**Geometrical Optics**

At high frequencies, or more specifically when the wavelength is small compared to other geometrical features, sound behaves as if it travels along rays, just like light rays. A flashlight beam is a good example of a bundle of light rays. This aspect of wave behavior is called geometrical optics because one can analyze things (such as telescope mirrors) by geometrically tracing rays, rather than solving a wave equation. The audio spectrum is generally too low in frequency for ray behavior to be obvious, but the imaging of loudspeakers discussed in the image analysis section is one good example of ray behavior of sound.

The theoretical basis for ray behavior is developed in this section. The derivation is similar to the one for the electromagnetic case presented in Born and Wolf, section III.

**Historical/Philosophical Notes**

The ray behavior of light led Newton to speculate
that light consisted of particles. In the absence of other forces, particles
travel in straight lines just like rays, so this was a reasonable inference.
Later experiments convinced physicists that light was a wave, not a particle.
The orthodox view since the 20th century is that light is a particle __and__
a wave. In fact, according to quantum mechanics, __everything__ is a particle
and a wave, including you and I (e. g. see Polkinghorne).

Prior to quantum mechanics there was Newtonian, or classical mechanics. It is interesting to me that this physics of sound site begins with classical particles - molecules - and assuming purely classical behavior of these particles we derive the sound wave equations. So we have particles behaving like a waves. Now we end up with waves behaving like particles.

One final note: a cornerstone of quantum mechanics is Schrödinger's equation. When he created this equation he was at least partially inspired by the derivation of geometrical optics from Maxwell's equations (see Polkinghorne). He just went in the opposite direction, from a short wavelength approximation to a wave equation.

**Derivation of Geometrical Optics**

Assuming e^{jωt }time dependence, the
acoustic equations (24) and (25)
can be written

The vector **r** defines a location in the medium,
p is the sound wave pressure, and the vector **u** is the wave velocity (mean
molecular velocity). The propagation constant k=2π/λ,
where λ is the wavelength, and Z_{0 }the wave impedance. All terms
are the same as defined in the Plane Wave Section.
We look for a solution of the form

In this form the functions **u** and p describe
a wave amplitude envelope. For a plane wave, **u** and p are simply constants.
The part of the wave function that varies rapidly in space, from wave crest
to wave crest, is represented by the exponential phase term. For a plane wave
traveling in the x-direction s(**r**)=x. Substituting this trial solution
into (1) yields

The "approximately equal to zero" is the geometrical
optics approximation. For small wavelengths k>>1. As long as the derivatives
of p and **u** in equation (3) are not too large, when they are divided by
k they will be close to zero. For a non-trivial solution of equation (3) the
determinate of the system must be equal to zero, which yields

**Characteristics of Geometrical Optics Propagation**

The function s(**r**) is called the *eikonal*.
A surface s(**r**)=constant is a surface of constant phase, also called a
wavefront. The wave propagates along rays that are perpendicular to the wavefront.
The top part of equation (4) shows that the average molecular velocity **u**
points along a ray in the direction of propagation. The bottom part of equation
(4) says that the ratio of pressure and velocity magnitudes is exactly the same
as a plane wave.

The pressure acting on a small area of the wavefront,
moving with velocity **u**, does work that corresponds to a power flow exactly
the same as a plane wave. The direction of power flow is along a ray.

If we move along a ray a small distance ds, the vector
**r** will change by d**r**, where |d**r**|=ds, and d**r** points
in the direction of the ray. Therefore on a ray **r** is a linear function
of the distance along the ray, meaning that it is a straight line. In the derivation
above we have implicitly assumed that the speed of sound is constant in the
medium, and in this case rays are straight lines. If the speed varies, the rays
are no longer straight, exactly like the case of a variable index of refraction
for light.

The geometrical optics propagation of sound is very similar to the propagation of a plane wave, except in one important respect: for a plane wave the wave amplitude is constant along a ray. Here the amplitude can vary. If one imagines a bundle of rays that form a tube, the power flowing in the tube will be constant. Power is inversely proportional to the tube cross-sectional area. If the rays diverge amplitude will decrease, and vice-versa.

**Validity of the Geometrical Optics Approximation**

The geometrical optics approximation breaks down
when the derivatives of the functions **u** and p in equation (3) become
large. If the radius of curvature of the wavefront is large compared to a wavelength,
the approximation will generally be valid. An example of when this breaks down
is the case of a plane wave normally incident on a parabolic reflector. A geometrical
optics analysis yields rays reflecting from the paraboloid that converge to
a point at its focus. The ray tube area goes to zero at the focus, and according
to geometrical optics, the power becomes infinite. But as one approaches the
focus the radius of curvature of the wavefront approaches zero. When it is less
than a wavelength the geometrical optics approximation is no longer valid.

If a sound wave is reflected from a surface where
the radius of curvature of all features on the surface are large compared to
a wavelength, then geometrical optics will generally provide an accurate solution.
Rays are traced by reflecting them from the surface as if it were a mirror.
Surface features that are very much smaller than a wavelength are not a problem
- the wave cannot "see" these features. Surface features roughly the size of
a wavelength, and features like edges, will cause wave diffraction. There is
an extension of geometrical optics called the *geometrical theory of diffraction*
which can deal with some of these problems, and I might add a section on that
topic some day.

Back to the Image Analysis Section