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Old 04-30-2008, 12:01 AM   #1
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Default A Guide to Subwoofers (Part II): Standing Waves & Room Modes

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THE SCIENCE BEHIND SUBWOOFER PLACEMENT

Prepared By: Big Daddy

For Part I of A Guide to Subwoofers, Click Here

BASIC FACTS ABOUT SOUND WAVES

Sound is a sine wave that travels through matter such as air, water, and steel. Sound does not travel in vacuum. A sound wave will spread out after it leaves its source, decreasing its amplitude or loudness. The relationship between speed, wavelength, and frequency is: Speed = Wavelength x Frequency.

The speed of sound in dry air is approximately 344 meters/second, 1127 feet/second, or 770 miles per hour, or one mile in 5 seconds at room temperature of 20C (68F) and sea level. Sound travels faster in liquids and non-porous solids than it does in air, traveling about 4.4 times faster in water than in air. The speed of sound in air varies with the temperature and humidity such that sound travels slower on cold days, but is nearly independent of pressure. The speed of sound is dependent on air (or any other gas) and is not dependent on the amplitude, frequency, or wavelength.

Longitudinal and Transverse Waves: Sound waves are an example of longitudinal motion and water waves are an example of a combination of both longitudinal and transverse motions. Here is a demonstration of longitudinal and transverse waves.


Longitudinal Waves


Transverse Waves


Water Waves
Animation courtesy of Dr. Dan Russell, Kettering University

Natural Frequency
A sound wave is created as a result of a vibrating object. Nearly all objects, when hit, struck, or somehow disturbed, will vibrate. When each of these objects vibrate, they tend to vibrate at a particular frequency or a set of frequencies. The frequency or frequencies at which an object tends to vibrate with when hit, struck, or disturbed is known as the natural frequency of the object.

The natural frequency of an object depends on the properties of the material the object is made of (this affects the speed of the wave) and the length of the object (this affects the wavelength). Since frequency = speed / wavelength, a change in either speed or wavelength will result in a change of the natural frequency.

Resonance
Assume that two tuning forks are mounted on a box and assume the forks have the same natural frequency (e.g., 256Hz). Suppose the first tuning fork is struck with a rubber mallet and it begins vibrating at its natural frequency of 256Hz. These vibrations set its sound box and the air inside the sound box vibrating at the same natural frequency of 256Hz. Surrounding air particles are set into vibrational motion at the same natural frequency of 256Hz. This sets the second fork into vibration with the same natural frequency. This is an example of resonance - when one object vibrating at its natural frequency forces the second object with the same natural frequency into vibrational motion.


Formation of Standing Waves
Sound reflects back and forth between two parallel surfaces. At certain frequencies the incident and the reflected sounds interfere to form “standing waves” in which those frequencies can be amplified, and what we hear at those frequencies depends on where we and the speakers are located. Standing wave are non-traveling vibrations (the sound waves themselves are not stationary and are continuously bouncing back and forth).

Let us consider the effect of a sound wave in a small room when it is reflected from the walls. A wave can be thought of as an upward displaced pulse (peak) followed by a downward displaced pulse (trough).The waves from the source and the reflected waves will cause destructive interference (positive cancels negative) in such a manner that there are points of no displacement (standing still). Such point are called Nodes (N). There will also be positions along the medium resulting from constructive interference that will vibrate back and forth between a maximum upward displacement to a maximum downward displacement. These positions are called Anti-nodes (AN). You can examine the following standing wave animations:


Here is an animation showing the standing wave patterns that are produced on a medium such as a string on a musical instrument. This type of medium would be said to be fixed at both ends. Standing Waves For A Musical Instrument.

Fundamental Frequency and Harmonics
Resonance is a common cause of sound production in musical instruments. As was mentioned before, when an object is forced into resonance vibrations at one of its natural frequencies, it vibrates in a manner such that a standing wave pattern is formed within the object. These patterns are only created within the object or instrument at specific frequencies of vibration; these frequencies are known as harmonic frequencies. At any other frequency, the resulting disturbance of the medium is irregular and non-periodic (non-repeating). These standing wave patterns represent the lowest energy vibrational modes of the object. While there are countless ways by which an object can vibrate (each associated with a specific frequency), objects favor only a few modes or patterns of vibrating. The favored modes (patterns) of vibration are those which result in the highest amplitude vibrations with the least input of energy. Objects are most easily forced into resonance vibrations when disturbed at frequencies associated with these natural frequencies.

