6.4 Boundary conditions

Separation of variables — and almost every other PDE technique — produces a family of allowed solutions parameterised by some constant. The boundary conditions are what cut that family down to the actual physical answer. They are also the most common source of errors in PDE work: a problem with the wrong boundary conditions is, mathematically, a different problem, and will give the wrong modes, the wrong reflection coefficients, the wrong steady-state, the wrong everything.

This lesson lays out the four families of boundary conditions you will meet across the bookshelf, the physics that selects each one, and a worked example showing what happens when you choose wrong.

What a boundary condition is

A PDE describes the relation between a field and its partial derivatives at every interior point of a domain. By itself, the equation has infinitely many solutions — any combination of mode shapes, any choice of $F$ and $G$ in the d’Alembert formula, any harmonic interior. The boundary conditions specify the value of the field (or some derivative of it) on the boundary of the domain, and together with the PDE they cut the infinite family down to a unique solution.

For an elliptic PDE that is the whole story. For a parabolic PDE you also need an initial condition; for a hyperbolic PDE you need two (initial value plus initial velocity). But the boundary conditions are the part that depends on what the walls of the domain are like.

What kind of condition is physically appropriate depends on what the field represents and what the boundary represents:

• A fixed string end holds the displacement at zero.
• A closed pipe end holds the air’s longitudinal velocity at zero (the air can’t move through a wall) — but the pressure at that point is at its maximum.
• An open pipe end holds the pressure at atmospheric (perturbation = 0) — but the displacement of the air at that point is at its maximum.
• A thermally-conducting boundary holds the temperature at the wall temperature.
• An insulated rod end holds the flux (heat current) at zero — i.e., $\partial_x T = 0$.

The same boundary that is Dirichlet for one variable is often Neumann for another. Naming the field whose boundary condition you are stating is half the battle.

The four families

Dirichlet — value fixed

A Dirichlet boundary condition specifies the value of the field at the boundary:

usually $\alpha = 0$ when the equation is homogeneous (so the problem stays homogeneous). Physical examples:

• A string clamped at both ends. Transverse displacement is zero at both fixed points.
• A closed organ pipe end (for the pressure field). Wait — no: at a closed end the pressure is at a maximum, not zero. The Dirichlet “pressure = 0” condition is for an open end, where the pressure has equilibrated with the atmosphere outside. Be careful which end is which.
• A thermally-conducting boundary. Wall temperature is specified; conduction holds the boundary fluid temperature equal to it.
• A grounded conductor. Electrostatic potential is fixed at zero (or at the ground potential).

The corresponding mode shapes vanish at the boundary. For the clamped string on $[0, L]$ these are $\sin(n \pi x / L)$ — the modes we derived in 6.3.

Neumann — normal derivative fixed

A Neumann boundary condition specifies the normal derivative of the field:

with $\beta = 0$ in the homogeneous case. Physical examples:

• A free string end. The slope of the string at a free end carries no force (no tension perpendicular to the string), so $\partial_x u = 0$ at the free end.
• A closed organ pipe end (for the pressure field). At a hard wall the air’s longitudinal velocity must vanish — and since the velocity is proportional to $\partial_x p$ via Euler’s equation, this is Neumann for $p$: $\partial_x p = 0$.
• An open organ pipe end (for the longitudinal displacement field). The air is free to move there, so $\xi$ is at a maximum, $\partial_x \xi = 0$.
• An insulated rod end. No heat current flows out, so $\partial_x T = 0$.
• A conductor’s surface in electrostatics for the normal electric field $\partial_n V$, set by the surface charge density.

The corresponding mode shapes have zero slope at the boundary. For a free–free string on $[0, L]$ they are $\cos(n \pi x / L)$ — the same frequency spectrum as the clamped case (modulo the $n = 0$ rigid translation), but the spatial shapes are cosines rather than sines.

Robin (impedance) — linear combination fixed

A Robin boundary condition specifies a linear combination of $u$ and its normal derivative:

at the boundary. Robin is the generic boundary condition: Dirichlet is the special case $\beta = 0$ and Neumann is $\alpha = 0$. Physical examples are everywhere:

• An acoustic impedance termination. A real wall is not perfectly hard and not perfectly soft. The local relationship between pressure and normal velocity at the wall is the wall’s acoustic impedance $Z$, so $p = Z\, v_n$ — and since $v_n \propto \partial_n p$, this is a Robin condition on $p$. Real acoustic terminations always look like this; perfect-Dirichlet and perfect-Neumann are mathematical idealisations.
• Newton’s law of cooling at a thermal boundary: heat flux out is proportional to temperature difference, $-k\, \partial_n T = h\, (T - T_\text{outside})$. This is Robin for $T$.
• A resistive electrical boundary between a conductor and a load impedance.

The corresponding mode shapes solve a transcendental quantisation equation rather than the clean $k L = n \pi$ of the Dirichlet or Neumann cases — the impedance ratios enter the equation and the modes are generally not exactly sinusoidal.

