Why is resonance such a widespread phenomenon?

Question

It is easy to mathematically describe the motion of a mass which is attached to a spring and also pushed around by a sinusoidal force. We get a differential equation of the form:

$$\frac{\mathrm{d}^2x}{\mathrm{d}t^2}=-\omega^2x+A\cos(\omega't)$$

Applying the Fourier transform to this and integrating to obtain the energy, it is possible to see that the energy of the system peaks when $\omega'=\pm\omega$. This is resonance.

But resonance is observed in many situations which are not easy to describe mathematically. If you sing at a wine glass at the right pitch, it might resonate and crack. If an earthquake shakes a building at the right frequency, it might resonate and collapse. If winds shake a bridge at the right frequency, it might resonate and collapse.

Question: Is there a very general mathematical result that explains why resonance is such a widespread phenomenon? I suspect that even if there were such a result, it wouldn't be of much interest to physicists and engineers, for it would only confirm what they already know and consider obvious. This is why I am asking this question on MathOverflow.

Supplementary questions:

1. What can we assume about the form of the equation? In each of the examples I gave, there is presumably some partial differential equation that describes the system. But I don't think this equation is necessarily linear, and we also cannot assume that energy is conserved, because resonance is observed even in damped systems (e.g. systems with friction).

2. Wave equation: How can we prove that resonance in a system described by the wave equation $\nabla^2 f=\partial_t^2f$ happens when the driving frequency matches the eigenvalues of $\nabla^2$? What relates resonant frequencies to eigenfrequencies?

Edit (14/9): clarifications / further questions:

For convenience, let us limit our discussion to the wave equation.

Suppose the state of a system (e.g. the air in a closed room) is described by a function $u(\mathbf{x},t)$ (e.g. the air pressure at time $t$ and point $\mathbf{x}$ in the room), which obeys the wave equation:

$$\partial_t^2 u=\nabla^2 u$$

As Michael Engelhardt explains in his answer, the qualitative content of the wave equation is that the system is an ensemble of harmonic oscillators which are each coupled to their nearest neighbours. The restoring force that each oscillator exerts on the surrounding oscillators is (to a good approximation) linear.

Eric Towers explains in a comment that these are empirical truths which cannot be deduced through purely mathematical reasoning. I understand this, and I apologise for not making it completely clear what I was looking for.

Nature give us the premise that the system is described by the wave equation, and also the boundary conditions (e.g. if $u$ is air pressure, then there is no pressure gradient perpendicular to the walls of the room).

The rest of our discussion can proceed along purely mathematical lines.

Normal modes are eigenfunctions of the relevant linear operator (e.g. Laplace operator), and these eigenfunctions form a complete basis for the space of all solutions. This is a purely mathematical fact.

If our system is driven by external forces, the equation that describes it is now:

$$\partial_t^2 u=\nabla^2 u + f(\mathbf{x},t)$$

where $f$ describes the external forces.

I would like to see the following questions answered in mathematical terms.

Questions:

1.Mathematically, how do we describe the 'excitation' of a particular normal mode? What conditions must $f$ satisfy in order for $u$ to be, or approach asymptotically, a normal mode?

2. Why does resonance happen exactly when the system is in a normal mode? How can the correspondence between "resonance" and "normal mode" be seen mathematically?

3. Can resonance happen in only some regions of the system? How can this be expressed mathematically? For example, if a driving force is applied to the system in a very localised manner (e.g. a speaker playing a sound in the corner of a big room).

Answer 1

Most systems you see around you are subject to a restoring force (otherwise, they'll go find an equilibrium elsewhere). Most restoring forces are linear as long as you're not too violent with the system (that's just Taylor expansion around the equilibrium). All systems you see couple to external forces (otherwise, how would you ever detect that they're there). So the elements contained in your basic resonance equation are ubiquitous.

The fact that most of the systems you see (glass, building, bridge) are multi-dimensional doesn't change this qualitatively. They all have normal modes, which each behave one-dimensionally.

Wave equations arise in systems that consist of many nearest-neighbor-coupled harmonic oscillators. Such a system, again, can be decomposed into its normal modes, which can each be resonantly excited. The eigenvalues of the Laplacian provide the wavenumbers that are related to the resonant frequencies through the wave speed.

Answer 2

Resonance is so universal because Fourier analysis is so universal, and because physical equations of motion are second-order.

Michael Engelhardt already explained why Taylor expansion around stable minima means that we can describe oscillations as linear, and that multidimensional systems oscillators can be expanded into normal modes, so it suffices to look at one-dimensional ones.

Here I want to address two possible misconceptions the OP may have. The first misconception is that damping is a sign of non-linearity. This is not so, since damped oscillations are described by the linear homogeneous equation $$ \frac{d^2}{dt^2}x+2\gamma\frac{d}{dt}x+\omega^2x = 0 $$ whose solutions decay exponentially like $e^{-\gamma t}$ (there are three cases depending on whether $\gamma>\omega$, $\gamma=\omega$ or $\gamma<\omega$, with the system actually "oscillating" as in repeatedly crossing $x=0$ only in the last case).

The second misconception is that resonance is somehow specific to driving by a sinusoidal force. This is not so, since in the inhomogeneous equation $$ \frac{d^2}{dt^2}x+2\gamma\frac{d}{dt}x+\omega_0^2x = f(t) $$ we can always Fourier expand $f$ and write the equation in frequency space as $$ \left(-\omega^2+2i\gamma\omega+\omega_0^2\right)x(\omega) = f(\omega) $$ Then, as long as $f$ has Fourier components $\omega\approx\omega_0$, the Fourier components $\omega\approx\omega_0$ of $x$ will blow up due to the expression in brackets being almost zero.

The last point about Fourier transforms also explains why resonance frequencies are related to eigenvalues of a differential operator, since they are simply the frequencies under which the system oscillates in the absence of a driving force, and the linear homogeneous equation above can of course be rewritten as $$ \left(\frac{d^2}{dt^2}+2\gamma\frac{d}{dt}\right)x = -\omega^2 x $$ which is an eigenvalue equation for the operator in brackets.

Answer 3

A model independent way to describe a resonance is through the frequency dependent scattering operator $S(\omega)$. Causality requires that this object is analytic in the upper half of the complex $\omega$-plane. In the lower half it may have poles, say at $\omega=\Omega-i\Gamma$. This gives a resonant response when excited at a frequency within $\Gamma$ of $\Omega$.

Concerning the relation between resonant frequencies and eigenfrequencies: At frequencies near resonance one can expand near the pole, $$S(\omega)=(H-\omega+i\Gamma)(H-\omega-i\Gamma)^{-1},$$ with $H$ a Hermitian operator (so that $S$ is unitary). The resonant frequency $\Omega$ then corresponds to an eigenvalue (eigenfrequency) of $H$.

The operator $H$ does not need to produce a simple wave equation. For example, this description of resonant scattering applies to nuclear physics, where the resonance is a bound state of strongly interacting elementary particles.