What is acoustic resonance


A standing wave is the vibrational state of a coherent system. As
The vibrating string of a musical instrument or the
Air column taken in an organ pipe. Although these waves have a wavelike appearance,
but are not moving anywhere; d. H. they vibrate in time but not locally - they are stationary.
The string is with a length L. between two fixed ends clamped in on which always
a vibration node is located. Between these two nodes (zero crossing) there is a
Maximum, which is also called the antinode or wave antinode. Shown here are the
Natural vibrations of an idealized string that are perfectly even and complete
to be assumed to be pliable, d. that is, they do not impose any vibration resistance on a bend
No damping by internal friction is assumed, the stiffness being zero.
The frequencies of the natural vibrations form a harmonic series of partial tones. These partials
are also called harmonics or partials. L. = Length of the string,
c = Speed ​​of sound. The 1st harmonic is the fundamental wave.

1st harm.L. = λ/2 = π        f1 = c / (2·L.)
2nd harm.L. = λ        f2 = 2·f1 = c / L.

3rd harm.L. = 3 λ / 2       f3 = 3·f1 = 3·c / (2·L.)
4th harm.L. = 2 λ        f4 = 4·f1 = 4·c / (2·L.)
Note: The distance between two nodes or two bellies
(antinodes) is
λ/ 2. The distance between a node (zero crossing)
and a belly (maximum) is λ/4.
Note, because there is often confusion:
belly at the sound pressure (pressure antinode) is to
at the same time (!)
node in the deflection of the air particles
(particle displacement node).
Mostly it is forgotten to inform the viewer of the images,
what it is: the sound pressure or the
Movement of deflection of air particles. That's a difference.


2. Standing waves (acoustic resonance) at Pipes (Whistle)

1st harm.L. = λ/ 2 = π     f1 = c / (2 · L.)

2nd harm.L. = λ                  f2 = 2 · f1 = c/L.

3rd harm.L. = 3 λ/ 2          f3 = 3 · f1 = 3 c/ (2 ·L.

Open pipe
Natural oscillations of a column of air in an open pipe (tube, flute).
At the open end there must theoretically be a Sound pressure- nodes lie,
if you ignore the final correction.
Reason: At the end of the pipe the pressure wave is infinite because of the
The cross-section of the pipe is almost completely reflected (softer
Pipe end and 180 ° phase rotation of the reflected wave).
The sound pressure is zero (pressure knot) because the impedance is at
Pipe end is zero (no resistance, but reflection).

1st harm.L. = λ / 4 = π  ·1 / 4     f1 = c/ (4 · L.)

3rd harm.L. = 3 λ/ 4                   f3 = 3 · f1 = 3 c/ (4 · L.

5th harm.L. = 5 λ/ 4                   f5 = 5 · f1 = 5 c/ (4 · L.

Tube closed on one side on the left
Natural oscillations of a column of air in a tube (flute) closed on one side;
also closed pipe. At the closed end there must theoretically be a
Maximum (Sound pressurebelly) lying.

There is misunderstanding when there are waves in the air because the signals are sinusoidal
Nodes and the bellies of the sound deflection, the particle velocity of the
Sound, sound pressure or sound density. Sound engineer with
Microphones are only interested in the behavior of the sound pressure in the air.
Note: A knot in the sound pressure corresponds to a belly of the sound deflection!

3. Room fashions (Standing waves) for reverberant parallel walls

Where do I put my speakers? Where is the best listening position? What to do with the acoustic damping?
You should know the following basics for the acoustics of sound waves at low frequencies in rooms.
This applies to the recording room as well as to the playback room = control room or the living room; please refer:
Room modes and string vibrations

The room resonances that form between the boundary surfaces of a room,
is called "standing waves" or spatial eigenmodes, also modes for short. They arise
if a multiple of half the wavelength (λ/ 2) between the boundary surfaces
of a room. A standing wave is a prerequisite for a room mode.
We sound engineers are interested in the behavior of the Sound pressurebecause through his
Effect our eardrums and the microphone diaphragms are moved; please refer:
The effect and the cause.
Room modes are nothing more than standing waves that are deep in every room
Train frequencies. Non-parallel walls are no remedy.

A standing wave as a sound pressure distribution; here:
3rd harmonic (2nd harmonic)
- 3 knots and 4 bellies

Room fashions (Natural vibrations) as Sound pressure distribution

The technical term is called a fashion, so the fashion and the plural are several fashions. So forget about mode and modes.