Consider a guitar string vibrating at its natural frequency or harmonic frequency. Because the ends of the string are attached and fixed in place, the ends of the string are unable to move. These ends become points of no displacement or nodes. In between these two nodes there must be at least one point of maximum displacement or anti-node. The most fundamental harmonic for a guitar string is the harmonic associated with a standing wave having only one anti-node positioned between the two nodes on the end of the string. This harmonic will have the lowest frequency and the longest wavelength. This is called as the fundamental frequency or the first harmonic of the instrument. Similarly, the instrument can produce harmonics at higher frequencies and are called second (two times the frequency of the first), third (three times the frequency of the first), and so forth. The following is the simulation of the four modes of a vibrating string.


The frequency associated with each harmonic depends on the speed and the wavelength of the waves. The speed is affected by the properties of the medium (tension of the string, thickness of the string, material composition of the string, etc.). The wavelength of the harmonic is dependent upon the length of the string. Variations in either factor will result in variations in the frequency at which the string will vibrate. When two sound waves that are one octave apart (i.e., their frequencies have a ratio of 2:1) are combined, they will sound pleasant.

SUBWOOFERS AND THE EFFECT OF STANDING WAVES

Low frequency wavelengths are much longer (e.g., 56.5ft at 20Hz, 22.6ft at 50Hz, and 11.3ft at 100Hz) than higher frequency wavelengths (e.g., 3.8ft at 300Hz, 1.1ft at 1,000Hz, and 1 inch at 13,000Hz). This is important, especially below 150hz or so. Above 150hz, the waves are small enough that they are not affected by the room size as much. They bounce around every which way. Standing waves only become a significant problem at lower frequencies (below 100 Hz) because we normally set the crossover frequency around 85Hz. Long wavelength bass frequencies travel back and forth bouncing off the walls.

In general, at most frequencies, the decay of sound waves is rapid, but when a sound’s wavelength is precisely twice the size of a room dimension (e.g., length), the waves from both directions reinforce each other at the wall boundaries and cancel each other in the midpoint of these two boundaries, creating a resonant condition. Like most other resonant conditions, standing waves produce a fundamental tone (the lowest-frequency resonance the space will support) and a series of harmonics. If the fundamental frequency is 25Hz, there will be other, progressively weaker ones at 50Hz, 75Hz, 100Hz, 125Hz and so on. Each of these harmonics causes a high energy peak points in the room, with a null (low energy) midway between each adjacent pair of peak points.

Standing waves in a room are called room modes or room resonance modes. The crests (high points) of the standing waves and the troughs (low points) between them represent what happens when a single subwoofer generates the long wavelengths of bass. Those peaks and dips in bass energy do not change unless you change the dimensions (length, width, and height) of the room and the frequency of the bass tones. Even if you did alter these, you would be left with a whole new set of standing waves to deal with.

Type of Room Modes
The sound waves interact with the room boundaries (walls, floor, and ceiling) and create standing waves or room modes. The standing waves are different between floor and ceiling, side walls, and end walls, unless any of these dimensions are the same (the worst kind of room is a perfect cube). There are three basic types of modes: axial, tangential, and oblique. Examples of these modes are shown in the following diagrams:


It is important to remember that these diagrams are over simplification. Remember sound waves will zig-zag around the room, and that sound sources are not directional like flashlights. A speaker is more like a bare lightbulb, or a light bulb in a box. Because of this zigzag, room modes are actually a range of frequencies centered around the number given in our calculations. In addition, tangential and oblique room modes are the most difficult to visualize.

To gain some understanding of the room modes and standing waves, it will be very helpful to consider a one-dimensional acoustic space like a long narrow pipe. If both ends of the pipe are closed, then it becomes similar to a one-dimensional room. Unless otherwise stated, all diagrams are created by Big Daddy.


Now position a subwoofer at one end of the pipe and connect it to a frequency generator. At the other end of the long pipe, put an SPL meter to measure the sound pressure. Start by feeding very low frequency signals to the subwoofer, you will notice no reading on the SPL meter. However, as you increase the frequency of the sound waves fed to the subwoofer, you will reach a point where the reading on the SPL meter jumps to a high point. This is the first mode and is called the fundamental resonant frequency or the first harmonic frequency of the one-dimensional room (pipe).

