Critical Distance and Acoustics

What is Critical Distance ?

In sound reproduction, achieving a balanced listening experience depends on the interaction between the direct sound from the speakers and the reflected sound bouncing around the room (often referred to as the diffuse field).

This ideal 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 personal taste, with some favoring a 60/40 or 40/60 mix.

The critical distance is the distance from the speakers where the direct and reflected sound levels become roughly equal. It’s influenced by two key factors:

Additional Considerations:

The formula can be found in Appendix A.

Modern Speaker Design: Beyond the 120-Degree Coverage Myth

Nowadays, many loudspeakers prioritize achieving a strict 120-degree sound dispersion (directivity) across all frequencies.

However, this “one-120°-coverage-fits-all” approach ignores psychoacoustics (how we hear) and the great impact of listening distance.

In most rooms, a 90-degree horizontal constant directivity provides a more even distribution of sound energy, creating a well-balanced listening experience.

This approach recognizes the concept of critical distance. Conversely, 120 degree dispersion in a typical listening environment will excite sidewalls too many (creating too much reflected sound) and will make the sound less and less balanced and more and more aggressive the further we go away to the speakers.

Here is a guide for listening distance and type of horn with coverage (most of our horns are availables in several coverage) we can use:

help

Direct, Diffuse Field and Modal regime

Understanding the behavior of sound waves in a room is crucial for achieving a good listening experience. Here, we’ll differentiate between two key concepts and explore some additional factors influencing low-frequency response:

Direct and Diffuse Field

It’s important to note: While the diffuse field is considered relatively constant within the listening zone, the overall sound pressure level (SPL) in a room, including the contribution from the diffuse field, will still decrease with distance from the speakers following the inverse square law. This decrease is primarily caused by the direct sound attenuating with distance, while the diffuse field level remains more stable within the listening area.

In addition to the concepts above, two other factors significantly influence how sound behaves in a room, particularly at lower frequencies:

The consequence of these modes is the creation of resonance frequencies and a type of reverberation that’s specific to the room itself. Even small changes in speaker position can significantly alter this response.

Schroeder Frequency:

The Schroeder frequency is a critical transition zone, often with a wide range, that separates the modal regime from the diffuse field. Below the Schroeder frequency, the modal regime dominates, leading to the chaotic response described earlier. Above the Schroeder frequency, the diffuse field becomes more prominent, with sound reflecting more randomly throughout the room and creating a smoother response.

Room Size and Schroeder Frequency: The size of the room significantly impacts the Schroeder frequency. Larger rooms have a lower Schroeder frequency. In very large rooms, the Schroeder frequency can even fall outside the audible range of human hearing. This means that even low frequencies will exhibit a more diffuse character, which is often considered ideal for listening experiences. In these large rooms, the density of modes in the low frequencies becomes high enough to create a diffuse response.

The formula can be found in Appendix B.

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:

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.

The formula can be found in Appendix C.

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.

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.

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:

Appendix A: 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.

Appendix B: Schroeder frequency 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)

Appendix C: TPI formula

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)