The corners that can be done with mirrors

A wall made by reflection puts one image behind a plane and the mirror that is a circle puts one inside a cylinder. Both work for the same reason, and the reason is not about fluids at all: the boundary is the fixed set of a reflection, the reflection generates a group of order two, and the flow can be made symmetric under it.

Put two walls together at an angle and there are two reflections. Two reflections generate a dihedral group, and the whole question is whether that group is finite.

When it closes, and how many #

Reflecting in one wall and then the other rotates by twice the angle between them. So repeating the pair rotates by $2\alpha$ each time, and the orbit returns to its start exactly when $2\alpha$ is a rational multiple of $2\pi$ with the right parity — which for a wedge of interior angle $\alpha$ means

$$\alpha = \frac{\pi}{n}, \qquad n = 1, 2, 3, \dots$$

and the group then has $2n$ elements. The images sit at $\theta = 2\pi k/n \pm \theta_0$, alternating in sign, and the count is exactly $2n$: two for a flat wall, four for a right angle, six for sixty degrees.

Both walls come out as streamlines to $2.5\times10^{-16}$ at every $n$ tried, which is round-off rather than a residual. The construction is exact and it is finite.

And when it does not #

At any other angle the reflections never return.

Take $\alpha = 1.1$ radians, which is 63 degrees — a perfectly ordinary corner, three degrees from one that works. Reflect the vortex repeatedly in both walls and the set of image positions grows without bound and becomes dense on the circle. The computation records nine images after four rounds, seventeen after eight, twenty-five after twelve, thirty-three after sixteen, and no sign of stopping.

The interesting number is what that buys. The worst normal velocity on the walls is 0.086 with nine images and 0.152 with thirty-three — it does not fall, and it does not fall because the truncated set is not an approximation to anything. There is no convergent infinite image system to truncate.

That is the refutation this essay carries, and its shape is worth noticing: the failure is not slow convergence, it is the absence of convergence, and a reader adding images and watching the residual wander would have no reason to suspect anything except impatience.

Why the count is what it is #

The arithmetic is worth doing once, because it makes the whole family memorable.

A vortex at angle $\theta_0$ inside a wedge $0 < \theta < \alpha$ is reflected in the wall at $\theta = 0$ to $-\theta_0$ with the opposite sign. That image is then outside the wedge and has to be reflected in the other wall, at $\theta = \alpha$, giving $2\alpha + \theta_0$ with the sign flipped back. Continue and the positions are

$$\theta = 2k\alpha \pm \theta_0,$$

with $+$ carrying $+\Gamma$ and $-$ carrying $-\Gamma$. The set closes when $2k\alpha$ is a multiple of $2\pi$ for some integer $k$, which needs $\alpha = \pi/n$, and it then has $n$ of each sign.

That is why the answer is a whole number and why it is exactly $2n$: the construction is the orbit of a point under a group, and a group’s orbit has the size of the group.

The flow exists anyway #

None of which means the problem has no solution. It has one, in closed form, at every angle.

The map $\zeta = z^{\pi/\alpha}$ takes the wedge to a half-plane, where one image is enough. Carrying the half-plane’s solution back gives a complex potential for the wedge, and the wall condition comes out at $10^{-16}$ or better at every angle tried, including 1.1 radians, 0.4 and 3.0.

So what fails at a general angle is not the physics and not the existence of a solution. It is the representation. The method of images is a statement about the symmetry group of the domain, and a domain whose group is infinite has a flow and no mirrors.

The comparison worth running is between the two routes where both exist. At $\alpha = \pi/3$ the mirrors and the map agree to $6\times10^{-16}$ at two hundred and forty points inside the wedge, which is the check that the finite construction is not merely closing the walls but solving the same problem.

That agreement is also the reason to trust the map at angles where the mirrors are unavailable. A method that reproduces another exactly wherever the other exists is a method that can be used where it does not.

The path, which needs no images at all #

There is a second closed form here and it is prettier than the first.

A vortex in a domain moves along a level curve of the Kirchhoff–Routh function, which for a simply connected domain mapped to the half-plane by $\zeta = f(z)$ is

$$H \propto -\ln\frac{2\,\mathrm{Im}\,f(z_0)}{|f'(z_0)|}.$$

Substituting $f = z^{\pi/\alpha}$ and cancelling, the level curves are

$$r\sin\frac{\pi\theta}{\alpha} = \text{constant}.$$

At $\alpha = \pi$ that is $y =$ constant, which is the familiar statement that a vortex runs parallel to a wall. At $\alpha = \pi/2$ it is $2xy =$ constant, the hyperbola in a right-angled corner. At every other angle it is a curve that needs no image construction to write down.