Continue raising the frequency of the signals and the meter drops back to normal for a while, but finally peaks again. This next frequency is evidence of the second resonance mode and is called the second harmonic frequency. The frequency of this second resonance will be exactly twice that of the first resonance. If we increase the frequency of the signal some more, we will find the third resonance mode which will have exactly three times the frequency of the first fundamental resonance mode. This harmonic series can continue as we increase the frequency.

If we move the location of the speaker or the SPL meter, we will get a new set of harmonic frequencies. However, if the subwoofer is moved to the middle of the pipe (low pressure zone), the odd numbered resonances will not be stimulated and will disappear. If we move the subwoofer to a position one third from either end of the pipe, only the third, sixth, ninth, and so on harmonics can be stimulated. If we move the speaker to the one quarter point of the pipe from either end, we will find only the fourth, eighth, twelfth, and so on harmonics.

In a closed pipe, which has been stimulated into its first resonance condition, we will find that the sound is very loud at either end of the pipe and very quiet at the halfway point, the middle. These loud areas are called sound “pressure zones”. If the subwoofer is placed in either of these pressure zones, it can pump up the resonant condition. However, if it is not placed in a pressure zone, it cannot pump up the resonant mode.

The second harmonic of a closed pipe has three pressure zones, one at either end and one in the middle. If we place the subwoofer in any three of these pressure zones, we will stimulate the second harmonic. However, if we place the subwoofer in the middle pressure zone, we cannot stimulate the first resonance but we can still stimulate the second one. Let us now plot the sound pressure as a function of distance, and remember that one wave moves from left to right and the other moves from right to left and polarity changes each time we cross a null.


Important Facts About Subwoofers, Listeners, and Standing Waves
  • Subwoofers are sound pressure generators. They will reinforce the room modes when they are located in high pressure regions of the standing waves.
  • Ears respond to sound pressure. When our heads are located in the high pressure regions of the standing waves, the room modes will be most audible.
  • If the subwoofer is placed in the null areas, the corresponding modes will disappear.
  • If you move your head to the null areas, you will not hear a lot of deep bass.
  • Room dimensions and subwoofer location create room modes.
There is another factor that limits the remaining options for speaker placement. The pressure zones are spread out and not pinpoint-sized. For all practical purposes, the subwoofer should be located at least 25 percent away from the end of the pipe to best avoid stimulating any of its first three harmonics. There is no location towards the middle of the pipe that suits a subwoofer position, as the pressure zones there are overlapping.

Calculating the Resonance Modes of a Home Theater Room and Subwoofer Placement
Axial Modes are the strongest and the most important, and the easiest to compute. Tangential Modes are about half as loud, and Oblique Modes are about a quarter as loud. They tend to be the least important, but if an oblique room mode occurs near another mode, that frequency may still be a problem. It is best to calculate all room modes to see where any overlap may be.

A room can be approximated by three intersecting pipes. These pipes would lie along the three room axes: front to back, side to side, and floor to ceiling. For most rectangular home theater rooms, it may be sufficient to calculate only the axial modes of the room.

Since a room can enforce a wave twice as long as it is, the first fundamental frequency can be calculated by using the formula: Standing Wave Frequency = Speed of Sound / 2*Distance Between Boundaries. If we multiply this frequency by 2, we will get the second harmonic frequency and so on. Usually it is necessary only to look at the first three or four modes because the crossover frequency for most home theater rooms are set around 80Hz-100Hz. Let us now calculate the axial modes for a 15ft W x 20ft L x 8ft H room.

Width
The first resonance frequency: 1130ftps / 2x15ft = 37.7Hz.
The second resonance frequency: 37.7 x 2 = 75.4HZ.
The third resonance frequency: 37.7 x 3 = 113.1HZ, ignore, because it is above the roll-off frequency of 85Hz.
The subwoofer has to be placed at least 25 percent away from the wall (15x0.25=3.75ft) because of the first harmonic, but that is the point of minimum of the second harmonic. Therefore, the subwoofer can be placed anywhere between 3.75ft (minimum of the second harmonic) and 7.5ft (minimum of the first harmonic) away from either wall.

Length
The first resonance frequency: 1130ftps / 2x20ft = 28.3Hz.
The second resonance frequency: 28.3 x 2 = 56.6HZ.
The third resonance frequency: 28.3 x 3 = 84.9HZ.
Since all three harmonics are below the roll-off frequency of 85Hz, we should place the subwoofer in a position that avoids the maximum and minimum of the three waves – at least 25% (20 x0.25=5ft) from either end walls.

