What a chain of oscillators is

A chain of oscillators is simply a set of point masses lined up and interacting with each other through harmonic forces.

• Point masses means that we study the motion of the masses without worrying about their shape and size. We do not consider, for example, their rotations or the space they occupy.
• Harmonic forces are forces that obey a simple law: the force between two points is directly proportional to the distance between them.

For example, calling ${\displaystyle x_1}$ the position of point 1 and ${\displaystyle x_2}$ the position of point 2, the expression of a harmonic force exerted by point 2 on point 1 can be

${\displaystyle F_{21}=-k(x_1-x_2)}$

We observe that:

1. The force can be either attractive or repulsive: if ${\displaystyle k>0}$ the two masses attract each other. Indeed, the force on particle 1 is always directed towards particle 2.
2. If ${\displaystyle x_1=x_2}$ the force is zero. So the equilibrium position is reached when the masses are in contact. Of course two real masses cannot occupy the same position, but the problem does not arise for two point masses.

Another expression for the harmonic force is

${\displaystyle F_{21}=-k(x_1-x_2-d)}$.

In this case ${\displaystyle d>0}$ is a length that shifts the equilibrium position of the masses. Indeed, in this case the force between the two masses is zero not when the masses are in contact, but when they are at a distance d

${\displaystyle x_1-x_2=d}$.

For example, if we think of the harmonic force as the force exerted by an ideal spring, this second expression is more useful, because it takes into account that the spring has a finite length at equilibrium: precisely d.

Why we study this system

The physical system we study here is only apparently exotic. There are many good reasons why we want to study its secrets.

1. First of all, it is a simple system compared with oscillating mechanical structures such as bridges, houses, engines, musical instruments, crystals, etc., and for physicists this is already a good reason.
2. The harmonic force is an excellent approximation of "real" elastic forces, at least as long as the deformations involved are small.
3. The harmonic force also approximates other types of non-elastic forces, such as the interactions between atoms, under certain particular conditions.
4. If we obtain a wave equation, we will be able to transfer all the properties of waves to the new system, without worrying too much about the details of its structure.
5. The chain is a discrete system, not a continuous one. This means that it has a finite number of components, and therefore that it can be computed on a computer. In other words, while the wave equation for continuous systems must be discretised in order to be solved numerically on a machine, this equation can be used directly. For an example implementation see the numerical simulation program for the Fermi-Pasta-Ulam chain.

The last point is very important for two reasons:

1. The numerical simulation of oscillations and waves in continuous systems always comes down, ultimately, to computing a discrete system.
2. The structure of matter on a small scale is actually discrete, because it is made of atoms and quanta.

Deriving the equation

• First of all we must write the expression of the force acting on the mass located at position x. At any given instant t it equals the sum of the forces exerted by the two "springs" with which it interacts

${\displaystyle F(x+d)+F(x-d)=dF(x)=k[\xi (x+d)-\xi (x)]+k[\xi (x-d)-\xi (x)]}$

• The equation of motion for the mass located at point x is

${\displaystyle dF(x,t)=ma(x,t)=m\frac{d^2}{d{t}^2}x(t)=m\frac{{\partial }^2}{\partial t^2}\xi (x,t)}$

• Equating the two expressions we find

${\displaystyle k[\xi (x+d,t)-\xi (x,t)-\xi (x,t)+\xi (x-d,t)]=m\frac{{\partial }^2}{\partial t^2}\xi (x,t)}$

• Now we can introduce the characteristic quantities of the chain, rather than of a single component mass; in particular, let ${\displaystyle L=Nd}$ be the length of the chain, ${\displaystyle M=Nm}$ its mass, and ${\displaystyle K=k/N}$ its stiffness constant.

${\displaystyle \frac{{\partial }^2\xi (x,t)}{\partial t^2}=\frac{K{L}^2}{M}\frac{\xi (x+d,t)-2\xi (x,t)+\xi (x-d,t)}{d^2}}$

A limiting case: from the discrete chain to the continuous medium

The limiting case in which a very large number of masses is concentrated in a finite length L is very important. It corresponds to considering the system as a continuum rather than as a discrete set of oscillators. In short, we are moving from a microscopic view of an object to a macroscopic one. The object in question thus becomes a continuous elastic medium, such as a long spring or a thin bar. If we have done our calculations correctly, the equation for the system of oscillators must reduce to the wave equation, the same one that holds for longitudinal waves in the bar and for transverse waves on the string.

Mathematically, crowding the points together means reducing d and increasing N so that L and M stay constant. So we must compute the limit of the oscillator equation for

${\displaystyle N\to \mathrm{\infty },d\to 0}$.

We obtain

${\displaystyle \frac{{\partial }^2\xi (x,t)}{\partial t^2}=\frac{K{L}^2}{M}\frac{{\partial }^2\xi (x,t)}{\partial x^2}}$,

and we immediately see that the ratio ${\displaystyle \frac{K{L}^2}{M}}$ corresponds to the square of the propagation speed of waves in the continuous medium. Since ${\displaystyle \frac{M}{L}=\rho }$ is its density (the mass per unit length), we easily find two equivalent expressions for the speed

the first contains only macroscopic quantities

${\displaystyle c^2=\frac{K{L}^2}{M}=\frac{T}{\rho }}$.

where ${\displaystyle T=KL}$ has the dimensions of a force: it is the force that would double the length of the chain, if Hooke's law held that far. In the equation T takes the same place as the tension of the string, but its physical meaning is different: the chain carries longitudinal waves, and T corresponds to the product YA of Young's modulus and the cross-section of a bar. The second expression contains only microscopic quantities

${\displaystyle c^2=\frac{k{d}^2}{m}}$,

and relates the microscopic properties k, d and m of the single oscillator to the speed of waves in the macroscopic medium.

NOTE: we have also obtained a rule telling us how to approximate a second derivative in the continuum with a simple sum of terms in the discrete case

${\displaystyle \frac{{\partial }^2\xi (x,t)}{\partial x^2}\to \frac{\xi (x+d,t)-2\xi (x,t)+\xi (x-d,t)}{d^2}}$

This expression is the basis of numerical simulations of waves.

Harmonic waves in the discrete chain

Without taking the continuum limit, we can look directly for harmonic solutions of the chain equation, of the form ${\displaystyle \xi (x,t)=A\sin (qx-\omega t)}$, where q is the wavenumber. Substituting and using the addition formulas for the sine we find

${\displaystyle m{\omega }^2=2k[1-\cos (qd)]=4k{\sin }^{}\left(\frac{qd}{2}\right)}$,

that is

${\displaystyle \omega =2\sqrt{\frac{k}{m}}|\sin \left(\frac{qd}{2}\right)|}$.

• For long waves (${\displaystyle \lambda \gg d}$, that is ${\displaystyle qd\ll 1}$) the sine can be approximated by its argument, and we obtain ${\displaystyle \omega =\sqrt{k/m}dq=cq}$: the chain behaves like the continuous medium.
• When the wavelength becomes comparable with the distance d between the masses, the frequency grows less than proportionally to q: the speed of the waves decreases as the frequency increases, and the chain is dispersive.
• The frequency cannot exceed the maximum value ${\displaystyle {\omega }_{max}=2\sqrt{k/m}}$, reached for ${\displaystyle \lambda =2d}$, when each mass oscillates in antiphase with its neighbours. Waves of higher frequency cannot propagate in the chain.

The same frequencies appear in the program of the Fermi-Pasta-Ulam chain (variable omegak2).

The page on the Fermi-Pasta-Ulam chain has a Matlab program that solves the wave equation for a chain of oscillators, even in the presence of non-harmonic forces.

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