The closed form is checked against an independent route: the vortex is advected by the velocity its own image system induces on it, nine hundred steps of RK4, over a trajectory travelling nearly four radii, and the invariant is held to fourteen figures.

What that tells anybody about corners in general #

The path formula is worth reading for what it does at the extremes, because it says something about corners this collection has met before.

Near the apex, $r\sin(\pi\theta/\alpha)$ is small, so a vortex on any given level curve stays away from the corner: it comes in, turns, and goes back out along the other wall, and the closest approach is set by the constant. A vortex cannot be driven into a corner by its own images.

And the speed along the path involves $\pi/\alpha$, which diverges as the corner closes. A vortex in a narrow wedge is moving very fast, because it has a great many images very close to it. That is the same divergence nothing turns a sharp corner is about from the other side — the exact theory’s velocity at a salient corner is infinite — and it is the same $z^{\pi/\alpha}$ producing it.

The one case where the images are also the theory #

There is a corner where the two routes are not merely equal but the same statement, and it is worth identifying because it is the case everybody learns first.

For $\alpha = \pi$ — a single flat wall — the map is the identity, so the conformal route is the image route: one reflection, one image, nothing further. That is why the method of images feels elementary. It is elementary in the one case where the map is trivial.

For $\alpha = \pi/2$ the map is $z^2$, and the four images are the four preimages of the half-plane’s two. For $\alpha = \pi/n$ they are the $2n$ preimages. So the finite image system is not a separate technique at all: it is what the conformal solution looks like when the map is a power with an integer exponent, which is precisely when the preimages are finite in number.

That reframing is the useful one to carry, because it makes the failure at other angles obvious rather than surprising. $z^{\pi/\alpha}$ with an irrational exponent has infinitely many branches, so the preimage of a single image point is an infinite set.

What is really being asked when a boundary is imposed #

Stand back from the wedge and the general question is worth stating, because this collection imposes boundaries constantly.

A boundary condition on a potential flow is a requirement that a harmonic function have a prescribed normal derivative on a curve. There are three ways to meet it. Reflect, when the curve is the fixed set of a symmetry the equation respects. Map, when the curve is the image of an easier one under a conformal transformation. Or distribute — cover the boundary with sources and vortices and solve for their strengths — which is what a panel method does and which works for any curve at all.

The three are in increasing order of generality and decreasing order of elegance. Images are exact, finite and available almost never; maps are exact and available whenever a map is known; bodies made of nothing is the site’s essay about the third, where a body is not a body at all but a distribution of singularities whose stagnation streamline happens to close.

The wedge is the cleanest place to see the boundary between the first two, because the same problem sits on one side of it or the other depending on whether a number is $\pi$ over an integer.

Why the sign alternation matters #

One detail of the image system deserves a paragraph, because getting it wrong produces a plausible picture that satisfies neither wall.

The images alternate in sign round the circle: $+\Gamma$ at $2\pi k/n + \theta_0$ and $-\Gamma$ at $2\pi k/n - \theta_0$. That is not a convention. A reflection reverses orientation, so it reverses the sense of a vortex, and an image whose sign was not flipped would make the wall a source line rather than a streamline.

The consequence for a wedge of odd $n$ against even $n$ is worth noticing. For $n$ even, each wall’s reflection maps the set to itself with the signs consistent; for $n$ odd it still does, because the group has $2n$ elements and the sign is determined by whether an element is a rotation or a reflection. What fails at a non-integer $n$ is exactly this: an orbit that never closes cannot be consistently two-coloured, and there is no way to assign signs at all.

So the failure is not merely that there are too many images. It is that the sign assignment the boundary condition requires does not exist.

The limit, and the discontinuity in it #

Now the question these essays share. Take the wedge angle to zero.

The image count is $2n = 2\pi/\alpha$, which diverges: a ten-degree corner needs thirty-six images and a one-degree corner three hundred and sixty. The flow does not do anything dramatic — the conformal map handles a thin wedge as easily as a fat one, and the path formula stays a formula.

