For this week’s assignment, you will conduct a full model analysis of your room. · On paper, calculate the first two axial modes in each axis for your studio or listening space. · Scan

Room Modes (You may need an online mode calculator for this assignment)

In lesson 2, we looked at the ways that various objects vibrate, resulting in audible sound. The frequencies of the various modes of vibration of a sting, membrane, bar, or column of air depend largely on its dimensions. Let's take a closer look at modes of a string. As you might remember, they are harmonics of each other because the string has only one dimension. The first mode is one half of a wavelength; the second is two halves, and so on. As you probably know, changing the tension of the string (e.g. tuning a guitar) changes the frequencies of its modes. The wavelength hasn't changed, but the frequency has. Remember that the relationship between frequency and wavelength depends on the speed of propagation in the medium. The equation for wavelength is:

Standing Waves

When the vibration in a string or air column is induced, the resulting oscillation is called a standing wave. It gets this name because the nodes and antinodes don't move—there is essentially no propagation. When a wave traveling along a string reaches a fixed end, it reflects out of phase and begins traveling the opposite direction. The standing wave is the result of the combination of these two waves of the same frequency traveling in opposite directions in the same medium. This animation shows two sine waves adding to create a standing wave.

A node is formed half of a wavelength from the end. This combination of a wave with its reflection could occur at any frequency. Consider, though, that this will happen at each end of the string. This means there will be a node one half wavelength from the other end as well. A standing wave is formed when the node created by both ends are at the same point along the string. In the case of the first mode, these nodes, each half a wavelength from an end, actually land at the opposite end of the string.

Modes in a Room

Acoustical standing waves can exist in rooms. Instead of the nodes and antinodes representing the deflection (or lack thereof) of at string, they represent the sound pressure at various points in the room. The boundaries of the room (walls, floor, and ceiling) are like the ends of the string—they reflect the sound back into the room. Unlike a string, however, the boundaries of the room act as antinodes—the point where the fluctuation in pressure is greatest. For this reason, the closest antinode is only ¼ of a wavelength from the surface, compared to ½ of a wavelength in the string. Here are graphs the compare the first three modes in a string with those in a room:

Let's look at an example of a room mode. If two walls in a room are 4 m apart, the first mode, or the lowest resonant frequency, will be that for which 4 m is half of a wavelength. To find the frequency for which 4 m is a whole wavelength, we would use this equation:

To find the frequency for which 4 m is half a wavelength, we simply divide this frequency by 2: 86 Hz divided by 2 equals 43 Hz. The first mode for this dimension of this room is 43 Hz. The second will be 86 Hz, the third 129 Hz, and so on. Small rooms, such as recording studios and control rooms, suffer from the effects of room modes more than large theaters or concert halls. The low-order modes (1st, 2nd, 3rd, etc.) in large rooms are for frequencies that are below 20 Hz. Not only could we not perceive these modes, but also there are also few sources of energy at these frequencies to induce resonance. In small rooms, modes lead to pressure variations from one part of a room to another, relative to the locations of the nodes and antinodes. Also, as we'll discuss later in the lesson, placement of low-frequency loudspeakers relative to the locations of the nodes and antinodes can affect the level of excitation of the modes. Room modes are a fact of life, and cannot be avoided in small rooms. However, the goal in designing a small recording or sound reproduction is to distribute these modes throughout the low frequency spectrum, and avoid having several modes at the same or very similar frequencies. This can be accomplished by choosing room dimensions carefully.

Types of Modes

Axial

The mode described in the example we did earlier is known as an axial mode. This is a mode that exists along one of the three axes (length, width, and height) of the room. Axial modes may exist between walls or between the floor and ceiling. These modes are the easiest to predict and tend to be the most prominent.

Tangential modes involve four surfaces instead of two. Because of this, they have half of the energy of axial modes, which equates to a difference of 3 dB. The illustration below shows the tangential modes in a simple rectangular room.

Oblique modes involve all six surfaces in the room. They have a quarter of the energy of axial modes, which equates to a difference of 6 dB. The illustration below shows one possible oblique mode in our simple room.

Calculating Room Modes

The frequencies of all three types of modes can be calculated using the following equation:

The variables are as follows:

• c is the speed of sound in air

• L, W, and H are the length, width and height of the room.

• p, q, and r represent the mode numbers for each of the axes of the room, where p represents the L axis, q the W axis, and rthe H axis.

We'll use this room to do some example calculations.

The type of mode to be calculated depends on the values of nL, nW, and nH.

For an axial mode, only one of the three is a non-zero number. For example, to calculate the first axial mode for between the floor and ceiling, dimension H, we would set r to 1 (for the first mode), and both p and q to 0.

To calculate a tangential mode, two values of n will have non-zero values. Remember that tangential modes involve two sets of parallel surfaces, and thus two axes. Let's calculate the first tangential mode for our sample room that includes all four walls. In this case, only r will be 0, as the mode doesn't include the ceiling and floor.

Oblique modes include all six surfaces, and thus all three axes. Let's calculate the first oblique mode in this room.

In each of the examples above, we have calculated the first mode in each axis. This means that half of a wavelength of the resulting frequency exists between each successive reflecting surface. For tangential and oblique modes, any combination of values of p, q, and r is possible. For example, let's calculate the tangential mode shown in the diagram below.

Along the long axis of the room, L, a whole wavelength develops between reflections. This represents the second mode. However, only half of a wavelength develops along the W axis. This is the first mode. To calculate the frequency of this tangential mode, we set ­pto 2 and q to 1. The value of r is 0, as this mode doesn't involve the floor or ceiling.

