Sound field properties relevant for low frequency reproduction
This will be an excerpt from a text I made 5 years ago, intended for a series of technical articles on bass and calibration.
We need this to understand the "Case: fixing the bass in a small room", where I will describe how a bass system was set up and calibrated to achieve really good bass - it is more than just frequency response.
Sound Field and Subwoofers
Measuring the response of a subwoofer loudspeaker system does not tell the full story about how it sounds.
Especially at lower frequencies the power intensity and particle velocity of the sound field are also very significant for perception, through tactile sensations.
Intro
Having a proper low frequency reproduction is something that really makes a huge difference in a sound system.
If the bass is not done right, it really does not matter how good the rest is, the total experience will be lacking.
Good bass is addictive - it has punch, impact, a sense of power and huge scale, and it does not sound boomy or resonating.
Today more and more people find that a subwoofer system with multiple units are required to get a decent frequency response and tame room modes. These systems are then set up by measuring the frequency response and analyzing, and making adjustments to subwoofer locations and equalizer settings to make it right.
When the frequency response measures reasonably flat, all decays are under control and there is sufficient output without excessive distortion, then it will sound perfect, right?
Wrong.
You also need to consider intensity and particle velocity of the sound field.
I have not seen anything about sound intensity in any of the multiple subwoofer set-up guides already written, they focus only on getting the frequency response right, and what happens to the intensity field is pure luck.
I will present a brief explanation for sound fields and intensity here.
Part 1 - Properties of the sound field explained
What is sound intensity and particle velocity - definitions, how to calculate and examples of measurements on real loudspeakers.
A full understanding of the physics behind this is not necessary to benefit from the next parts, but it might be useful to know some of the terminology.
So you can skip this part if you have no interest in the physics behind this, that is perfectly all right.
Sound and pressure and particle velocity
SPL - Sound Pressure Level - is what is normally considered the measurement unit for sound. It is defined as pressure relative to a reference level, and usually presented in logarithmic scaling, the decibel - dB. This is what is measured using a standard microphone, and what is considered to be descriptive for how the sound is perceived - or heard. Pressure is what we hear.
Sound can be observed as a disturbances traveling through air, where both pressure and particle velocity changes around an equilibrium. This particle velocity is what we are interested in learning more about here - what is it, how is it related to pressure and other properties of the sound field.
The particle velocity must not be confused with speed of sound - the traveling speed of sound waves trough air.
Sound Intensity
Sound intensity is a measure of how powerful a sound field is, defined as acoustic power per unit area:
I=P[W]/A[m2]
Which is the same as the product of sound wave particle velocity and sound pressure:
I=p*_u [W/m2]
We then realize that sound intensity requires a particle velocity potential, also that this velocity potential has direction and that there is a phase relationship between pressure and velocity potential.
It is also obvious that since total acoustic power is constant, and the area surrounding a radiating source increases with distance, then the intensity will decrease with increasing distance from the source.
Now we will look into the properties of the sound field, and how particle velocity, pressure and intensity are related.
For a spherical harmonic wave propagating in free space we can write:
Phi=(A/r)e^j(wt-kr),
where Phi is the velocity potential and k = w/c0 is the wave number.
Since the particle velocity v is:
v=dPhi/dr
Differentiating Phi we find v:
v=-A/r(..k) osv.
And p can be found to be:
p=-A/r osv.
From this we see that the particle velocity increases close to the source and for lower frequencies, relative to pressure.
In free field far away form the source the pressure - velocity relationship is fixed:
p/v lim->88 = c0d
Coming next: Active and reactive sound fields
Active and reactive sound fields
Active field - We define the active part of the sound field as the part which transfers energy from one location to another:
I = p*v cos a
Reactive field - a sound field that does not transfer any energy:
J = p*v sin a
We see that the phase relationship between particle velocity and pressure defines whether there is transfer of sound energy or not - active or reactive sound field.
In a sound field where there is no transfer of energy, such as inside a sealed lossless room, the net intensity will always be zero.
A standing wave is another example of a reactive sound field.
A dipole loudspeaker is an example of an acoustic transmitter producing a sound field with a very large reactive part.
When the transfer of acoustic power is complete and pressure and velocity are in phase, it is an active sound field.
A horn loudspeaker is an example of an acoustic transmitter producing an active sound field.
Next: Practical implications of sound field properties
How it works in more practical terms
In the near field close to a source pressure and velocity are 90 degrees out of phase.
Near field means that the distance to the source is small or comparable to the wavelength, the wavefront is spherical, so that the velocity potential kind of 'leaks' sideways.
In the far field pressure and velocity are in phase, as the wavefront now approaches a plane wave.
The fact that the particle velocity and the pressure both decrease linearly as a function of distance from the source means that the intensity will decrease with the power of 2 to the distance.
Close to the source, however, this situation changes.
Since the acoustic impedance increases for decreasing frequency close to the source, the velocity relative to pressure also increases, and the reactive intensity is larger.
Now, what does close to the source mean.
If near field is defined as d << L/2pi, we see that the term close depends on frequency, and that in normal small rooms we will always be in the near field at the lowest frequencies.
For a standing wave in a room the net intensity is zero, and pressure and velocity are 90 degrees out of phase.
The active intensity is zero, but the reactive intensity is high.
In a diffuse sound field - a room with lots of reflections - the net intensity is zero.
Reactive and active sound intensity are both zero.
In an active sound field where velocity and pressure are in phase, the amplitude of velocity relative to pressure is fixed, as the acoustic impedance is constant.
To achieve higher particle velocity amplitude it is necessary to create some kind of reactive sound field.
In all practical situations in real rooms reflections will affect both phase and amplitude of the velocity potential relative to pressure, so that intensity will be highly influenced by the room and quite different from the free field situation.
Next: Examples of sound field measurements