What is discontinuous is the method’s existence. At $\alpha = \pi/18$ the image system is finite, exact and thirty-six vortices long. At $\alpha = \pi/18 + 10^{-6}$ it does not exist at all. The two flows differ by a millionth everywhere.

That is a different shape of residue from the rest of these essays, and it is worth having in the set. The usual pattern is that a limit removes something and leaves a quantity behind that diverges or refuses to vanish. Here the limit is perfectly smooth in the physics and destroys a technique, discontinuously, on a set of angles of measure zero.

What it says about the exact theory’s toolkit #

The collection’s account of ideal flow is a toolkit: superposition, images, maps, distributions, theorems about far fields. It is worth being clear which of those are properties of the equations and which are properties of particular geometries.

Superposition is the equation’s — Laplace’s equation is linear, and flows add up is that fact and its one exception, which is that the pressures do not. The far-field theorems are the equation’s too.

Images are the domain’s, and they exist for a wall, a circle, a sphere, a wedge of $\pi/n$, and essentially nothing else. Conformal maps are the domain’s as well, but two-dimensionally so: three dimensions are kinder records the price of losing them, which is that a sphere’s surface speed is $1.5U$ rather than $2U$ and that no complex-variable theory exists at all above two dimensions.

A reader who remembers images as a general method will eventually meet a corner at 63 degrees and spend an afternoon adding vortices.

What a reader should take from a method that nearly always fails #

It would be easy to finish with a shrug — images are a party trick, conformal maps are the real machinery — and that is not quite right either.

The image construction has one property the map does not: it is local and physical. Each image is a vortex, its contribution is a velocity, and the sum can be read as a superposition of things a reader can point at. That is why it survives in teaching and why it survives in practice for the two or three geometries it covers, which happen to be the two or three that come up: a wall, a circle, a right-angled corner.

The map is more powerful and less interpretable. Nothing in the $\zeta$-plane corresponds to an object in the flow; the solution is correct and there is nothing to point at.

And there is a third thing the wedge teaches, which is about the collection’s habit of asking what a result is a property of. The number $2n$ is a property of a group. The wall condition being satisfied is a property of the flow. The two coincide on a set of angles and nowhere else, and a reader who has noticed the coincidence has understood the method better than one who has memorised the recipe.

Bodies made of nothing is the essay where this collection first builds a body out of singularities and finds that the body was never there — only a streamline that happened to close. The wedge is the same lesson about a technique: the images were never there either. They are the group, written as vortices.

What the same question looks like in three dimensions #

A short note on the generalisation, because the two-dimensional answer is unusually tidy and the three-dimensional one is famous for a different reason.

Reflections in two planes meeting at an angle generate a dihedral group in three dimensions as well, and the finiteness condition is the same: the angle must be $\pi/n$. What changes is that there are many more ways to arrange planes, and the finite groups of reflections in three dimensions are exactly the symmetry groups of the regular solids and their relatives — the same classification that governs which crystals can exist.

So the corners a three-dimensional image method can handle are the corners of a crystallographic arrangement, and everything else has an infinite orbit. The restriction is more severe rather than less, because a three-dimensional corner has more angles to get right.

And there is no conformal map to fall back on. Three dimensions are kinder records the price of that from the other side — a sphere’s surface speed is $1.5U$ rather than a cylinder’s $2U$, and the whole complex-variable toolkit is gone.

Limits recorded rather than smoothed over #

The wedge is a two-dimensional corner with straight walls. Nothing here says anything about a three-dimensional corner, where the group of reflections is a spatial one and the finite cases are the crystallographic ones.

The vortex is a point. Everything in what a point vortex is not applies here too: the images strain the vortex, and a real core near a corner deforms. The path formula is a point vortex’s path.

And the truncation experiment is not a proof. Showing that the residual fails to improve over thirty-three images at one angle is evidence that the construction does not converge, not a demonstration that it cannot. The demonstration is the group-theoretic one, and it is a sentence rather than a computation: the orbit of a point under a dihedral group of infinite order is infinite.

The residue #

A method that is exact when it applies and absent when it does not is a strange object to have in a toolkit, and the wedge makes it visible.

What the limit throws away, as the corner closes, is not accuracy. It is generality: the set of angles on which the finite construction exists has measure zero in the set of angles a corner can have, and the construction’s exactness on that set is the reason nobody notices.