# Room Acoustics Case Study – Small Room

by Pat Brown

## This article is the first of a series that applies these principles to actual spaces. This article is Room Acoustics Case Study – Small Room.

In a previous article I showed how the room’s frequency response magnitude can be divided into zones. Each zone has its own requirements for measurement and acoustic treatment. Please review the article if you need a memory refresh on this topic. I’ll review some of the concepts here, but not all of them. This is the first of a series of articles that will apply these principles to rooms of various size.

Acoustical Size
Rooms can be classified by their physical size, but what is the “acoustical size” of a room? The answer to this question determines the effective methods for making room measurements, applying acoustical treatment, and tuning the sound system.

First, we must estimate the transition region between “small room behavior” (room modes dominate) and “large room behavior” (specular reflections dominate). There are two approaches.

The Schroeder Frequency FS
The Schroeder Frequency FS is determined using the room’s reverberation time RT. Here’s the formula:

where

RT is the reverberation time in seconds

V is the room volume in cubic feet or cubic meters

k is 2000 (SI) or 11,885 (US)

Equation 1 – The Schroeder Frequency FS

Critical Frequency FC
As the RT gets shorter, the validity of FS is reduced. In rooms with a very short RT a simpler method calculates the critical frequency FC. In principle FS ≈ FC. Both describe the transition of a room from modal to specular behavior. FC does not require knowledge of the RT. The smallest room dimension will produce the highest frequency modes and FC considers only that dimension. The formula is:

where c is the speed of sound in feet or meters

RSD is the Room’s Smallest Dimension in feet or meters

Equation 2 – The Critical Frequency FC

FS is preferred in reverberant spaces. FC is preferred in dead spaces. They should be in rough agreement so one can provide a sanity check for the other. Both metrics indicate a “transition region in wave behavior,” not a single frequency of abrupt change (Figure 1). There are room modes above FS and specular reflections below it.

Figure 1 – It is important to “think acoustically” when evaluating a room. This includes understanding that the numerical results of calculations are approximate and often indicate a “region of transition” rather than a sharp change.

In room acoustics work, measured data is always better than calculated data. A full acoustic evaluation of a spaces requires the room impulse response RIR. Let’s start with a room (Figure 2) and the Room Impulse Response RIR (Figure 3).  The dimensions are approximately 24 ft x 16 ft x 12 ft. The ceiling height is sloped with an angle change. This is not a shoebox. The RIR can be collected with a number of contemporary measurement systems. I used an open loop method to collect this one. The measurement process required about 3 minutes total, including setup.

Figure 2 –  The room wireframe showing the center channel, subwoofer, and test position. The RIR is unique for every position in this room.

Figure 3 – The room impulse response for the main listener position (courtesy ARTA™).

Listening Room
Figure 4 shows the frequency response magnitude of the RIR for a listening room. It is determined by applying the Discrete Fourier Transform DFT to the RIR. This was done in ARTA™ using the entire RIR length.

Figure 4 – The frequency response magnitude of the RIR, achieved by applying the Discrete Fourier Transform DFT to the RIR. FC and 4FC as determined by calculation are shown. Some of the room modes are marked with red dots. Note the increase in modal density with increasing frequency. Below FC the modes can be considered individually. Above 4FC the modes must be considered collectively.

Small or Large?
This room lacks a statistical reverberant field. This diminishes the validity of FS but it doesn’t prevent us from doing the calculation.

This places 4FS at about 450 Hz.

FC is a simpler approach since it doesn’t require an RT measurement. It is based solely on the room’s smallest dimension (floor-to-ceiling distance in this case) and is probably more appropriate for this space. Since the ceiling is sloped, I’ll use an average height.

This places 4FC at about 1350 Hz.

Dual-Domain Analysis
The RIR can also be presented as a Cumulative Spectral Decay CSD plot (Figure 5). This helps one visualize the decay of the modes vs time. In general, the longer the decay the more bothersome the mode.

Figure 5 – This is the top view of a waterfall plot produced by opening and processing the RIR in RoomCapture™.

This room lacks a statistical reverberant sound field due to its high absorption. We can still calculate and present the 1/1-octave band RT (Figure 6) from the RIR. In general, if the mid band RT is less than 1.6 s the reflected energy will not produce a diffuse field. The decaying sound energy in this space is not reverberation. This invalidates our arsenal of statistical room acoustics measures and methods. They simply don’t apply in small, dead rooms because they are global in scope (assume a uniform acoustical condition). Their use is popular due to their simplicity, but real-world acoustical investigations require the collection of the RIR at listener positions of interest. This allows the observation of specific acoustic events such as the rear wall reflection or a particular room mode.

In this listening space the modal dominance extends well into the speech range. Time domain metrics like the RT would only apply above about 1200 Hz. Figure 6 shows the RT (T30) above 4FC is less than 0.3 s making in a relatively “dead” space. This is a “small room” (mode dominated) for most of the audible spectrum.

Why does it matter? The mid-band RT of this space is about 0.3 s. This suggests that the room is quite dead, and that reflected energy should not be a problem. But, if a 150 ms specular reflection arrives at the listening position (well short of the 0.3 s RT) it may be devastating to the listening experience. Rooms like this require seat-specific modal analysis below 4FC and seat-specific specular reflection analysis above 4FC. The RT reveals little about the room’s behavior. Acoustic panels may be added in an effort to treat a problem, but unless placed in the reflection path may only make the problem worse.

