# Membrane absorber

Membrane absorbers (also known as plate resonators , colloquially also known as plate oscillators ) are constructions based on the spring-mass principle that serve to absorb sound .

## construction

Membrane absorber (sectional view)

The basic structure consists of a membrane ( plate ) that serves as a mass (for example made of wood, metal, glass, plaster, etc.), and an air cushion behind it, which acts like a suspension .

With careful selection of the materials, the vibration of the membrane is dampened by its own weight (or inertia ) in such a way that sound absorption takes place.

The sound absorption by natural resonance described here must be differentiated from the absorption by elastic materials (wool fibers, rubber , etc.) or by dissipation in slowly deformable sound media (sand, water, etc.). Both modes of absorption are often combined to form plate vibrators that are supported by an elastic filling.

## calculation

The sound absorption of the membrane absorber is most effective in the frequency range of its natural resonance. This results approximately from the following numerical equation :

${\ displaystyle f \ approx {\ frac {600} {\ sqrt {d '\ cdot {\ tfrac {m} {A}}}}}}$

With

• Frequency of the natural resonance in Hz${\ displaystyle f}$
• Thickness of the air cushion in centimeters${\ displaystyle d '}$
• mass per unit area of the membrane in .${\ displaystyle {\ frac {m} {A}}}$${\ displaystyle \ mathrm {\ frac {kg} {m ^ {2}}}}$

A typical example of a plate absorber is a double or multi-glazed soundproof window construction : Two window panes of different thickness (i.e. different natural resonance) mutually hinder the development of vibrations in a certain calculated frequency range (usually the low frequencies of road noise ).

### Derivation

The above Numerical value equation can be derived from the following physical size equation (see Helmholtz resonator ):

{\ displaystyle {\ begin {aligned} f & = {\ frac {\ omega} {2 \ pi}} \\ & = {\ frac {1} {2 \ pi}} {\ sqrt {\ frac {k} { m}}} \\ & = {\ frac {1} {2 \ pi}} {\ sqrt {{\ frac {\ rho \ cdot A ^ {2} \ cdot c ^ {2}} {V}} { \ frac {1} {m}}}} \\ & = {\ frac {c} {2 \ pi}} {\ sqrt {{\ frac {\ rho \ cdot A ^ {2}} {d \ cdot A }} {\ frac {1} {{\ tfrac {m} {A}} \ cdot A}}}} \\ & = {\ frac {c \ cdot {\ sqrt {\ rho}}} {2 \ pi }} {\ frac {1} {\ sqrt {d \ cdot {\ tfrac {m} {A}}}}} \ end {aligned}}}

With

• Frequency of natural resonance${\ displaystyle f}$
• Radial frequency of natural resonance${\ displaystyle \ omega}$
• Mass of the membrane ${\ displaystyle m}$
• mass per unit area of the membrane${\ displaystyle {\ frac {m} {A}}}$
• Spring constant of the air cushion ${\ displaystyle k}$
• Area of the air cushion and the (vibrating) membrane${\ displaystyle A}$
• Volume of the air cushion${\ displaystyle V}$
• Thickness of the air cushion in meters${\ displaystyle d}$
• Airtightness ${\ displaystyle \ rho \ approx 1 {,} 25 \, \ mathrm {\ frac {kg} {m ^ {3}}}}$
• Speed ​​of sound .${\ displaystyle c \ approx 330 \ mathrm {\ frac {m} {s}}}$

Inserting the material constants (density and speed of sound of the air) and evaluating the pre-factor finally provides the numerical value equation (with ). ${\ displaystyle {\ frac {c \ cdot {\ sqrt {\ rho}}} {2 \ pi}}}$${\ displaystyle d '= 100 \ cdot d}$

In addition to the natural resonance frequency, other important factors for sound-absorbing structures are:

• Bandwidth : The range in which frequencies above / below the natural resonance are absorbed (equivalent to filters also called quality factor or Q factor). Basically, the lighter the membrane and the larger the volume of the air cushion, the greater the bandwidth. (Example: A 5 mm thick hardboard in front of a 3 cm thick air cushion results in the same natural resonance frequency as a 3 mm thick hardboard in front of a 5 cm thick air cushion. The difference between the two construction methods lies in the bandwidth: The construction with the lighter membrane / plate is Experience has shown that a bandwidth of over 1.5  octaves is to be expected, whereas with the construction with the heavier membrane, a bandwidth of just under an octave.) The bandwidth is difficult to calculate, as precise knowledge of the different physical properties of the materials used is necessary which can only be determined by complex (costly) measurements.
• Equivalent absorption area : In the case of membrane absorbers, this is to be equated with the physical (actual) surface. B. in Helmholtz resonators depends exclusively on the natural resonance frequency, regardless of the actual size / surface of the resonator.

## application areas

Membrane absorbers are mainly used in room acoustics to shape the reverberation of a room in appropriate frequency ranges; in other words, to specifically absorb those frequencies that correspond to the natural resonance frequencies of the membrane absorber concerned. Membrane absorbers are particularly suitable for absorbing low-frequency sound components (20–400 Hz). They are used in particular in recording studios , auditoriums , concert halls and other rooms in which high quality acoustic performances are desired.