some gets reflected back.

Effective impedance of soundwave in air gets lower with lowering of frequency becose longer wavelenght has more time to apply pressure on surface and overcome inertia of its mass.This means lower frequencies should get to the other side of wall more easily,isolation ability of wall should drop.

Isolation of wall is frequency dependent,if we go from highest to lowest frequency region.There are 4 regions: coincidence region,mass law region,resonance region and stiffness controlled region.The top three regions are easy for me to understand,its the behaviour of stiffness region that I dont understand.

The whole point of this question and reason I dont understand it is becose in stiffness region,isolation actualy increases with lowering of frequency! Take a look at these graphs   This is completly unexpected from my point of view,I was thinking that the lower the frequency,the harder it is to reflect it,the lower the isolation yet this stiffness region acts exactly opposite,isolation increases with frequency lowering,this goes against everything I learned about sound isolation and reflection.

1. Why? Why does the transmission loss/isolation/reflection increases with lowering frequency?

2. Does the transmission loss keeps increasing to infinity? Does it gradualy decrease to the point its flat line or does it reverse at some super low frequency and starts falling down?

3. Does this apply to other geometrical shapes too or is this just some specific aspect of wall? Will sphere made from metal have this weird stiffness region too where isolation increases with lowering of the frequency?

1) "At low frequencies the partition tends to move as a membrane exhibiting bending motions. The more resistance there is to this bending motion, that is, the greater the panel stiffness, the higher will be the low frequency transmission loss obtained" (Noise Control in Building services, Sound Research Laboratories Ltd. Pergamon press. p107-108)

2)The smaller the frequency the bigger the wavelength (assuming speed of sound is constant) - maybe if it gets extreme enough it will seem like a constant pressure bending the partition for a time period and then gone - like a steady wind? idk. Also humans and other critters can only hear and feel notice vibration within a certain frequency range so like a low enough frequency might not count as a sound?

3)I think at this frequency you could kind of assume you're working with near constant force or pressure and just look at how your object would bend in response to those?

There's a wikipedia page called "Room Acoustics" which describes four ways in which sound behaves in a room based on frequency. These seem to correspond to the four sections of the sound transmission graph as well as with what my textbook from answer 1 says... Hope it helps • I believe all sound frequencies induce bending motion in the wall,not just low frequencies.Low frequencies should bend the wall more becose they induce pressure of one polarity for longer time,they can push against the mass inertia and stiffness of the wall longer so they should penetrate the wall more easily yet exact opposite is true. – wav scientist Apr 6 '18 at 11:01
• I'm struggling to get my thoughts in a row here - apologies: – Ayan Booyens Apr 6 '18 at 11:12
• Ok so if you look at the partition as a system where sound is converted to vibration which is then re-radiated as sound on the other side (no holes for actual sound to penetrate through). like your output can be a different phase or amplitude but otherwise is same as input. Then maybe your bode plot looks like a high-pass(?) filter where the low frequencies are <the opposite of amplified>. Like this graphs' Y axis is (input - output) whereas a bode plot is (output/input). like for a low enough frequency or long period its not a sinusoidal input anymore. I'm grasping at straws a bit here sorry – Ayan Booyens Apr 6 '18 at 11:26
• I dont understans what you mean.If wall sound transmission vs frequency graph was filter,it would be alot more complex curve than simple high/low pass.It works roughly as low pass at medium and high frequencies,with dips and peaks caused by resonances and coincidence,at low frequency it becomes high pass filter.Why it changes from low pass to high pass at low frequency is mystery to me. – wav scientist Apr 6 '18 at 11:40
• I've gotten myself confused too. sigh. I mean like wall sound transmission vs frequency is not a bode plot because its output minus input (i.e. transmission loss) to frequency. but it might look like a filter if it was the ratio of transmitted to transmitting plotted to frequency. Like I think that for all practical purposes at low frequencies the period is so long that the system doesn't react like a second order LTI or whatever but can be treated like a static bending problem. Where a stiffer wall would bend and transmit less. but stiffness might hv all kinds of damping effects in HF range – Ayan Booyens Apr 6 '18 at 12:37

At low frequency, I will stop thinking about sound waves and consider static pressure instead, as @Ayan wrote.

1. The isolation/reflection increase because the wall resist displacement. At high frequencies, the air displacement by sound is small ($$\mu$$m - mm); the wall can follow this displacement. At lower frequencies, to have a pressure wave you need large displacement of air (cm - metres); the wall is stiff, doesn't move much, so doesn't transmit the sound. A sliding wall will move and equalize the pressure, while if it is fixed, contains the pressure: you may imagine two rooms separated by a sealed wall; in one room you have $$P_1$$, in the other $$P_2$$. The difference in the two pressures is held by the wall, and represent the isolation/reflection in sound pressure.
2. With frequency going to zero, I expect zero transmission. Zero frequency means infinite displacement of the air, but the wall, under finite pressure, flex only a small defined amount, thus displacing only a small, fixed amount of air on the other side; and the lower the frequency, with finite displacement, the lower the sound level. As before, there can be some transmission if the second chamber has small volume: the wall flex and compress the air in the second chamber, representing sound transmission, even at zero frequency
3. It does not depend on the shape of the wall/container, but on its stiffness. Also, you need no holes in the wall, otherwise the pressure will escape. As example, in a sealed chamber put those three items: a rubber balloon; a rigid and sealed steel sphere; the same steel sphere, with a small hole. If you put pressure (low frequency sound) in the rubber balloon, it will inflate, with internal pressure almost the same as external. The sound is passing through. When you pump air in the steel sphere, it hold it and expand just a little: the pressure/sound is not going through. If there is an hole, the air escape until there is same pressure in and out: the sound is going through.