Critical Distance, Far & Near Field

Direct & Diffuse Field and Modal regime

Before talking of Critical Distance we must understand some notions, for peoples that just want to know what to use, you can skip “Calculating Your Ideal Listening Distance”.

Direct Field or Near Field:

This refers to the sound that travels directly from the source to the listener without any reflections from the walls or other surfaces. The intensity of the direct field decreases with distance following the inverse square law (meaning it weakens by 6 dB for each doubling of distance).

The transition formula between Near and Far field is addressed here : Wavefront Propagation

Diffuse Field or Far Field:

A diffuse sound field is one that is homogeneous and isotropic (exhibits the same properties in all directions) throughout the volume of a room at any given time.
It is made up of an infinite number of plane waves propagating in all directions.
There are no major room-related anomalies to be observed in the measurements.
Unlike the direct field, the level of the diffuse field is constant, so there is no attenuation.
This explains the variation between direct and diffuse fields as a function of distance: the further away you are, the more the diffuse field predominates.

In this case, the reflections are very important but localized and isolated in frequency.
They coexist with absorptions close in frequency, which gives a chaotic response, typically in the low frequencies in our living rooms.
This gives rise to what is commonly called modes, typically large peaks of +/-15dB between 30 and 80Hz in our living rooms due to the dimensions of the room relative to the wave lengths involved.
This results in resonance frequencies and reverberation that are specific to the room and where any change in the position of the speaker will vary this response.

Schroeder frequency:

It’s a transition zone, which can be quite wide, below which the modal regime is dominant and above which the diffuse field is dominant.
The larger the room, the lower the Schroeder frequency, until it falls outside the audible band in very large rooms, offering only diffuse reverberated sound even in the low frequencies.
In a sense, in these large rooms, the density of modes in the low frequencies is sufficient to be diffuse, which is ideal.

Basic Formula:

Fs = 2000 * √(T/V)

Where:

Fs is the Schroeder frequency in Hertz (Hz)
T is the reverberation time of the room in seconds (s)
V is the volume of the room in cubic meters (m³)

Time Period of Integration (TPI), the brain interpretation:

The time period of integration (TPI) is the time window within which our auditory system perceives sound reflections as part of the direct sound rather than distinct echoes. This phenomenon is known as the “fusion effect.”

The TPI depends on several factors:

TPI Formulas:

The TPI can be estimated using various formulas, including:

Schroeder’s Formula: TPI = 0.6 * (c / f) * (1 + (d / r))
Blauert’s Formula: TPI = 1.5 * (c / f) * (1 + (d / r))^0.5
Houtgast’s Formula: TPI = 4.2 * (c / f) * (1 + (d / r))^0.4

Where:

c is the speed of sound (approximately 343 m/s)
f is the sound frequency in Hertz (Hz)
d is the distance between the sound source and the listener (in meters)
r is the radius of the listener’s head (approximately 0.0875 m)

Additional Factors:

The TPI is influenced by various factors beyond the distance and frequency:

Thévenot’s Fusion Curve:

This curve illustrates the relationship between the TPI and the level difference between the direct sound and its reflection.
A higher level difference allows for a longer TPI, meaning that reflections can arrive later without being perceived as echoes.

Haas Effect

The Haas effect is a psychoacoustic phenomenon related to the TPI.

If two identical sounds arrive very close together (under 35 milliseconds), our brains perceive them as a single sound coming from the direction of the first sound to arrive.

This allows sound engineers to increase sound level in the back of a room without affecting perceived sound direction by using delaying signal sent to a relay speaker.

In room accoustics in our reverberated field the early reflection on the near wall can create another “virtual” source and enter in Haas effect, it can be negative if the amplitude of this virtual source is too high.

What is Critical Distance and how to define it ?

In sound reproduction, the ideal listening distance aims to achieve a balance between the direct sound from the speakers, the direct field, and the reverberated sound reflected by the room, the diffuse field.
This balance is often described as a 50/50 ratio of direct to reverberated sound. However, this is a guideline, and preferences can vary depending on the music and your taste, 60/40 or 40/60.

Key Points:

Calculating Your Ideal Listening Distance

Experiment with different listening distances to find the sweet spot that sounds best to you.
Consider the size of your room and the type of speakers you have.
Take into account your personal preferences for a more direct or spacious sound.

Critical Distance Formula:

The critical distance, which is the theoretical distance for achieving a 50/50 balance, can be calculated using the following formula:

dc = √(αQ / 50)

Where:

dc is the critical distance in meters
α is the Sabine absorption equivalent in m²/s (explained below)
Q is the directivity factor of the speaker (a value representing how the speaker spreads sound)

Directivity factor represents the speaker’s tendency to focus sound in specific directions compared to an ideal omnidirectional source radiating sound equally in all directions.

High Q: Indicates a more directional speaker, concentrating sound energy in a narrower beam.
Low Q: Indicates a more omnidirectional speaker, radiating sound more evenly.

Understanding Q is crucial in the formula as it influences the critical distance (dc):

Higher Q (more directional): Leads to a greater critical distance. This means you can sit further away from the speaker and still maintain a good balance between direct and reflected sound (50/50 ratio).
Lower Q (less directional): Results in a shorter critical distance. You need to sit closer to the speaker to achieve the desired balance.

Remember:

Sabine Absorption Equivalent (α):

The Sabine absorption equivalent (α) represents the total sound-absorbing capacity of a room. It essentially measures how effectively a room absorbs sound energy and prevents reflections. A higher α value indicates greater sound absorption, leading to a shorter reverberation time (less echo) and a potentially quieter environment. Conversely, a lower α value means more reflections and a longer reverberation time, resulting in a noisier and potentially less comfortable space.

Calculating Sabine Absorption:

The total Sabine absorption of a room can be calculated by summing the product of the surface area of each material and its corresponding sound absorption coefficient (SAC).

Total Absorption (α) = Σ (Surface Area x Sound Absorption Coefficient)

Σ (sigma) represents the summation over all materials in the room.

Sound Absorption Coefficient (SAC):

The sound absorption coefficient (SAC) is a value between 0 (perfectly reflective) and 1 (perfectly absorbent) that specifies how well a particular material absorbs sound at different frequencies. Different materials have different SAC values, and understanding these values is crucial for calculating the total Sabine absorption of a room.

Conclusion and additional considerations:

The critical distance formula is most accurate in ideal conditions with a well-established diffuse sound field (even distribution of sound throughout the room). In real rooms, this might not always be the case.

The formula is also only strictly valid if the listening position is directly on the speaker axis.

Remember, there’s no one-size-fits-all solution. Experiment, consider the factors mentioned, and personalize your listening experience, there is no golden rule because every room acoustics is different, but we often see: