They are some of the most common experiences we share: both the
sounds we hear and the light we see are waves. At the same time
they are subtle and pervasive: we shall see that they lie at the heart
of the fundamental physics.
The propagation speed of a wave (denoted c) is the rate at which a
given peak of the wave travels. The wavelength (l) is the distance
between peaks, and the frequency (n = w / 2 p, where w is the
"angular" frequency) is the number of peaks per unit time. The period
(T = 1 / n) is the time between two peaks (recall periodic functions
in math), and the wave number (k = 2p / l) is proportional to the
number of peaks per unit distance. (The factors of 2p in the angular
frequency and wave number are necessary so that the trigonometric
functions we will be using will be periodic in the period and
wavelength, respectively.) Here we see a wave train on a string as a
function both of time and of distance:
(since c depends on the frequency, waves of different frequency tend
to disperse). Therefore for this wave, c is 1.6 m / s. For sound waves
in air, c is equal to the Sqrt ( 1.4 R T / (28.95 g / mol)); for
electromagnetic waves, c is 1 / Sqrt (me). The relationship between
the speed of light and the permittivity and permeability was one of
the first clues that light is an electromagnetic wave.
Finally, the amplitude (A) of a wave is the distance from the resting
(equilibrium) state of the medium to the peak of the wave. The wave
above has an amplitude of 2 m. The energy of a wave is proportional
to A 2, and its intensity (I) is the power it delivers per unit area (P /
4 p r 2 for a spherical wave).
There are two main types of solutions to the wave equation, as we
indicated above: travelling and standing waves. Travelling waves
have the form
y = A cos ( k x - w t + d).
The (dimensionless) argument of the cosine function is called the
"phase", and is of the form b (x - c t); we therefore see that it
describes a wave which translates to the right (for positive c) in time
(recall translation of functions from algebra). Since
cos (a + b) = cos (a) cos (b) - sin (a) sin (b),
y = A [cos (k x - w t) cos (d) - sin (k x - w t) sin (d)].
d is called the "phase angle", and effectively allows us to specify the
relative "starting point" of the wave at time zero. By experimenting
with various values of d (ie., 0, p / 2, p, 3 p / 2, 2p), we see that we
can produce waves which have any given initial value (between - A
and A) at time zero The other main type of solution to the wave
equation is the standing wave, with the form
y = f (x) g (t).
The Doppler Effect
You have probably heard of the Doppler Effect in conjunction with
recent improvements in radar technology: "Doppler Radar" is capable
of measuring the velocities of winds, and is instrumental in the
identification of tornados. The basic principle is familiar to you
when driving as well: the pitch (frequency) of the horn or siren of an
approaching vehicle is higher than when it passes you and recedes.
This "Doppler Shift" in the frequency also has an important medical
usage in the measurement of the speeds of moving fluids inside the
The "received" wavelength is related to the "source" wavelength by
= l s (c - u s) / c,
the received frequency is related to the source frequency by
n r = n s c / (c - u s).
The same principle applies when the source is stationary but you are
approaching it at a speed u r. Now the received wavelength is
related to the source wavelength by
= l s c / (c + u r)
(since the moving receiver now determines the period of the wave)
and the received frequency is related to the source frequency by
n r = n s (c + u r) / c.
If both the source and receiver are moving and u s and u r are the
speeds with which they are approaching each other (respectively),
the Doppler Shift is
n r = n s (c + u r) / (c - u s).
The resultant shift is
n r = n s (c + u) / (c - u).
This allows us to determine the speed of the blood as
u = c (n r - n s) / (n r + n s).
Refraction & Reflection
The speed of sound in air depended on both the molecular weight
and the temperature of the air. We also noted that the speed of light
depended on both the electrical permittivity and the magnetic
permeability of the medium. It is a general characteristic of waves
that their phase velocity depends on the medium in which they
travel. While we will shift our attention to light, we note that
everything we will be discussing still applies to other types of
Nature is fundamentally "lazy". We have seen that particles in a
potential field will move to the position of least energy and that
electricity takes the path of least resistance. Waves are no different.
