# Wave Phenomena

### Visualizing Wave Number and Phase Angle

The frequency of a wave is often expressed in terms of "cycles per second", although its proper unit is Hertz (Hz). This makes it easy to understand the effect of increasing or decreasing the frequency: more or fewer cycles occur in the same period of time.

The wave number, which is simply the number of cycles per unit length, is somehow more difficult to intuit, even though we have simply replaced time by distance. And the phase angle, while simply a constant angle added to the argument of the sine or cosine function describing the wave, can be even more intimidating.

This applet allows you to visualize the effect of changes in wave number and phase angle:

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

The horizontal axis extends from -2π to +4π. And if you think of the horizontal axis as time instead of distance, the wave number scroll bar acts as a frequency scroll bar.

### Listening to Beats

When two notes start at the same time, they add together such that the frequency you hear is the average of the individual frequencies, and the volume cycles from 0 to maximum and back again at a rate which is the difference between the two frequencies. This volume variation is called beats.

This applet allows you to hear beats:

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

### Playing with Standing Waves

A standing wave is a wave formed by adding together identical waves traveling in opposite directions. They occur between obstructions, where the amplitude (or height) of the wave must be zero, or inside tubes or pipes. When the wave is fixed at the obstruction (or closed end of a pipe), the boundary condition is "fixed" (or "closed"). When the wave is allowed to vibrate freely, such as at the open end of a pipe, the boundary condition is said to be "free" (or "open").

Standing waves are described by their fundamental frequency and mode (an integer starting with 1, which corresponds to the fundamental). Two waves can be in phase, so that they reach their minimum and maximum values simultaneously, or out of phase, so that one reaches its maximum every time the other reaches its minimum, or they can have any relative phase angle.

This applet allows you to view standing waves and superposition of two waves (when turned on, the red wave is the sum of the black wave and the blue wave); the fundamental is fixed, but boundary conditions, modes and relative phase angles are adjustable:

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

Note that the blue wave is drawn after the black one, so that if they have the same mode and are in phase, you will only see the blue one. Also notice that if the boundary conditions are not the same at each end, only odd integer modes are allowed.

### Constructive and Destructive Interference

When two waves add together in phase, we say that they "constructively interfere". When they add together out of phase, we say that they "destructively interfere". The places where constructive and destructive interference occur depend on the difference in travel distance from the sources of the waves: for two waves at the same frequency and in phase, if the difference in travel distance is a whole number of wavelengths (the length of a single wave cycle), constructive interference occurs. For the same two waves, if the difference in travel distance is an odd number of half wavelengths, destructive interference occurs.

This applet allows you to visualize constructive and destructive interference in waves from two sources (located at bottom center):

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

### Diffraction Patterns

Diffraction is one of the quintessential wave phenomena: a source of light shines on a screen with two slits; the light passes through those slits and falls on a screen some distance away. Each slit acts as a source of light waves (this also works with sound), and the two sources are at the same frequency and in phase (they are coherent). The interference pattern which is observed on the screen is call a "diffraction pattern", and depends on the width and separation of the slits, the wavelength of the light, and the distance to the screen.

These applets allow you to view diffraction patterns; the first shows a visual approximation of the pattern, and the second plots the intensity at the screen.. Wavelength and experimental geometry can be varied.

### How to "see" Infinity in a Mirror

A mirror can give you a glimpse of 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. It turns out 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.