How a chain of oscillators becomes a field

A field describes many local degrees of freedom with one function. A stretched string makes the idea tangible: every small piece can move, and neighboring pieces pull on one another. When we care about waves much longer than the distance between those pieces, their displacements can be described by a smooth function. The field equation keeps the motion that survives at that resolution.

That last condition matters. A continuum description earns its usefulness by making accurate predictions for a chosen range of wavelengths. We can see both its success and its failure in a chain of springs.

Start with a piece you can pull

Put identical masses along the horizontal coordinate $x$, with spacing $a$. Let $q_j(t)$ be the transverse displacement of mass $j$, positive upward. Both $x$ and $q_j$ have units of length. Write the mass of one piece as $\mu a$, where $\mu$ is mass per unit length. The spring between adjacent pieces has stiffness $\kappa/a$, where $\kappa$ has units of force. A small spring to the support supplies an additional restoring force with angular frequency $\Omega$.

The two neighboring springs pull a displaced piece toward their average position. Newton’s law gives

The dots mean time derivatives. The first derivative is the rate at which displacement changes; the second is its acceleration. The combination in parentheses measures curvature along the chain. It vanishes for a straight, uniformly tilted row of displacements.

The on-site springs are optional. They will become useful because they give even a very long wave a nonzero oscillation frequency.

A traveling wave reveals the approximation

Try a sinusoidal displacement $q_j=A\cos(kja-\omega t)$. Here $A$ is an amplitude, $k=2\pi/\lambda$ is the wave number, $\lambda$ is the wavelength, and $\omega$ is angular frequency. Advancing one site shifts the phase by $ka$. Substituting the trial wave into Newton’s law and using the addition formula for cosine gives

This relation between frequency and wave number is the dispersion relation. It tells us which rhythm accompanies each spatial pattern. A Fourier expansion means building a more complicated displacement by adding these sinusoidal patterns. Because the equation is linear, each pattern evolves independently.

For a long wave, the phase changes little from one site to the next. Then $\sin(ka/2)$ is close to $ka/2$, and the relation becomes

The figure compares these two predictions in reference units. They agree near the left side because neighboring sites see nearly the same phase. Near the right side, the lattice resolves the wave’s alternating peaks and troughs, and its frequency follows the full sine function.

A numerical example

Take $a=0.100\,\mathrm{m}$, $c=2.00\,\mathrm{m\,s^{-1}}$, and $\Omega=3.00\,\mathrm{s^{-1}}$. A wave with $\lambda=2.00\,\mathrm{m}$ has $k=\pi\,\mathrm{m^{-1}}$. The discrete equation gives about $6.94\,\mathrm{s^{-1}}$; the continuum expression gives about $6.96\,\mathrm{s^{-1}}$. The difference is roughly a third of a percent.

Keep the material the same and shorten the wavelength to $0.200\,\mathrm{m}$. Adjacent sites now have opposite phases. The discrete frequency is about $40.1\,\mathrm{s^{-1}}$, and the continuum prediction is about $62.9\,\mathrm{s^{-1}}$. The continuum approximation applies to substantially longer waves.

Replace many coordinates with one function

Define a smooth displacement $\phi(x,t)$ that agrees with $q_j(t)$ at the lattice sites. Taylor expansion expresses a nearby value using the value and successive derivatives at the chosen position. The linear terms in the expansions at $x+a$ and $x-a$ have opposite signs and cancel; the quadratic terms add. Keeping the next surviving term gives

The symbol $\partial_x$ means differentiating with respect to position while holding time fixed. The first retained term gives the continuum field equation,

The omitted fourth derivative explains the discrepancy we just calculated. For a Fourier mode it brings an additional factor of wave number squared relative to the leading gradient term. Reducing the lattice spacing helps a fixed long wave; it does not make every wave on the lattice equally accurate.

We have changed the description from a list of coordinates to a function over space. We still have a classical system. Quantization will come after we understand its independent modes and conserved quantities.

The same equation follows from energy

The Lagrangian is kinetic energy minus potential energy. For the smooth field it is the spatial integral of a density,

An integral adds the contributions of small intervals. Each term in this density has units of energy per unit length. The gradient term stores energy when neighboring displacements differ; the final term stores energy when the field moves away from the support.

To find the equation of motion, vary the field by a small amount that vanishes at the time endpoints and at fixed spatial boundaries. Integration by parts reverses the product rule inside an integral: it moves a derivative to the other factor and supplies an endpoint term. Here that endpoint term vanishes. Requiring the first-order change of action to vanish for every allowed field variation gives

The three terms become $\mu\partial_t^2\phi$, $-\kappa\partial_x^2\phi$, and $+\mu\Omega^2\phi$. Dividing by $\mu$ recovers Newton’s-law result. The action is useful because the same method extends to several fields and exposes their symmetries.

Compare two mode shapes

Open the field-modes Lab, hold the occupation at one and compare mode numbers one and three. Count the internal nodes, the positions that stay at zero displacement. Predict the frequency and wavelength ratios before changing the mode. In this fixed-end, massless example the third mode has two internal nodes, three times the frequency and one third of the wavelength. The numerical spring chain above is a separate example with an on-site restoring force and finite-spacing corrections.

Exercises

What changes when the masses get heavier?

Double $\mu$ while keeping $\kappa$, $a$, and $\Omega$ fixed. What happens to the frequency of a nonuniform mode? What happens to the uniform mode?

How would you test a continuum calculation?

A program matches the continuum dispersion for one wavelength. Design a second check that could reveal a lattice error without changing the physical wavelength or length of the chain.

Can a single coordinate describe the field?

Suppose every point is displaced by the same amount at every time. Why can this special motion be described by one oscillator, even though the full field has many degrees of freedom?