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I am studying the normal modes and I am confused because there are several names that I am not sure if they are the same physical phenomena or they are different things that are related. I would like to know the definitions of each one and how these concepts are related to the resonances of a room. I read the chapter on standing waves in the 9th edition of Serway and Vuille's book and I know that standing waves are waves created by the interference of two waves in opposite directions and produce nodes and antinodes. This book explains this phenomenon with a string model, but this model is a bit different from a room model and doesn't mention normal modes, so how does this relate to normal modes, room acoustic resonances, room modes, and forced vibrations?

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  • $\begingroup$ Please only ask one question per post — only ask several if they are so closely related that it wouldn't make sense to split them up since they cannot reasonably be answered separately. That way, users that might be able to answer one question but not the others still can provide useful, complete answers to a question. In this particular case, while it might seem that you are asking a single question, notice that for someone to answer this question they must know about all six (if I counted right) terms you mentioned. $\endgroup$ Jan 28 at 5:20

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All mechanical systems (and you could consider an acoustical system as a mechanical one. This is justified in the use of electro-acoustico-mechanical analogies) have at least one preferred way of "behaving" when oscillating. This has to do with their normal response when they are excited too, which is of course the topic of forced excitation. These terms are clarified below.

Quoting Harris' Shock and Vibration Handbook:

Mode of vibration: In a system undergoing vibration, a mode of vibration is a characteristic pattern assumed by the system in which the motion of every particle is simple harmonic with the same frequency. Two or more modes may exist concurrently in a multiple-degree-of-freedom system.

This means that, according to the authors, a mode is the pattern of vibration. This is consistent with the fact that for a specific geometry, you will always (in the linear regime) get the same pattern for a specific mode, regardless of the frequency that excites the mode. For example, the modes of a rectangular plate or a parallelepiped room have the same patterns (different for each case of course) no matter what their scale is. Of course scale does affect the frequency that excites the mode. Based on that and quoting the same book again

Natural frequency: Natural frequency is the frequency of free vibration of a system. For a multiple-degree-of-freedom system, the natural frequencies are the frequencies of the normal modes of vibration.

and the normal mode of vibration is (again quoting the same reference)

Natural mode of vibration: The natural mode of vibration is a mode of vibration assumed by a system when vibrating freely.

Key word here is freely. This means (as mentioned in the beginning) that the natural modes (excited at the natural frequencies) are the ways (at the frequencies) that the system "likes to" (or has the "tendency") to vibrate. Since acoustics, like vibrations can be described by the "unified framework" of oscillations the same principles apply there too.

Now, according to the same authors, there is a specific definition for the normal modes and this is

Normal mode of vibration: A normal mode of vibration is a mode of vibration that is uncoupled from (i.e., can exist independently of) other modes of vibration of a system. When vibration of the system is defined as an eigenvalue problem, the normal modes are the eigenvectors and the normal mode frequencies are the eigenvalues. The term classical normal mode is sometimes applied to the normal modes of a vibrating system characterized by vibration of each element of the system at the same frequency and phase. In general, classical normal modes exist only in systems having no damping or having particular types of damping.

Now, coming to resonances. As I have already done multiple times in this answer I will, once more, quote the same old reference for a definition of resonance. This is

Resonance: Resonance of a system in forced vibration exists when any change, however small, in the frequency of excitation causes a decrease in the response of the system.

For the sake of completeness I will also quote a term that is not used so often. This is antiresonance and according to the same source it is

Antiresonance: For a system in forced oscillation, antiresonance exists at a point when any change, however small, in the frequency of excitation causes an increase in the response at this point.

As you can see, when we refer to resonances we are already talking about forced excitation of a system. As you may already know, when we are forcing oscillations to a system we can "arbitrarily" pick the frequency of excitation. The frequency at which a resonance exists/happens is called resonance frequency (or frequency of resonance). The same definition from the source is

Resonance frequency: Resonance frequency is a frequency at which resonance exists.

Now, coming to the part we didn't touch so far, the standing waves. According to the reference, a standing wave is

Standing wave: A standing wave is a periodic wave having a fixed distribution in space which is the result of interference of progressive waves of the same frequency and kind. Such waves are characterized by the existence of nodes or partial nodes and antinodes that are fixed in space.

Of course you may already know that, but on the other hand you should consider how would it be possible for a system of infinite extent to have natural modes (free vibrations) or exhibit resonances (forced vibrations). I don't think you could have such a thing (please correct me if I am wrong). This means that in a sense you can think of the standing waves as the natural/normal modes of an acoustical system. These two "things" (modes and standing waves) seem to have similar characteristics (according to the definitions presented above). Both exhibit fixed distributions in the spatial variables which constitutes them, at least for practical reasons, similar if not identical.

Normal Modes

To explain the normal modes (answering to your comment), you first of all have to consider that they are modes. As stated above, a mode is a specific spatial distribution pattern (in the room acoustics case, most often of pressure or displacement but it is not restricted to only those quantities).

Now, to make it clearer, modes are excited at specific frequencies, no matter if we are talking about free or forced oscillations. The only difference being that when a system is freely vibrating it does so only at the normal frequencies (in the stade-state) whereas in forced oscillations it vibrates at the frequency of the external force. Nevertheless, when the frequency of external forcing function coincides with one of the natural frequencies you achieve resonance and you can observe the mode. If those modes (regardless of whether they are excited at a freely or forced vibrating system) are (linearly) independent then we talking about normal modes.

As already mentioned in the answer, the normal modes are the eigenvectors and the frequencies they happen are the eigenvalues of an oscillating system is stated/modeled as an eigenvalue problem.

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  • $\begingroup$ Right, I catch pretty much everything, but the normal modes still baffle me. We can say that the normal modes are excited by normal frequencies, right? if so, what are the normal frequencies? Are those frequencies the ones produced by forced vibrations? I mean, those vibrations that we arbitrarily choose to excite the system. $\endgroup$
    – Caeta
    Jan 21 at 0:39
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    $\begingroup$ I have edited my answer to include some more explanation of the normal modes. Please let me know if this is adequate and makes sense. If not, I may add more information. $\endgroup$
    – ZaellixA
    Jan 21 at 11:06
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    $\begingroup$ So, if I understood, normal modes are also called natural modes, and they can be excited by forced vibrations (the system will vibrate at the forced vibration frequency) or free vibrations (the system will vibrate at natural frequencies also called normal frequencies). But whatever what frequency the system is vibrating the normal modes are only excited by oscillations at the natural or normal frequencies of the system. Something still it is not clear to me is what does it mean that normal modes are "independent" of other modes? $\endgroup$
    – Caeta
    Jan 21 at 19:48
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    $\begingroup$ Yes, that is correct! Natural modes will be excited either by principle when the system is freely vibrating or by an external force when its frequency coincides with the natural frequency corresponding to this mode. Now, regarding what independence means, for all practical cases, it is when one normal does NOT affect and is NOT affected by any other. This means that if the system vibrates at one normal mode and you ADD more frequencies to the forcing function, then the response of this mode will stay the same (unaffected) no matter how many more frequencies/modes you will excite (cont.) $\endgroup$
    – ZaellixA
    Jan 21 at 23:50
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    $\begingroup$ the system at, at the same time. This means that you can simply use linear superposition of modes to find the resulting spatial distribution of the quantity at hand (its "shape" in space). Thus the first mode of vibration will stay exactly the same if you decide to excite the system at its fourth normal mode at the same time. Hope this makes sense. $\endgroup$
    – ZaellixA
    Jan 21 at 23:52

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