Height
The first resonance frequency: 1130ftps / 2x8ft = 70.6Hz.
The second resonance frequency: 70.6 x 2 = 141.2HZ, ignore, because it is above the roll-off frequency of 85Hz.
The third resonance frequency: 70.6 x 3 = 211.8HZ, ignore, because it is above the roll-off frequency of 85Hz.
The vertical position for a subwoofer is anywhere in the middle half of the room, keeping it at least 25% (two) feet away from either the floor or ceiling.

So, a 15ft W x 20ft L x 8ft H room will have the smoothest bass if the subwoofer is located 2ft from the floor or 2ft from the ceiling (6ft from the floor), between 3.75ft and 7.5ft from the side walls, and five feet from the end walls. This is done to avoid the coupling of the subwoofer to room modes.

Unfortunately, most people do not have an unlimited options with regard to the placement of a subwoofer in their living room, so the benefit from one of these calculations is limited. However, if one is building a dedicated home theater room, then one should pay more attention to these calculations.

Dr. Floyd Toole, formerly of National Research Council of Canada and currently a Vice President and researcher at Harmon International has developed a simple Excel Program to calculate axial room modal frequencies.

Here are other calculators: Calculator1, Calculator2, or Calculator3.

Also, see animations at the beginning of Post #2.

Location, Location, Location
The location of your subwoofer in the room creates the standing wave modes. And the modes are what determine whether your listening position gets great bass or poor bass. If your chair or sofa happens to be located in one of the troughs of the standing waves, you are not going to hear much deep bass. But if you get up and walk a few feet back, or to the left, or to the right, chances are you will hit one of the peaks and the bass will be very strong, perhaps too much of a good thing. An equalizer will solve some problems, primarily those related to peaks. A null is an entirely different situation and no amount of boost can fill a room-induced null. Think of it as a water drain. No amount of water can fill a drain.

The following analysis is based on the work of Dr. Floyd Toole. All diagrams are created by Big Daddy.

One Subwoofer
Let us consider the width modes. One subwoofer close to a wall is in the high pressure region of all the width modes and energizes all of them.


What happens if the subwoofer is moved to the location of the first pressure minimum (green minimum)? That particular mode is not energized and will disappear. What then happens if it is moved to the next null (magenta minimum)? That mode will disappears, but the other one returns. Subwoofer location determines which of the room resonances is activated, and which ones are not activated.

Optimum Position for One Subwoofer: If the subwoofer is placed in the wrong position in the room, we hear “room booms” instead of music. Bad speaker positions are those that allow the speaker to stimulate room resonance (modes). The best position for one subwoofer would be in the anti-mode region of all the room resonances.


Position of the Listener
Similarly, the location of the listener determines which modes will be heard. Just like a subwoofer against a wall energizes all room modes, a listener with his/her head against either wall hears all of the modes. There will be probably be too much bass and very tiring.

If the listener is moved forward to a null position, no sound will be heard from that mode. Different positions mean that different frequencies will be heard with very different loudness. This is true for subwoofers as well as listeners.

Other Techniques for Placement of One Subwoofer: Subwoofer Crawl Placement
This is an effective technique for the placement of a subwoofer. Play a CD or a DVD that has lots of low bass and move your subwoofer to the normal listening position. Go to the spot where you would like to place the subwoofer. Now sit down in that place and listen to the recording. If it sounds reasonably good in that position, go ahead and put your subwoofer there. If the bass does not sound good, try other available locations to find the location that provides the best sounding bass. Remember that moving the subwoofer as little as 6"-12" can have a noticeable impact on its performance. The following video by Axiom Audio demonstrates the subwoofer crawl technique.

Using an SPL Meter and Test Tones: It is more accurate to use an SPL meter and low frequency test tones. You can buy a digital or an analog SPL meter from Radio Shack for less that $50. If you do not know how to use an SPL meter, read the thread on Calibrating Your Audio with an SPL Meter, Sticky Under Receiver Discussion.