Periodic — wrap-around

A periodic boundary condition identifies opposite faces of the domain:

The domain has no free boundary — it wraps around, like a ring. Physical examples:

• Born–von Kármán conditions in solid-state physics, used to define electron states in a crystal of size $L$.
• A ring resonator in acoustics or optics — sound or light that can travel around the ring repeatedly.
• The angular coordinate in any problem with cylindrical or spherical symmetry, where $\theta$ and $\theta + 2\pi$ are physically the same point.

The corresponding mode shapes are the full Fourier set $\{e^{i k_n x}\}$ with $k_n = 2 \pi n / L$ for integer $n$ (positive and negative), or equivalently $\{1, \cos(2\pi n x / L), \sin(2\pi n x / L)\}$.

Boundary conditions on a 2-D or 3-D domain

In one spatial dimension the boundary is two points. In two dimensions it is a curve; in three it is a surface. The same four families apply at every point of the boundary, but you can have different types on different stretches. A typical example: a rectangular plate with one edge held at a fixed temperature (Dirichlet), an adjacent edge insulated (Neumann), and the remaining two edges in contact with a cooling bath (Robin). The interior temperature distribution that emerges from those mixed boundary conditions can be quite intricate; that is what the elliptic problem is for.

When you set up separation of variables on a higher-dimensional domain, you separate the spatial variables further — $u(x, y, t) = X(x)\, Y(y)\, T(t)$ — and each variable’s boundary conditions select its own ladder of modes. The full mode catalogue is the outer product: $\{X_m\}_{m=1}^\infty \otimes \{Y_n\}_{n=1}^\infty$.

A cautionary worked example: wrong BC, wrong modes

Imagine an organ pipe open at both ends, like an orchestral flute. We want its acoustic modes.

The physically correct setup. Pressure is the natural field. At an open end the pressure perturbation is approximately zero — the air at the opening is in contact with the atmosphere, which buffers any fluctuation. The boundary conditions on pressure are therefore Dirichlet at both ends:

Separation of variables gives mode shapes $p_n(x) \propto \sin(k_n x)$ with $k_n = n \pi / L$ and frequencies $\omega_n = n \pi c / L$. The fundamental is $\omega_1 = \pi c / L$ — the standard open–open flute formula. Half a wavelength fits between the two ends in the lowest mode.

A common mistake. Suppose we instinctively write boundary conditions in terms of the displacement field $\xi$ rather than the pressure, and then carry over the same Dirichlet statement: $\xi(0, t) = \xi(L, t) = 0$. This says the air at each open end cannot move, which is the opposite of the truth — the air at an open end is most free to move. We have just specified the physics of a closed–closed tube while believing we were specifying an open–open one.

The mode shapes that come out of this wrong setup are $\xi_n(x) \propto \sin(k_n x)$ with the same $k_n = n \pi / L$. The frequencies are the same as the correct open–open case, by accident of symmetry. So a frequency-only sanity check would pass. But the spatial shapes are wrong — they have nodes where the correct shapes have antinodes — and any subsequent calculation that uses the field’s behaviour at the boundary (intensity, impedance, radiation pattern, coupling to a player’s lip) would carry the wrong sign or magnitude.

The correct displacement statement. The right boundary condition for $\xi$ at an open end is Neumann: the air can move freely, so $\partial_x \xi = 0$. With Neumann on $\xi$ the mode shapes are $\xi_n(x) \propto \cos(k_n x)$ with $k_n = n \pi / L$ — pressure nodes at the ends become displacement antinodes, exactly as physics demands. Same frequencies, correct shapes.

The moral. Before you write a boundary condition down, name the field whose boundary value you are stating and ask which of the four families fits the physics at that point for that field. Pressure at a free surface is Dirichlet; displacement at the same surface is Neumann. These are dual statements of the same physical fact, but they translate into mathematically different equations. Get the type right.

The full acoustic-tube treatment is in Sound 7.6 — Tube modes, which has tables of the modes for every endpoint combination (closed–closed, open–closed, open–open, with and without flanges).

Why the BC choice matters for completeness

There is one further reason to care about the boundary conditions, which only becomes apparent when you try to use the modes for a calculation. A solution by separation of variables looks like

with the $X_n$‘s the mode shapes selected by the BCs. To match arbitrary initial conditions $u(x, 0) = f(x)$, you need to expand $f$ in the basis $\{X_n\}$. That expansion will work — for any sufficiently well-behaved $f$ — if and only if the $X_n$‘s form a complete orthogonal set on the domain.

This is not automatic. It is automatic for the four standard families above, because each leads to a self-adjoint spatial operator (the formal name for “an operator with nice eigenfunctions”) and the Sturm–Liouville theorem then guarantees the eigenfunctions are complete and orthogonal. Choose a non-standard boundary condition that is not self-adjoint and the mode catalogue may not span the space of physical initial conditions — leaving the technique stranded.

In the rest of the bookshelf you will always be in the standard cases. This is mentioned here so that the reason separation of variables works is visible, not just the mechanics.

The next lesson, 6.5 — Modes and mode sums, develops the orthogonality, the Fourier-projection step that pulls out the coefficients $\{c_n\}$, and the modal-density bookkeeping that becomes important when there are many modes within a band of interest.