Figure 6 – The RT can be measured for each octave band, but the room reflections are only reverberation-like above 4FC.

Playback Systems
Rooms of this size abound, but they rarely need sound reinforcement. The modal dominance in the speech range and the dense specular reflections can provide natural amplification for acoustic sources. Playback systems typically consist of hi-fi loudspeakers or studio monitors, with a “sweet spot” established for the listener (e.g. recording engineer or audiophile). Special attention is given to avoid very early reflections from near the loudspeaker that can grossly corrupt its direct sound field. These solutions include in-wall mounting and/or “near field” monitors in close proximity to the listener.

Acoustical Treatment
Appropriate acoustical treatments include absorptive panels and scattering surfaces above 4FC. These can include fiberglass, mineral wool, and fibrous boards (TectumTM). For a given thickness (e.g. 2-inches) their effectiveness will diminish with decreasing frequency.

Membrane and/or resonant absorbers are required below FC. These can include bass traps, resonators, and diaphragmatic panels. Such treatments involve movement, such as the vibration of a membrane. Active bass traps can be electronically tuned to a specific frequency range. In short, typical 1 – 2-inch-thick fiber panels will be ineffective at low frequency in this space.

A surface with irregular geometry will scatter the sound. Diffuser panels are an effective treatment above FC, scattering the reflected sound rather than absorbing it. They can maintain some “liveness” that would be lost if porous absorption were used.

Absorption Types
Historically a passive absorber consists of a soft, fuzzy material that converts the sound energy striking it into heat. Its acoustic impedance is similar to the air, so when placed on a hard wall the effect is like the sound wave keeps going rather than being reflected back into the space. This is the dominant form of acoustical treatment in the marketplace.

Traditionally, an active absorber involves movement. Rather than converting the energy into heat, the sound energy moves a membrane, converting the sound energy into mechanical energy.

A resonant absorber is tuned to a specific frequency. The sound energy excites this resonance, which stores the energy and dissipates it over time. The Helmholtz resonator serves as an example.

The terminology has evolved to include new technologies. Some resonant absorbers can be electronically tuned to a specific frequency. Since the required electronics is active, these are commonly termed “active” absorbers. Since membrane absorbers do not require electronics, they are passive. See the problem? The terms have become contradictory and confusing. Here are the terms and definitions that SynAudCon is migrating towards.

Porous Absorber – A passive absorber that converts sound energy into heat. Most effective above FC.

Membrane Absorber – A passive absorber that converts sound energy into motion. Most effective below 4FC.

Resonant Absorber – A passive absorber that uses a tuned cavity to store sound energy. Most effective below FC. The Helmholtz resonator is an example.

Active Absorber – A membrane or resonant absorber that uses active electronics. Most effective below FC. Active bass traps can be tuned for a specific room.

Since these products are expensive, it is important to deploy them properly. Thin, porous panels applied to a “boomy” podcast studio will result in a boomy podcast studio. The resonance may even be more pronounced after treatment since the environment is more acoustically dead. It’s the acoustician’s job to determine and apply the appropriate treatment for the appropriate frequency range.

Acoustical Treatment In This Space
The living room has Tectum panels on the ceiling, framed to be aesthetically pleasing. These, and the carpet serve to control room reflections above 4FC. The rear wall is completely covered with 12-inch deep QRD diffusors. These are most effective in the transition region between FC and 4FC – a vital part of the spectrum for this space. The front wall is diaphragmatic, adding some absorption below FC. The two front corners of the room have doorways that reduce the sound pressure, effectively acting as bass traps.

Figure 7 – The surface treatments in this space are specific to the listener positions.

The region around 30 Hz is devoid of sound in this space due to the modal distribution. The room geometry simply doesn’t support it. Low frequency performance in small rooms is determined primarily by the room’s geometry, sub placement, and listener position. It can be tamed with acoustical treatment.

Acoustic Modeling
Modeling software can be effectively used in this space to map the direct field at any frequency. It’s the best way to make sure that the Left, Center, and Right channels provide adequate coverage. Regarding the room’s acoustical performance, the use of ray tracing in this space is limited to 4FC and higher. This is useful for examining the specular reflections produced by the room’s geometry at high frequencies (short wavelengths), allowing the designer to make changes to the loudspeaker aiming and surface treatments while still at the drawing board stage of the project. Computer modeling is required to handle the uneven distribution of absorption in the space (Figure 8).

Room modeling software is a powerful tool, but geometric acoustics methods must be confined to the octave bands above 4FC.

Figure 8 – The room as modeled in CATT-Acoustic. The colors represent different materials, and therefore different absorption values. Modeling is the preferred room/system design method above 4FC.

Conclusion
An auto mechanic or home builder has specific tools for specific purposes. Once they identify the task or the problem, they whip out the tools that they need to solve it. In the same way, audio and acoustics practitioners have a toolkit that they pull from when designing and tuning systems. We have various ways to collect the RIR and an assortment of methods for analyzing it in both the time and frequency domains.

In future articles I will apply these same tools to other rooms.  pb