You can download free test tone generators from the following sites:
RealTraps - Test Tone CD
Test Tone Generator Free Download
Tone Generator Software - Create Audio Test Tones, Sweeps or Noise Waveforms

Similar to the listening subwoofer crawl technique mentioned above, start with the subwoofer located in the listening position and set the meter where you want to put the subwoofer. Set the SPL meter to 75dB setting and play the test tones from 20Hz to 120Hz in increments of 1/6 octave and write down the SPL levels. Move the subwoofer and repeat the experiment. Use an Excel worksheet and plot the SPL readings versus frequencies for different locations. Use these measurements to find the best possible placement. Of course this means having the flattest response.


Problems Associated with One Subwoofer and Standing Waves
With some care in placement of a single subwoofer and the listening location, one listener can experience fairly smooth and deep bass in a rectangular room. Unfortunately, other listeners seated elsewhere in the same room will hear different bass response, which may be significantly irregular. Trying to reduce some of the largest peaks (too much bass) at one or two frequencies is possible with careful placement and equalization for one location and one listener. But attempting to apply equalization for multiple locations is usually ineffective. There are far too many problems in a small home theater room that cannot be solved with one subwoofer. Using two subwoofers is preferable as you will get a better bass performance and will have less of a problem with standing waves, since the bass will originate from two locations.

MULTIPLE SUBWOOFERS

Considerable research has been done by Dr. Floyd Toole, Todd Welti, Sean Olive, Allan Devantier and others into the behavior of deep bass in different rooms. The following suggestions are based on their work.

Two Subwoofers Against the Opposite Walls Across the Width of the Room
Two subwoofer against opposite walls will cancel the odd-numbered modes, leaving only one active mode. In the diagram below, blue and green modes will be eliminated, leaving behind only the magenta mode.


Remember that the sound pressures on opposite sides of a null in a standing wave have opposite polarity. If one side is decreasing , the other will be increasing. If we use one subwoofer in a room, this does not matter. A single subwoofer against a wall will energize all the room modes. However, if we place another subwoofer with the same polarity against the opposite wall, the first and third modes will have opposite polarities at the subwoofer locations and the subwoofers will behave in a destructive manner, cancelling the odd-numbered modes. This will leave only the second (magenta) mode across the width of the room.

Two Subwoofer Move to the Null Position of the Second (Magenta) Mode
If we move the subwoofers to the null locations for the second mode, they will still be in opposite polarity regions for the odd-numbered (blue and green) modes, and as a result width modes are significantly reduced, if not eliminated. What this means is that everyone across the width of the room will hear the same bass sound.


What we have accomplished is the elimination of room modes by selecting different locations for the subwoofers and the listeners. We can do this for length, height and other mode types. The distribution of the room modes will look a lot less cluttered. We may still need to equalize in the sense of changing the frequency response of the system for everyone in the room to hear smoother and more uniform bass.

Watch this video on the benefits of using Multiple Subwoofers by Axiom Audio.

Multiple Subwoofer Placement (Rules of Thumb)
In most circumstances two subwoofers will perform better than one. While you might assume this is for added SPL, the greatest benefit will actually be smoother bass response. Two subwoofers are easier to place and result in a flatter frequency response and creation of a much larger “sweet spot” for everyone in the room to hear smoother and more consistent bass.

For maximum output, some experts suggest that you put a single subwoofer in a corner for maximum output and place a second one in a less reflective area to smooth out the response. You can use the “crawl around the room” technique as described above for determining the location of the second subwoofer, except in this case, look for the minimum amount of bass output.

Dr. Toole suggests that in a rectangular room you should put one subwoofer close to the front wall in the middle, and another subwoofer at the back of the room in the same relative position. THX recommends placing them in the middle of the left and right walls. Dr. Toole also recommends some equalization to flatten the bass response so that all the seats in the primary listening area hear solid and even bass.

For Better Results, Use Four Subwoofers
Four subwoofers were found to be most effective when two subwoofers are placed at the middle location of the front and back and two subwoofers at the middle location of each sidewall, opposite each other. Placing one subwoofer in each of the room’s four corners was also found to be similarly effective.








The 25% Subwoofer Positioning: This solution is suggested by Todd Welti at Harmon International: “You shrink the whole room by 25% and put the subwoofers at the corners of that virtual room. Of course you get incredible performance, but that is not practical for most people. But if you use two or four subwoofers in the corners or the wall midpoints, you can get pretty good performance.”

Cont'd in the next post.

Last edited by Big Daddy; 01-03-2013 at 04:12 AM.
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