If we recall the most general expression for the phase of an oscillator, we see at once that waves can also travel:

y(x,t) = A sin (kx + ωt + δ).To simplify things, we will set the phase angle equal to zero and rewrite this equation as

y(x,t) = A sin (k (x + (ω/k)t)).Compare this to the form of a function which undergoes

y(x) -> y(x + a)moves the function to the left a units:

This shows us that our traveling wave is simply an ordinary sine wave which is being translated by an amount depending on time. As we saw in the last section, w/k is the speed of the wave which in this case is traveling to the left:

A wave which travels to the right is described by

y(x,t) = A sin (kx - ωt).We say that a traveling wave

**Huygens' Principle** states that each point on a wave front acts like a source for the next wave front. It is as if there is a tiny harmonic oscillator at every point in space and the vibration of one oscillator along a traveling wave causes the next one in the direction of travel to begin vibrating:

Each oscillator generates a hemispherical wave disturbance in the direction of travel, which we have shown here in cross section as semicircles. The wave front is tangent to the semicircles at any given time.

We found that the energy of an oscillator is proportional to the square of its amplitude. For a spherical wave, that energy is evenly distributed on the surface of an ever-expanding sphere. We define theConsider theintensityto be the ratio of the power carried by the wave to the surface area of the spherical wave front:I = P / (4 π rFrom this we can see that while the total energy in the wave is constant (ignoring dissipative effects), the intensity drops off as 1 / r^{2}).^{2}.

We see that the angle of reflection is equal to the angle of the incident rays.

Do the following experiment as you are reading this: place a pencil in a glass of water. The pencil appears to be broken because the speed of light is different in the water than in the air: this is called **refraction**. If we write the speed of light in an arbitrary medium as

v = c / n,n is called the

while for light (of wavelength 589.3 nm),

- n = 1 in air;
- n = .06 in glass;
- n = .23 in water;
- n= 6.1 in rubber;

Since the speed of light is less in water than in air, the wave fronts change direction when crossing from one medium to the other:

- 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.

where again the wave fronts are drawn at equal times. The change in direction is described by **Snell's Law**:

nwhere the angles are measured from the normal to the interface between the two media:_{1}sin θ_{1}= n_{2}sin θ_{2}

Note that for any angle greater than the "critical angle"

θSnell's Law is not applicable; instead of refraction, there is_{ c}= arcsin (n_{2}/ n_{1})

It is the refraction of light which makes possible both vision and vision correction. A ray of light as it traverses corrective lenses and the eye is refracted:

The depth, shape and index of refraction of each medium determines the degree to which the rays of light are bent, and hence the

- as it enters the corrective lens;
- as it exits the corrective lens;
- as it enters the cornea;
- as it exits the cornea into the eye's lens;
- and finally the ray is refracted as it exits the eye's lens and enters the vitreous humour.

Recall the wave demonstration applet in the previous section. Set the black and blue waves to the same mode number and observe the red superposition wave as you vary the blue wave phase angle. When the phase angle is zero, the black and blue waves are said to be **in phase** and **constructively interfere**. When the phase angle is π or -π, the superposition wave has an amplitude of zero and the black and blue waves are said to be **out of phase**: they **destructively interfere**. The interference of **coherent waves** (which start out in phase) lies at the heart of the quintessential wave phenomenon: **diffraction**.

We start by illustrating how interference works in two dimensions with this applet, which shows the superposition of two waves from sources at the bottom of the image. The white spots are where the waves constructively superpose and are in phase, and the black spots are where the waves destructively superpose and are out of phase:

Suppose that a beam of light is incident on a surface with two slits in it (separated by a distance d). Huygens' principle states that the two slits will act as coherent sources. These correspond to the applet above, with equal frequencies and zero phase angle. When the light rays from these two sources interfere at the screen (a distance L away from the slits), one will have traveled farther than the other:

The angles are approximately equal, as are the hypotenuse and adjacent sides of the large triangle, if the angle is small. This is true since the length L is usually much greater than the distance between the slits. The difference in length of the two rays from the slits to their point of intersection is d sin θ. When this length difference is an integral number of wavelengths:

d sin θ = m λ,the phase difference is

k Δx = (2 π / λ) * m λso the waves are in phase, and we have constructive interference and a bright spot at the point of intersection (at x = L sin θ). When this length difference is a wavelength times an integer plus one half:= 2 π m,

d sin θ = (m + 1/2) λ,the phase difference is

k Δx = (2 π / λ) * (m + 1/2) λso the waves are π radians out of phase, and we have destructive interference with a dark spot at the intersection. If we label the minima and maxima with a number "a" as follows:= 2 π m + π,

we can write their locations as

x = a L λ / d.The first applet plots the intensity as a function of position, while the second applet provides a visual illustration of the actual diffraction pattern:

For constant wavelength, an increase in d moves the maxima closer together (the angle must decrease for a given m) and washes the pattern out. For a constant slit separation, increasing the wavelength spreads the pattern into a continuum (the angle must increase for a given m). In the limiting case, if d is less than λ, the condition for secondary maxima (m > 0) cannot be met. This illustrates the fact that **resolution** is limited by wavelength: you cannot distinguish two objects which are separated by less than one wavelength of the wave being used to image them.

One result of Huygens' Principle provides us with a unique opportunity to "see" infinity. Consider a spherical mirror, formed by "slicing" off a spherical segment of a reflecting sphere of radius r:

For a convex mirror (looking into the outside of the spherical segment), the **magnification** of the image (the ratio of the image size to the object size)
is equal to 1 when the object touches the mirror, and reduces to 0 as the object recedes to infinity; and the image is always "right-side-up".

But for a concave mirror (looking into the inside of the spherical segment), something very interesting happens. Huygens' Principle leads us to discover that the magnification M is given by

M = 1 / (1 - 2 O / r)where O is the distance from the object to the surface of the mirror. M is still 1 when the object touches the mirror, and the image is still right-side-up. But as the object recedes toward a distance r/2 from the mirror, the magnification approaches infinity (here, r = 2m):

As the object continues to recede, the image is *inverted* (M < 0), and slowly shrinks in magnification. Try this
with the inside of a highly polished serving spoon, or some other reflecting concave surface.

And in a coincidence of physics, the magnification of a concave mirror "mirrors" the form of the radial **singularity**
(where changes in radial distance go to infinity) of a static black hole of mass M:

dsHere, as an object approaches r=2M, a far-away observer sees any change in position as taking infinitely long: the inward-falling object never appears to gets closer than 2M,^{2}= dr^{2}/ (1 - 2 M / r)

The next section introduces the magnetic field.

©2013, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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