Acoustics - Resonance modes

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As we have already said, the frequencies we have just described are called resonance modes, and are frequencies whose wavelength is a multiple of the distance between two parallel walls. Specifically we call primary modes, modes whose wavelengths [Wavelength ] are double the distance between the walls.

There are three types of resonance modes:

  1. Axial: this type of mode is generated between two parallel surfaces (for example, two walls in a room, or in the space between the floor and the ceiling). The following figure illustrates a primary mode between two parallel walls:

    Environmental acoustics - Propagation of an axial mode in a room

    Propagation of an axial mode in a room

    Looking at the section of a room, as illustrated in the following figure, we notice the primary axial mode:

    Environmental acoustics - Side-view of an axial mode

    Side-view of an axial mode

    We can see how a primary mode's wavelength is double the distance between the two walls. A room has three primary axial modes: one covering its width, one covering its height, and one its length. Clearly there will also be secondary modes present (double the frequency of the corresponding primary mode), tertiary modes (triple the frequency) and so on, with amplitudes diminishing as frequency increases. This means that primary modes are the ones that most affect the acoustic response of an environment.

  2. Tangential: this type of mode is generated when the soundwave reflects on 4 surfaces. The following figure shows an example of a tangential mode:

    Environmental acoustics - Propagation of a tangential mode within a room

    Propagation of a tangential mode within a room

    This kind of mode can arise in the following conditions: between the four walls of a room, or between the floor, the ceiling and two walls, or the floor, the ceiling and the two other walls. A tangential mode's amplitude will be smaller than an axial mode's, because it involves four reflections rather than two, which means greater absorption of acoustic energy.

  3. Oblique: this mode is generated when sound is reflected off six surfaces in a room. Its amplitude is greatly reduced compared to the other modes. The possible path of an oblique mode is illustrated in the following diagram:

    Environmental acoustics - Propagation of an oblique mode in a room

    Propagation of an oblique mode in a room

    To calculate the modes' frequency in a room we'd use the following formula:

    Equation 15.1. Calculation of resonance frequencies of modes 

    Calculation of resonance frequencies of modes

    Before you panic, read the next lines, and you'll see that it's not as complicated as it seems. Let's begin by highlighting the quantities present in the equation:

    c = speed of the sound in the medium in question (seeing that we are on planet earth we'll take air as the medium through which sound travels. In our case: c = 344 m/s).

    l = room length

    w = room width

    n = room height

    n1, n2, n3 = indexes representing the mode.

    Let's take a look at some examples. Say that we wanted to calculate the frequency of the primary axial mode along the room's length (supposing the width to be l = 10 m). This mode is identified by the following set of three values:

    n1 = 1, n2 = 0, n3 = 0

    If we substitute these values in the formula we'll have:

    Equation 15.2. Calculation of a primary axial mode 

    Calculation of a primary axial mode

    We get a primary axial mode with a frequency value of 17.2 Hz, which is below our audible threshold, and therefore doesn't give us any problems at all.

    To identify the resonance modes the following notation is used:

    • The three primary axial modes are identified by the three triads: 100, 010, 001

    • The three tangential primary modes are identified by the three triads: 101, 110, 011

    • The only primary oblique mode is identified by the single triad: 111








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