They follow the path of least total elapsed time. The consequence of
this is that their path changes when they enter a different medium
(with a different characteristic phase velocity). These "lazy rules"
which nature follows are all specific examples of the "principle of
least action". The action is defined as the integral of the difference
between kinetic and potential energy with respect to time.
Everything in nature behaves the way it does because it follows the
path of least action. Another of nature's deepest mysteries is why
this must be so.
If we define a light "ray" as the "arrow" perpendicular to a moving
wave front, we can conveniently examine the changes in path across
the boundary of two media. That change in path is known as
"refraction", and is expressed in terms of "Snell's Law":
n 1 sin q 1 = n 2 sin q 2,
where n is the "index of refraction" of the medium. The index of
refraction is the ratio of the phase velocity in a standard medium,
ie., air for sound or vacuum for light, to the phase velocity in the
medium; it is always greater than one. The angles are measured from
the normal to the interface between the two media:
Note that for any angle greater than the "critical angle"
q c = arcsin (n 2 / n 1),
Snell's Law is not applicable; you get "total internal reflection",
which means that the ray (and hence the wave) does not cross the
If n 1 is the same as n 2, Snell's Law describes reflection from a
mirrored surface: the reflected angle equals the incident angle. Some
representative indices of refraction are:
n = 1 in air;
n = .06 in glass;
n = .23 in water;
n= 6.1 in rubber;
for light (of wavelength 589.3 nm),
n = 1 in a vacuum;
n = 1.0003 in air;
n = 1.333 in water;
n = 1.336 in vitreous humour (inside the eye);
n = 1.413 in the eye's lens;
n = 1.52 in crown glass;
n = 1.61 in flint glass, and
n = 2.42 in diamond.
We can use Snell's Law to understand the focussing of images by
lenses, including the lens of the eye. We categorize several types of
For each of these lens types, we can specify a "radius of curvature"
for each curved face. That radius is the radius of a circle which
would have that face as an arc. We also define an "object length" (O,
the distance from the center of the lens to the object which is to be
seen), an "image length" (I, the distance from the center of the lens to
the place where the image is to be viewed) and a "focal length" (f).
The focal length is the distance from the center of the lens to the
"focal point"; light rays from a sufficiently distant object will all be
approximately parallel as they enter the lens, and will converge at
the focal point:
The object is of course on the left in this picture, and the image is on
the right. Note that the image is inverted, as it will be with a
convex lens. These quantities are related by
1 / O + 1 / I = 1 / f.
For thin lenses in air with radii of curvature r 1 and r 2,
1 / f = (n - 1) (1 / r 1 - 1 / r 2)
(light enters the r 1 side and exits the r 2 side; r i is considered positive
if its center of curvature is on the right; if either side is flat, r is
assumed to be infinite). Note that the focal length will be negative
for concave lenses, since the focal point is actually on the same side
of the lens as the object:
We can also define the strength of a lens (in "diopters", if f is
measured in meters) as
S = 1 / f,
and vision correction is then a simple matter of addition:
Scorrective lens + Seye's lens = Scorrected vision.
This fact that the strengths of lenses add also allows us to construct
one type of lens from others. For instance, a plane-concave lens with
exit radius of curvature 10 cm can be constructed from a plane-
convex lens with exit radius of curvature 8 cm and a concave lens
with entry radius of curvature 8 cm (to "fit" the plane-convex lens)
and exit radius of curvature 10 cm. The strength of the composite
lens is simply the sum of the strengths of the component lenses.
We define the magnification of a lens as
m = I / O,
which is effectively the ratio of image size to object size.
A ray of light as it traverses corrective lenses and the eye is
1. as it enters the corrective lens;
2. as it exits the corrective lens;
3. as it enters the cornea;
4. as it exits the cornea into the eye's lens;
5. the width of the lens (and hence the radii of curvature and the
focal length) is controlled (within the elastic limts of the lens)
by muscles which stretch the lens "vertically";
6. and finally the ray is refracted as it exits the eye's lens and
enters the vitreous humour.