Forced Oscillation Explained with Violin String

In musical instruments, none of the excitations, whether by bowing, plucking, reed vibration, lip vibration, striking, etc, are sinusoidal methods. Therefore, they all, by Fourier's theorem are composed of multiple sine wave frequencies. If one or more of these frequencies corresponds to resonant frequencies of the instrument, the energy of those components will persist longer than other frequencies. If energy continues to be input to the instrument, the resonant frequencies will persist, giving rise to the continuing musical tone.

Even a brief strike, like a mallet blow, has a time behavior, and resembles either a square, sawtooth, or triangle pulse, can be viewed as a pulse of multiple frequencies. It need not be a multiple period train. A single pulse induces the forcing function. Whether it affects the object depends on the damping of the object. Consider the striking of a tympani (low damping) versus a snare drum (higher damping).

Some excitation methods, especially brass-instrument mouthpieces and reeds, begin the tone with multiple frequencies, then the reflected wave in the instrument creates a feedback loop, locking the lips or reed into resonant frequencies of the instrument. This is especially important for brass players, who can practice with a mouthpiece only playing a tune close to the actual pitches of the music. That way, they are adjusted to optimize the frequency feedback.

The violin (and others) with bowing action is excited by a "grab-release" action of the resin-coated horse hair on the string. The action very strongly resembles a continual sawtooth wave plucking action. The string's length, tension, material, and thickness determine the resonant frequencies of the string, while the sawtooth wave, by Fourier's theorem, inputs more than enough frequencies to the string. The ones which match the resonant frequencies persist, while the ones which don't match dampen very quickly.

So for the more precise explanation, you have a harmonic oscillator with some sort of drag term and some sort of force, $$ \frac{\mathrm d^2x}{\mathrm dt^2} = -2\lambda \frac{\mathrm dx}{\mathrm dt} - \omega_0^2 x(t) +F(t). $$ Exercise 1: prove that if $F(t)=0$ then this oscillates at frequency $\sqrt{\omega_0^2-\lambda^2}$ as it decays like $e^{-\lambda t}.$

Exercise 2: prove that if you drive with a periodic force $F(t) = A e^{i\Omega t}$ (or use sines and cosines if you are not yet comfortable with complex numbers as 2D rotations) then the response of the system is also periodic in the same $\Omega t.$ What do you get for the amplitude? When is that number highest?

Exercise 3: what does the differential equation look like in Fourier space? Exercise (2) drove the system with $F(\omega) = A\delta(\omega-\Omega)$ where $\delta$ is the Dirac delta-function, do you see the basic results of exercise 2 in Fourier space? If we view this result as multiplying the frequency-space force by a complex function, what does the graph of its absolute value look like? Especially, when the quality factor $Q=\omega_0/(2\lambda)$ is large, what do you see this graph looking like? Also, from your experience of guitars and how long a string “rings out” versus what frequency it plays, what sort of $Q$-factor is reasonable for musical instruments?

Exercise 4: what is the Fourier transform of a normal distribution with standard deviation $\sigma$, i.e. $f(x) = e^{-x^2/(2\sigma^2)}/\sqrt{2\pi \sigma^2}$? What is its width?

Once you have these basic exercises completed, maybe you can complete the explanation yourself. How I would put it is, a short sharp force is like a Gaussian with a very small standard deviation. This makes it very wide in frequency-space, which we can interpret as its having lots of different frequencies. But we saw in exercise 3 that there is this susceptibility function with a peak around $\omega_0$ and so those are the frequency parts of the driving force that the system is most going to pick up, particularly if it is a high-$Q$ resonator. Its response is in this little peak around some frequency and it muffles all of the other components to zero. (Indeed if you look up $Q$-factor in Wikipedia you will see that a common definition is the width of the resonance peak in frequency space, divided by the frequency of the peak. So this is almost the definition of a high-$Q$ resonator.)

Caveats

While I think the above answers your question, it can be somewhat dangerous to assume that we automatically know everything from knowing the littlest thing. Note that this system is very simple, not having multiple modes at various frequencies, and in many other respects it is not “real.”

Probably the easiest generalizations are finite element models, so you set up a bunch of masses along some $y$-coordinate with positions $x_i(t)$ and then their restoring forces return like $$\ddot x_i = -2\lambda \dot x_i - \omega_0^2 (2 x_i -x_{i-1} - x_{i+1})$$ so the analysis gets tied up a little with this tridiagonal matrix $$\begin{bmatrix} 2&-1&0&\cdots\\ -1&2&-1&\cdots\\ 0&-1&2&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{bmatrix}$$ and the process of diagonalizing this matrix is more formally called finding the “normal modes” of the system. This “normal modes” analysis is not just the basis of mechanical engineering software, but I have seen it a lot in quantum chemistry calculations, and similar approaches can be helpful at other times, so learning differential equations that have some matrices thrown in can be very rewarding.

But you might also like the full 2d “string wave equation” which just replaces the above finite difference term with $$\frac{\partial^2y}{\partial t^2} = -2\lambda \frac{\partial y}{\partial t} + \omega_0^2\frac{\partial^2y}{\partial x^2}.$$ This launches into a discussion of wave equations more generally. In particular, you get these resonance peaks at $\omega_0,2\omega_0,3\omega_0,\dots$ but it all has to do with boundary conditions on the sides of the string, as you form standing waves on it.

If you own a guitar you may have noticed that the string wave equation assumes that the tension in the strings is held constant but actually the amplitude of real oscillations is enough to stretch the string, so the string rings out like “byoooowwww”, higher in pitch at the beginning and lowering pitch at the end. This is a fundamental nonlinearity that needs to be added to the string wave equation above. More generally a lot of linear instruments are better understood as linear-but-coupled-to-a-nonlinear-system, for which a really good article is provided by Fletcher (PDF warning).

The bow pulls the string till the tension exceeds the friction force and the string skips to its initial position, to be pulled again by the bow. This happens multiple times during one complete movement of the bow. The frequency of the resulting oscillations is determined by the eigenmodes of the string (normal modes in the OP language), which are typically the integer fractions of its length. It is by controlling the length of the string that the musician changes the frequency of the sound.

This is not what I would call forced oscillations - rather we are dealing here with generating oscillations (which in itself is an important thing, but often more complex mathematically than just a forced oscillator). Forced oscillations, on the other hand, are oscillations due to periodic force, which consequently occur at the frequency of this driving force rather than with eigenfrequencies of the oscillator. Same can be said about a tube/trumpet or a tuning fork/guitar. The oscillations are created differently in all these cases, but in none of them they are created by a periodic force.