Specific room modes are often designated by the three values of n. For example, the tangential mode calculated in the previous example would be the "2-1-0" mode for this room. The calculator allows you to enter your room dimensions (in either feet or meters), and calculates all of the modes up to 250 Hz. It orders them by frequency, and uses color codes to warn you when several modes are grouped closely together. You'll notice that the modes are identified using the three number arrangement described above.

Pressure vs. Velocity

Up to this point, all of the graphs of sound in air that we have looked at have represented pressure over time or distance. Here is a simple graph of pressure over distance in a standing wave.

The antinodes of this graph—the highest and lowest points—represent the locations of the largest fluctuation in pressure. However, the particle velocity at these pressure antinodes is actually the lowest of any point in the wave. The molecules move from one pressure antinode to the next. As they approach the new antinode, pressure builds, and they decelerate to a stop. They then reverse direction and accelerate away. The highest particle velocity is found at the points where the change in pressure is greatest; the lowest velocity is found where the change in pressure is lowest-at the pressure antinodes. As explained in the following video, the change in velocity is greatest at the pressure nodes.

Calculus and the Derivative

Calculus is the branch of mathematics that encompasses the study of change. One of the fundamental operations in calculus the derivative. The derivative measures the rate of change. A common example is that velocity is the derivative of distance. In other words, velocity is a measure of the rate of change of distance.

The relationship between particle velocity and pressure represents a derivative as well. Since velocity is proportional to the rate of change of pressure, we can say that particle velocity is the derivative of pressure. We could represent this using this equation:

The symbol Δ is the Greek letter delta, which is used to represent change. The equation above tells us that v is proportional to the change in pressure divided by the change in time.

Absolute Value of Pressure

In order to visualize the relationship between pressure and velocity in a room mode, it is often helpful to consider the absolute values of each. For the purposes of absorption of sound, the magnitude of each is of greatest concern, as absorbers behave the same regardless of the relative pressure (above or below steady-state) or the direction of molecular motion. When we calculate the absolute value, or magnitude, of a sine wave, the negative parts become positive. Here is what it looks like on a graph.

Here is a three-dimensional sketch that uses this technique to depict pressure resulting from a single axial mode in a room. This is the fourth mode, as it contains four half wavelengths. Notice that there are antinodes at the two boundary walls, as you'd expect with a room mode. Pressure is greatest at the antinodes.

Let's now add velocity (shown in red) to this graph. Remember that velocity nodes are at pressure antinodes, and vice versa.

Theories of Small Room Design

While there are many schools of thought on rooms designed for music playback through loudspeakers, almost all designers agree that first-order, or early, reflections should be eliminated at the listener's position. These early reflections may come from the walls, ceiling, recording console surface, or even the floor. As you learned in previous lessons, strong early reflections can cause problems in large rooms because of the potentially long time delay between the direct sound and these early reflections. In small rooms, the problem is that the delay between the direct and reflected sound is too short. Here's a simple diagram showing a loudspeaker and a listener. The green path represents the direct sound, and the red path represents a reflection from the side wall.

Let's say that the direct path is exactly 1 m, and the reflected path is 1.334 m. Using 344 m/s for the speed of sound, this means that the reflected sound will reach the listener 1 ms later than the direct sound. This is far too short for our ears to hear it as a discreet echo. However, let's look at what happens when the direct sound and reflected sound combine at the listener's ear. Consider what will happen at 500 Hz. Here is a plot of the direct sound. The period at 500 Hz is 1/500 = 0.002 s, or 2 ms.

Now, let's add our reflected sound, which arrives 1 ms later.

The delay of 1 ms causes the phase of the delayed sound to be exactly opposite that of the direct sound. The effect is that these will cancel each other, and effectively filter 500 Hz from the overall spectrum. This is known as destructive interference. It is the same effect that causes nodes in a standing wave. At 1000 Hz, this difference of 1 ms would amount to a delay of one complete cycle. The sum of the direct and reflected signals at this frequency would be as much as twice the amplitude of the direct sound. Here we have constructive interference. If we shift another 500 Hz to 1500 Hz, we again have a complete cancellation. This destructive interference will cause total cancellation at 500 Hz, 1500 Hz, 2500 Hz, and so on. At multiples of 1000 Hz, the signals will add constructively. The resulting frequency response curve looks like this:

This effect is known as comb filtering because the response plot looks like a comb. Comb filtering has an adverse effect on the clarity and accuracy of the sound reproduced by our loudspeakers. Because we have two ears that are in slightly different locations, the comb filtering at one ear will be different from that at the other. This helps our brains to separate the direct sound from the reflected sound, but the accuracy of the stereo image suffers. Sound from the left loudspeaker will bounce off of the right wall, and vice versa.

Reflection-Free Zone

The remedy to all of this is to create a reflection-free zone at the listener's location. This is done by determining the points on the walls, ceiling, and other surfaces from which sound would reflect on its way to your ears. In this simple sketch, the first reflections from each loudspeaker off of each side wall are shown.

By adding absorption at the points of reflection, the reflections are significantly reduced.

This should also be done on the ceiling and floor (in the form of a rug). We'll discuss the back wall in the next section. Ideally, this absorption should be effective down to 500 Hz or lower. A quarter of a wavelength at 500 Hz is 17.2 cm. This would be the ideal thickness for a porous absorber in these locations. A common mistake is to use very thin absorbers on the order of 2 cm. These would only be effective down to 4 kHz or so. There is a lot of sound below that in both speech and music.