# In general terms, what key elements are necessary for resonance to exist in a physical system?

I found a related question An Analogy for Resonance, and John Rennie gives a good explanation using description of the harmonic oscillator. But I'm really looking for an accurate and complete list of the specific elements that are required for a physical system to be 'resonant'.

I've looked on line and in physics text books, but nothing really addresses all types of systems in general. They rather give examples of specific physical systems like LC circuits where electrical and magnetic fields are involved, or mechanical systems where springs and masses are. But I would think a more generalized view of resonance - across all types of physical systems would require defining the key elements in terms of energy.

Is there generalized definition that provides a complete and concise summary?

• A good question. I'm wondering if this linked question is at all helpful? – D. Betchkal Mar 7 '18 at 5:10
• @D.Betchkal thanks for the link. Interesting but still not fully satisfying. I did add a comment to that question regarding the OP's "are there any holes?" question. – docscience Mar 7 '18 at 18:32
• @D.Betchkal cause I've found when I ask this question, everyone tends to point to linear models (like the answer below). But nonlinear systems can also exhibit resonance. So there must be a more fundamental answer. The "there has to be at least two independent "states" where some physical quantity is sloshing back-and-forth between those states" comment I believe is fundamental. But when he says "must at least second order" most often interpreted as "and linear" as well. But not necessarily so. – docscience Mar 7 '18 at 18:37

## 2 Answers

there has to be at least two independent "states" where some physical quantity is sloshing back-and-forth between those states. then the amount of physical storage of energy must exceed $\frac{1}{4\pi}$ of the amount of energy dissipated each cycle that this physical quantity is sloshed back-and-forth. this means the resulting differential equation (if this system is "linearized") must be at least second order and will have this form:

$$\frac{1}{\omega_0^2} \frac{d^2}{dt^2}y(t) + \frac{1}{\omega_0 Q}\frac{d}{dt}y(t) + y(t) = x(t)$$

and the $Q$ value must be greater than $\frac{1}{2}$. if that is the case, then $\omega_0$ is the resonant angular frequency.

• I suppose also you also need a source of energy - either an input or an initial imbalance in the states. I will sometimes see definitions that say the input needs to have frequency components that match the natural frequency of the system. So aren't these, in some form, also a key element that needs to be considered? – docscience May 22 '15 at 21:00
• Also - what about energy losses? If the system loses energy at a greater rate than it can gain energy, won't it be hard to build a resonant condition. Or does the size of Q in the linearized 2nd order model specify the loss? – docscience May 22 '15 at 21:06
• I would say an energy input would be a requirement for the system to be resonating, but not for being resonant. – Brionius May 22 '15 at 23:28

Let's first get some kind of definition of what resonance actually is. Wikipedia says the following:

In physics, resonance is a phenomenon that consists of a given system being driven by another vibrating system or by external forces to oscillate with greater amplitude at some preferential frequencies.

Essentially, some amplitude becomes bigger at a certain frequency.

nothing really addresses all types of systems in general

That's because it's not the primary goal of a physics text book to do this. If you want to understand systems in general, you will probably get more insight from a systems theory book.

They rather give examples of specific physical systems like LC circuits where electrical and magnetic fields are involved, or mechanical systems where springs and masses are.

That's because they can be described with the same differential equations, just using different symbols. You can substitute one for the other.

A very important thing in systems theory is the transfer function, which is defined as \$\frac{output}{input}\$ If you multiply this function with the input, you receive the output from the system. This function usually depends on the frequency.

Now this sounds suspiciously related to the definition of resonance above. If the transfer function has a value bigger than 1, the output (it's value or amplitude) will be bigger than the input. It's often the case that the transfer function has a maximum. In this maximum, the input is amplified the most, which is what's called the resonance.

Given a RLC row circuit as a system, one possible transfer function could be that of the overall voltage being the input and the voltage over the resistor being the output:

$$\frac{U_R}{U}=\frac{R}{R+j\omega L+\frac{1}{j\omega C}}$$

Another transfer function could be that of the overall voltage being the input and the current being the output:

$$\frac{I}{U}=\frac{1}{R+j\omega L+\frac{1}{j\omega C}}$$

You can see that both depend on $\omega =2\pi f$. To show resonance, they have to have a maximum that's bigger than 1

I'd say that resonance is not a property of the system per se, but a property of a pair of two of its properties, one being used as an input while the other is the output. If the transfer function, that describes the ratio between input and output has a maximum or peak at a certain frequency, then you have resonance.

• I'm really more interested in understanding the reason for resonance in all types of physical systems, so in a general way. Mathematical models help explain the phenomena, but I suspect there's a better, more fundamental way to define the essential elements that are necessary for resonance. And I believe it involves the common denominator across all systems - the flow of energy. Transfer functions, even nonlinear state space models probably too specific and restrictive to get at the heart of the question. – docscience May 23 '15 at 23:35
• @docscience what is too specific about a transfer function? How is "flow of energy" modeled in a non mathematical way? I have a feeling that you really want to find a nail for that "explain it with energy" hammer. – Name May 26 '15 at 16:33