Calculation of the three modes (room modes) - room resonances of rectangular rooms

Note: Room modes are only practical as low sound pressure frequencies
Disturbingly audible below 300 Hz. Lose higher modal frequencies
in importance, because their disturbing effect is caused by other room acoustics
Effects masked because the modal density is so high that it is not selective
Mode clusters occur more, which can be seen as a disturbing coloration of the sound
to make noticable.

Sound pressure maxima always develop on hard walls with the modes -
these are wave bellies or sound deflection minima. - "Loose Ends".

With the following formula, standing waves, i.e. room resonances (modes),
between the reverberant boundary surfaces of a rectangular room (more precisely:
cuboid), determine their frequency.
Where is:
f = Frequency of the mode in Hz
c = Speed ​​of sound 343 m / s at 20 ° C
nx = Ordinal numbers of the natural vibrations (room length) (1, 2, 3, ...)
ny = Ordinal numbers of the natural vibrations (room width) (1, 2, 3, ...)
nz = Ordinal numbers of the natural vibrations (room height) (1, 2, 3, ...)
L, W, H = Length, width and height of the room in meters
The sound absorption coefficient is on a reverberant reflective wall (boundary surface)
α = 0
A standing wave of the 1st order with the fundamental = f1 arises when half
Wavelength of the excitation oscillation exactly in the distance of the hard limiting
areas fits. The lowest frequency for the axial mode is:
f1 = c / (2 · L.)
Where is:
c = Speed ​​of sound (343 m / s at 20 ° C)
f1 = Fundamental frequency of the axial mode in Hz
L. = maximum distance between the boundary surfaces in meters.
The higher order standing waves are calculated from the integer multiples
1st order fashion.
f2 = 2 · f1
f3 = 3 · f1
f4 = 4 · f1
Pressure maximum
Zero crossing
21λ = 2· L.f1 = 1 · c / (2 · L.)1st harmonicFundamental oscillation
32λ = L.f1 = 2 · c / (2 · L.)2nd harmonic1st harmonic
43λ = (2 ·L) /3f1 = 3 · c / (2 · L.)3rd harmonic2nd harmonic
k + 1 kλ = (2 ·L) / kf1 = k· c / (2 · L.)k. Harmonics (k - 1). Harmonic
The alternating sound pressure is actually a longitudinal wave a),
in contrast to the transversal wave b), which can be better represented here.

Standing wave as sound pressure distribution between
the walls. A wave with open ends.

1st harm.L. = λ / 2 = π        f1 = c / (2·L.)
2nd harm.L. = λ        f2 = 2·f1 = c / L.

3rd harm.L. = 3 λ / 2      f3 = 3·f1 = 3·c / (2·L.)
4th harm.L. = 2 λ        f4 = 4·f1 = 4·c / (2·L.)

Here the alternating sound pressure becomes a sinusoidal oscillation and its harmonics
between two totally sound reflective, d. H. reverberant walls shown.
It is important that in the case of resonance there is always a
Sound pressure maximum (sound pressure belly) is located.
In many representations, the representation of the string vibration is incorrect
used, which always has a fixed knot at each end, such as B. here:

These are not representations
of Sound pressurevibrations
between two walls in
a small room ...
... but these are strings
vibrations (amplitudes).
A typical one
Misrepresentation with
Knots on the walls.
The representation of the
Sound deflection
was not wanted.
A string is clamped between two "fixed ends". A wave knot appears at the ends.
A wave between two hard walls is considered to have "loose ends" in sound pressure.
Therefore, a wave maximum or a wave belly appears as sound pressure on the wall sides.

See incorrectly: Standing wave - Wikipedia and Room fashion - Wikipedia
It's right here: From Wikibooks, the open-content textbooks collection

Spatial distribution of sound pressure amount p (red) and the speed of sound v (blue) for one
standing wave as total reflection on one reverberant Wall (schematic).
At a distance of λ / 4 from the wall is the sound pressure p = 0.
Since the absorption effect in a material increases proportionally to the speed of sound,
the absorber should be effective in the fast maximum d/ 4 as the distance to the wall
are located or should have an appropriate thickness. Much easier than that
Fast maximum is the sound pressure minimum measurable, which is exactly at the point of the
Fast maximum is at λ / 4.
Here is a nice animation of a standing wave - explanation through
Superposition of the incoming wave with the reflected wave. To do this must
one adjust: "reflection on loose end".

