**Babinet's
Principle**

**Scattering
from Disks and Apertures**

**Introduction**

Born and Wolf present two derivations of Babinet's principle for the electromagnetic case, an approximate version, and a rigorous version. The sound and electromagnetics approximate derivations are similar. The rigorous derivations differ in some respects, as discussed below. The primary focus in this section is on sound waves; the term "field" is used here as a generic term for a solution of a wave equation.

**Approximate Version**

Here we assume the "hard" boundary condition
for a screen or a disk: the normal component of molecular velocity **u**
is zero everywhere on the screen or disk surface. The approximation is that
on the remainder of the plane the velocity is the same as an undisturbed incoming
plane wave. In general, this approximation is accurate when the size of the
aperture or disk is large compared to a wavelength. Given this approximation,
the solution for the scattered field is given in the section on the general
boundary integral solution of the wave equations, equation
G18.

For problem A the the normal velocity is zero everywhere on the surface except for the aperture, where it is the same as a plane wave. For problem B the normal velocity zero on the aperture and the same as a plane wave everywhere else. Define problem C as the sum of the solutions of problem A and problem B. Thus the boundary field for problem C is simply the undisturbed plane wave, and the solution is the undisturbed plane wave for all z ≥ 0. Thus given the solution to A, the solution to B is the plane wave minus the solution to A. That, in a nutshell, is the approximate version of Babinet's principle.

A somewhat subtle point regarding scattering problems
involves the distinction between the "scattered field" and "total
field." The total field is equal to the scattered field __plus__ the
incident field. Consider a disk lying in the plane z=0, with the z-axis passing
through its center, and a plane wave incident from z=-∞. The disk scatters
a pressure pattern that is anti-symmetric with respect to z, i.e. exactly the
same scattered pressure in front, and behind, the disk, except for a sign change.
When the incident plane wave is added in to yield the total field, near the
z-axis it tends to cancel the scattered pressure for z ≥ 0, creating a
"shadow" region immediately in front of the disk. It tends to reinforce
the pressure for z ≤ 0. Babinet's principle states that the scattered
field of the disk equals the total field of the screen for z ≥ 0. The
field for z ≤ 0 is deduced from the symmetry properties of the scattered
fields.

The clearest example of this is the limiting case
of a screen with __no__ aperture. The screen scatters a plane wave in both
the positive and negative z-directions. When the incident plane wave is added
in, it exactly cancels the scattered plane wave for z ≥ 0. For z ≤
0 the total field consists of the incident and reflected plane waves. At z=0
the pressure is doubled on the z ≤ 0 side of the screen, and is zero on
the z ≥ 0 side of the screen. Babinet's principle says that the scattered
field of the complementary disk, which in this case has zero radius and zero
scattered field, equals the zero total field of the screen in z ≥ 0. It
is __not__ equal to the total field of the screen in z ≤ 0.

The solution to the piston problem does not involve
an incident field, and the resulting pressure is essentially the same as the
scattered field of the aperture for z ≥ 0. The normal component of the
velocity **u** equals the piston velocity at the face of the piston,
and as noted, the solution assumes that the piston excursion is negligible.
This differs from the aperture case where the velocity at z=0 really is the
velocity of the incident wave, as long as the size is large compared to a wavelength.

**The "bright spot"**

In the limiting case of a disk of zero radius the scattered field is zero, and the total field for z ≥ 0 is the incident plane wave. For a finite disk the total field still includes the incident plane wave. This causes the "bright spot" in the diffraction pattern of a disk. There is a shadow region in front of the disk, but exactly in the center the shadow is punctuated by a bright spot, in the case of light diffraction, or high pressure, in the case of sound.

**Rigorous Version I**

In electromagnetics the "scattered field" is generated by an electrical and/or magnetic current sheet flowing in the screen or disk. One normally does not think of sound waves arising from a current sheet, but the scattered field behaves as if this were true.

For the screen, problem A, we assume the usual "hard" boundary condition for sound waves: the normal component of molecular velocity is zero on the surface of the screen. For the disk, problem B, we assume the more-or-less theoretical "soft" boundary condition: the pressure is zero on the surface of the disk. These correspond to the electromagnetic cases of a perfect electrical, and magnetic, conductor respectively.

