6.5 Modes and mode sums

A mode is a special solution of a linear PDE on a bounded domain: a spatial shape that oscillates in time at a single frequency, without changing form. The boundary conditions (6.4) select a discrete catalogue of these shapes, and the technique of separation of variables (6.3) hands us the catalogue automatically. This lesson is about what to do with that catalogue.

Modes are the natural language for bounded-domain problems for three reasons. First, every solution of a linear PDE with linear boundary conditions can be written as a sum over modes, so the catalogue is complete. Second, distinct modes are orthogonal under a natural inner product, which means the coefficients in the sum can be extracted one at a time. Third, the spectrum of mode frequencies tells you almost everything you need to know about how the system responds to forcing — what frequencies it amplifies, what it filters out, what bandwidth it has at each peak. The connection between mode catalogues and Fourier analysis is direct: a mode expansion is a generalised Fourier series, and what carries over is exactly what we develop in Foundations 7.

Modes, made visible

Pick a 1-D tube with a chosen pair of boundary conditions and look at the modes.

Pick a boundary combination (closed–closed, closed–open, open–closed, open–open) and a mode number $n$. The plot shows the spatial shape $X_n(x)$ — the displacement of the air column at the instant of maximum displacement — along with the wavenumber and the frequency the mode carries. Three things to take from playing with this:

• Each boundary combination has its own discrete ladder of frequencies. The fundamental and the spacing between modes are not the same for a closed–closed tube as for an open–open one, and the closed–open tube interlaces modes that don’t exist in either symmetric case. The boundary conditions write the entire mode catalogue.
• Mode shapes are sinusoids inside the tube. The interior structure is sin or cos, depending on the boundary type. The boundary points themselves are either nodes (where the shape is zero) or antinodes (where it is at a maximum); each end is one or the other depending on whether the BC was Dirichlet or Neumann for this field.
• Mode number $n$ counts the half-wavelengths inside the tube. Higher $n$ means higher frequency and more wiggles per length. The fundamental $n = 1$ is the lowest-frequency mode the geometry can support.

The interactive shows one mode at a time. The actual motion of the air is a superposition — a weighted sum of many modes oscillating simultaneously at their own frequencies. The rest of this lesson is about that sum.

Orthogonality

The reason mode expansions work — the reason the coefficients in $u(x, t) = \sum_n c_n X_n(x) T_n(t)$ can be pulled out one at a time — is that distinct mode shapes are orthogonal under the natural inner product on the domain. For the clamped string of length $L$:

where $\delta_{mn} = 1$ if $m = n$ and $0$ otherwise. The relation can be derived from the trigonometric product identity $\sin A \sin B = \tfrac12[\cos(A-B) - \cos(A+B)]$ and direct integration — see the Foundations 1 integral toolkit. Analogous orthogonality relations hold for the Neumann case (cosines), for the periodic case (full Fourier basis), and for the Robin case (Sturm–Liouville eigenfunctions).

Orthogonality is not a coincidence. Each mode shape $X_n(x)$ is an eigenfunction (refresher →) of the spatial differential operator that appears in the PDE — for the wave equation, this is $-\partial_x^2$, and the eigenvalue is $k_n^2$. The operator with the standard boundary conditions is self-adjoint (the differential analogue of “Hermitian”), and a foundational theorem of linear algebra extended to function spaces guarantees that the eigenfunctions of a self-adjoint operator with distinct eigenvalues are orthogonal. This is the Sturm–Liouville theorem and the function-space generalisation of the spectral theorem →.

Fourier projection: how to extract a coefficient

Given a function $f(x)$ on $[0, L]$, suppose we want to write it as a sum of clamped-string modes:

To find the coefficient $c_n$ of a particular mode, multiply both sides by that mode shape and integrate:

Orthogonality kills every term except $m = n$, which contributes $c_n \cdot L/2$. Therefore

This is the Fourier-projection formula, the workhorse for matching initial data to a mode expansion. The same recipe — multiply by one basis element, integrate, divide by the basis element’s self-inner-product — extracts coefficients for any orthogonal basis. The cosine case (Neumann boundaries) gives the same formula with $\sin \to \cos$ and a factor of $2/L$ replaced by $1/L$ for the $n = 0$ mean term. The periodic case gives the standard Fourier series. The Robin case requires the appropriate Sturm–Liouville inner product.