The red wave is the incident sound wave, the green one is the reflected back
Sound wave. The blue is the standing wave that moves through the overlay
of both waves. A standing wave is required for a room mode,
however, standing waves are generally not space modes. These two
Terms are usually not presented correctly as synonymous.

The oscillations of sound pressure between parallel
hard walls - i.e. standing waves or room modes -
look different than the vibrations of strings.
While the oscillation in the first case has "loose ends"
(Wave bellies), it has "fixed" in the case of the string
Ends "(wave nodes).

We find it on reflective walls (interfaces)
always the maximum sound pressure, i.e. a wave belly.
That is also the principle of
Boundary microphones.

Two different types of boundary microphones

Lower cut-off frequency for the boundary microphone

Standing waves even with open and closed organ pipes

Comparison of the reflections in the reverberation room
and an extremely dampened room

Sound pressure reflections (modes) reverberant Walls like in a reverberation room.

Sound pressure reflections (modes) sound softening Walls, like in an anechoic room.

Spatial distribution of sound pressure p (red) and the speed of sound v (blue) in one
standing wave with total reflection at one sound softening Wall (schematic).
At a distance of λ / 4 from the wall is the sound pressure p = max.
A typical room (control room) with axial modes and sound pressure bulges on the walls
What are room fashions? - Portrayed correctly - Trikustik®
The picture on the left shows the vibrations of a string with knots on the outside.
The picture on the right shows the actual sound pressure distribution on the walls.   
This is only defined for 2D, which is usually sufficient. Under certain
Assumptions and roundings, however, similar statements can also be made for 3D.

Calculation of the three modes (room modes) - room resonances of rooms

Calculator for the spatial eigenmodes - Trikustik®
For rectangular rooms, this calculator determines the room eigenmodes with the
20 lowest natural frequencies and displays them in ascending order.

The hyperbaric effect

Low-frequency tones are also to be reproduced below the lowest possible room mode.
One uses the Pressure chamber effect for small with reverberant boundary surfaces
equipped rooms, whereby the room must be sealed pressure-tight.
See: Basics of room acoustics. The "dB Drag Racing" fans need this for theirs
Volume competitions in the car.
Absorption - reflection - diffusivity - combination
Attention: There is one on a hard reflection wall (closed end)
Sound pressure antinode but one at the same time
Displacement node of the air particles.
When speaking of standing waves, one must express it clearly
that we have amplitudes of the sound pressure changes
think that also move our eardrums. Most of the time it will
to forget. The sound pressure in air oscillates with the belly and
Knot - at the same time the deflection movement of the air particles oscillates
with knots and belly.
The relationship between the maximum sound pressure Δpm
and the maximum deflection amplitude of the air particles ξm is:
Δpm = (v ρ ω) ξm.
Amplitudes of the sound field sizes of a plane wave
Differentiate the amplitudes of the sound deflection of the air particles,
Sound pressure, sound velocity and pressure gradient.
Sound pressurebelly (sound pressure antinode) is deflectionnode
the movement of air particles (particle displacement node).
Deflectionbelly the movement of air particles is sound pressurenode.
In the state of rest a green air particle has the same position as in
vibrating state. The sound deflection of the air particles is located
is in the zero crossing during the sound pressure amplitude p
thereby a maximum (overpressure, compression) or a minimum
(Negative pressure, dilution) shows. When idle is a red one
Air particles at the maximum or minimum of the amplitude of the
Sound deflection and at the same time this is a zero crossing of the
Sound pressure oscillation. Thus the sound pressure is zero there;
d. H. the normal static air pressure is available alone.
Avoid the word "rash" jokingly with "sonic acne"
referred to as. Deflection of the air particles sounds better.
In pictorial representations of standing waves (space modes) between
two walls it is seldom stated whether the sound pressure as a pressure fluctuation in the air
or the deflection of the air particles should be shown.
However, this clarification is absolutely necessary.
The above illustration shows the important difference.
Left = sound pressure (abdomen) corresponds to right = deflection (knot).
To think about: The effect of standing waves reports to us
Hearing, and the sound pressure effectively moves our eardrums.