For a hard screen in the plane z=0, the scattered pressure is an odd function of z, and the scattered field z-component of molecular velocity is an even function of z. For a soft disk the symmetry is reversed.

Let ρ denote the cylindrical coordinate radius, and let a denote the radius of the hole and disk. A superscript I, S or T denotes incident, scattered, and total fields respectively; a subscript A or B denotes problem A or problem B. The variable u denotes the z-component of molecular velocity, and all references to "velocity" refer to the normal component. The variable p denotes pressure.

A rigorous boundary value solution must satisfy two conditions: (1) the scattered field must be a solution of the wave equations (the incident field is of course assumed to be a solution); and (2) the total field must satisfy the boundary conditions.

On the surface of the hard screen and soft disk the boundary conditions for the total field at z=0 are:

Away from the scattering surfaces the boundary conditions are that both velocity and pressure are continuous. The incident wave is continuous everywhere. The scattered fields that are an even function of z will be continuous at z=0, but the fields with odd symmetry must be zero at z=0, in the regions on the plane away from a scattering surface:

Suppose we have a solution for problem A. Comparing equations (1) and (2), if the incident field is the same for both problems, and if we have, at z=0

Then the fields of problem B will also satisfy the boundary conditions at z=0. If the incident field sources are in z < 0, we can set the scattered field of the disk equal to the total field of the screen for all z ≥ 0, and we have a solution to problem B for z ≥ 0. As noted above, the scattered fields for z ≤ 0 for problem B are then deduced from the symmetry relations for scattered fields.

This version of Babinet's principle is the sound
analogue of the electromagnetic version. Clearly the boundary conditions for
the screen and disk can also be interchanged. In the derivation given by Born
and Wolf the disk and screen are both perfect electrical conductors, and the
incident field polarization is rotated 90^{0} between problems A and
B. Babinet's principle is presented in the form of the total electric fields
in problem A plus the total magnetic fields in problem B. Using the symmetry
property of electromagnetic fields this is essentially equivalent to the derivation
given here. The problem in the case of sound is that there is no direct analog
for the case of a "hard" boundary condition for __both__ the screen and the
disk, which is really the most interesting case. That is the subject of the
next section.

**Rigorous Version II**

This derivation is unfortunately a bit messier than the previous one. We can represent any sound wave in the region z ≥ 0 as a summation of cylindrical waves, in terms of the potential function

where k is the free space propagation constant, f_{0}=1,
and β_{0}=0. If the values of β_{n }are equally spaced,
as the spacing approaches zero and N→ ∞ this summation approaches
a continuous wave spectrum that is a general representation of any sound wave
traveling in the +z direction. (See the section on Wave
Spectra). Pressure is proportional to φ , and the z-velocity component
is proportional to the partial derivative of φ with respect to z. For the
case of a_{0}=1, and a_{n}=0 for n > 0, equation (4) represents
a plane wave of unity amplitude propagating in the +z direction, which we denote
φ _{0}; this is the incident field. The boundary conditions for
a "hard" screen and disk at z=0 are

We again assume a solution for problem A, meaning
a set of coefficients a_{n} such that

A set of coefficients b_{n} is then a solution
of problem B if

Each term in the summation physically represents
a conical wave propagating at an angle θ_{n} with respect to the
z-axis, where

If the aperture is large compared to a wavelength,
the values of the coefficients a_{n} are negligible for large angles.
Therefore for waves propagating at small angles β_{n}/k <<
1 and equation (7) is satisfied for

To the same level of approximation the left hand side of equation (9) is the scattered field for problem B, and the right hand side is the total field for problem A, so this is the same as equation (3). This rigorously confirms the approximate result, and provides a quantitative indication of when the approximation will be accurate. Equation (7) could be considered to be an exact formulation of Babinet's principle for this case, in the sense that it provides an exact relation between the two solutions.

**Numerical Example**

Equations (6) and (7) can be formulated as matrix
equations, which can be solved for the coefficients a_{n} and b_{n}.
An example for an aperture and disk of radius 2 wavelengths is shown
in this figure. If the approximate Babinet's principle was exact for a hard
screen and a hard disk, all of the dots would lie inside the circles. They don't,
but they are fairly close even at this relatively small size. For larger radii
the agreement rapidly improves.

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Revised (and hopefully more understandable) version posted July 2006.