Once you have the $\{c_n\}$, you have completely specified the solution: each mode $X_n$ carries its own time evolution $T_n(t)$ (oscillating for the wave equation, decaying for the heat equation, time-independent for Laplace), and the full field is the sum.

Completeness

Orthogonality alone is not enough — there are plenty of orthogonal sets that fail to span the function space you care about. The other half of the Sturm–Liouville guarantee is completeness: any sufficiently well-behaved function on the domain can in fact be expanded in the mode basis, with the resulting series converging (in the mean-square sense) to the function. There is no “leftover” piece of $f$ that the mode catalogue cannot account for.

What “sufficiently well-behaved” means depends on the details — piecewise continuous with finitely many jumps is a typical sufficient condition — but for any function you are likely to encounter in physics, completeness holds. This is what lets us treat the modes as a basis, in the same sense that $\{(1, 0), (0, 1)\}$ is a basis for $\mathbb{R}^2$.

The deeper version of this fact, that the eigenfunctions of a self-adjoint operator on a function space form a complete orthonormal set, is the spectral theorem. It is one of the central results of mathematical physics and the reason quantum mechanics, separation of variables for PDEs, and Fourier analysis all share the same mathematical skeleton.

Modal density

Once you have a catalogue of modes you can ask: how many modes fall into a given frequency band? The answer is the modal density $D(\omega)$, defined so that the number of modes in $[\omega, \omega + d\omega]$ is $D(\omega)\, d\omega$. Modal density matters whenever the system has many modes inside the band of interest, which is the regime where the details of each individual mode stop mattering and statistical properties of the mode catalogue take over.

The simplest case: a 1-D string of length $L$ with clamped ends has modes at $\omega_n = n \pi c / L$, evenly spaced by $\Delta \omega = \pi c / L$. The modal density is

independent of frequency. In two dimensions (a rectangular membrane) the modes are labelled by two integers $(m, n)$, and the modal density rises linearly with $\omega$. In three dimensions (a rectangular cavity) it rises quadratically:

with $V$ the cavity volume. This is the foundation of room acoustics — above a certain frequency, called the Schroeder frequency, a typical room has so many overlapping modes that the field looks statistically uniform rather than mode-resolved. We develop this picture in Sound 7.8 — Room modes and modal density.

The same $\omega^2$ modal density of a 3-D cavity also underwrites the Rayleigh–Jeans law for blackbody radiation, Debye’s theory of specific heat of solids, and the Bekenstein–Hawking entropy of black holes. It is one of the most-recycled formulas in physics.

The forced-mode picture

So far we have built mode sums for the homogeneous PDE (no driving term). When you add a sinusoidal forcing — say a loudspeaker driving an acoustic cavity at frequency $\omega_d$ — each mode responds independently as a damped, driven oscillator at frequency $\omega_d$, with its own natural frequency $\omega_n$ and its own damping $\gamma_n$. The amplitudes follow the forced-oscillator formula from 5.3,

and the total response is the sum across modes weighted by how strongly the forcing couples to each. The mode with $\omega_n$ closest to $\omega_d$ dominates — provided $\gamma_n$ is small enough that the resonant peak is narrow. The system as a whole therefore acts as a bank of resonators, one per mode.

This is the central picture of frequency-domain acoustics, and one of the central pictures of hearing physiology: the cochlea is, to a useful approximation, a bank of mechanically resonant filters arranged along the basilar membrane (Hearing Ch 4), each one a mode of a slightly different damped, driven oscillator.

Where modes go from here

The mode picture and the Fourier picture are two faces of the same object. In Foundations 7 we develop the Fourier transform as the continuous limit of the mode catalogue — what happens when the domain becomes unbounded and the discrete ladder $\{\omega_n\}$ becomes a continuous variable $\omega$. The orthogonality relations become Dirac-delta normalisations and the mode sums become integrals, but the structure is the same. Anything you know about mode expansions transfers to the Fourier transform by replacing sums with integrals.

The final lesson, 6.6 — The heat equation and Laplace’s equation, shows how the same mode catalogue (selected by the same boundary conditions) governs parabolic and elliptic problems — with the time-evolution piece $T_n(t)$ being the only thing that changes between the three canonical PDEs.