Why do certain rooms/vessels respond to specific frequencies?

Question

Since I was a kid I was always wondering why is it that when I sing in a small room (i.e. bathroom!), whenever I touch a certain frequency, the whole room vibrates sympathetically.

What are the parameters that make an acoustic body (be it a room, cell or whatever) resonate to a specific note?

Is there a way to reproduce the resonance in other acoustic bodies? I'm wondering if we can encapsulate an instrument in a box that changes its size according to the played pitch to amplify it and make it resonate in this wild manner.

Answer 1

Because of dynamics called room modes.

Room modes are the collection of resonances that exist in a room when the room is excited by an acoustic source such as a loudspeaker. (...) each frequency being related to one or more of the room's dimension's or a divisor thereof.

To keep things simple, we will assume the room has 6 parallel walls (right prism or cube) and will just focus on one type of room mode: axial (because it's the simplest, and it is predominant over other modes). The calculations of the room modes of more complex shapes can get very complex, very fast.

The relevant parameters are the dimensions and shape of the room, and the wavelength of the sound. When the wavelength of the sound and the dimensions of the room coincide, there will be resonance. Similar to how a string will resonate to specific frequencies depending on its length (and tension, and materials, etc).

To make a room resonate:

1. Measure one dimension of the room: either height, length, or width. For the formula we are using we will measure it in meters.

2. Find the frequency that has a wavelength that is double the length of one of the three dimensions of your room. It's easier than it sounds, just use this formula:

   Frequency = 1/2 x 343 m/s / the dimension of the room that you just measured

   Where 343 m/s is the speed of sound in meters per second (an approximation, since it is not constant and depends on many factors) If you want to use other length unit, you can convert the speed of sound to your unit of choice, and use that unit in the measurement of the room and calculations.

3. Playback (or sing, if it's within your vocal range) that frequency inside that room.

For example, suppose there is a distance of 4 meters between the walls of your bedroom. The frequency we want would be 1/2 x 343 m/s / 4m = 42 Hz (that's between a E and an F).

Smaller rooms resonate to higher frequencies. That's why you noticed this in the bathroom using your voice, since a small room might have a room mode in the frequency range of the human voice. With one meter between walls: 1/2 x 343 m/s / 1 m = 171.5 Hz (very close to an F).

A given room also has higher resonating frequencies. If 42 Hz is a resonating frequency of your bedroom: then 42 Hz * 2 = 84 Hz, 42 Hz * 3 = 126 Hz and so on also are resonating frequencies of the room.¹ Because the lower resonating frequencies of a room are usually very low, these might be easier to make resonate.

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1. Actually, waves which “bounces” against perpendicular walls can also resonate, but the formula which corresponds to them is slightly more complicated. If l, w and h are the length, width and height of a room (in metres), and for any integers (whole numbers) n, p and q, the frequency F given by the following formula resonates:

F = 1/2 * 343 m/s * square_root ( n^2 / l^2 + p^2 / w^2 + q^2 / h^2 ).

Answer 2

JCPedroza's answer is correct for a square room, but I think it's worth pointing out that the shape of the room is not just it's dimensions. For example, a square room with an open window will act different than if the window is shut. In acoustics, we often model the response of a room as a circuit. If you break up the space into pieces, each piece can be represented as a circuit element. There are multiple analogies you can make, but as one example, a tube of air acts like an acoustic inductor, and a box of air acts like an acoustic capacitor. If you draw an equivalent representative circuit for the geometry you're interested in and work out the circuit analysis, you can figure out exactly what will happen.

So how does a resonance happen? Well, a big empty room is going to have a large acoustic capacitance, but it will also have a small acoustic inductance. When a capacitor and inductor are connected to each other, they begin to slosh energy back and forth between each other. There's one frequency where the sloshing reaches a maximum, and that's the resonance frequency. So that's why you will hear the resonance frequency of the room louder than you normally would (at omega = 1/sqrt(LC)). There are equations you can use to measure the acoustic inductance and capacitance of a room, but it's quite a lot of work.

Two rooms can have the same resonant frequency, but one may seem to ring longer than another. In circuit analysis, we call that the Q (quality) factor of a resonator, and it depends on the resistance in the circuit. You can add more resistance by putting a bunch of furniture or carpet or sound panels in the room, which will cause the ringing to stop much sooner, since they'll dissipate energy and prevent it from being sloshed back into the capacitor or inductor. In an electric circuit, that's dissipated as heat, and in a room with a sound wave, it's also dissipated as heat!

Answer 3

One form of the phenomenon is called the "standing wave". As it applies to rooms or acoustic spaces, it is sometimes refered to as "resonant room mode".

Here is a link to an article in Audioholics Magazine that discusses how these phenomena affect room acoustics.

Answer 4

I'm wondering if we can make an instrument that changes its size according to the played tone to amplify it and resonate it in this wild manner.

Well, most wind instruments work in that manner. There are rather few, like fanfares, that are mostly indeterminate regarding their resonances.

And natural trumpets don't have valves or holes, either, but still resonances.