If you have ever really observed waves closely, you have probably noticed their repetetitive nature. The Crests are always followed by troughs which are followed by another crest. All waves are generated by an oscillating object, whether a slinky, a tuning fork or an electron.

Simple Harmonic Motion

Objects that oscillate or vibrate at regular intervals are undergoing simple harmonic motion. The two most obvious examples would be a pendulum and a vibrating spring. In each case, there is an equilibrium position where the pendulum or spring is at rest. When either is moved away from their equilibrium position and released they move to return to their equilibrium position. Because of the force applied over a distance, energy is introduced to the system. When they return to the equilibrium position, rather than stop, they continue past equilibrium, stop and move back, again overshooting equilibrium. The energy of the system eventually dissipates due to friction and they stop oscillating.
 

Oscillating Pendulum - One Complete Cycle	Oscillating Spring- One Complete Cycle

Notice the repetitive and cyclic nature of this motion. In each case, there is some equilibrium position that the object vibrates around. It reaches some maximum displacement from equilibrium in both directions. Its speed, acceleration, kinetic and potential energies seem to vary at regular intervals.

We can describe several characteristics of this motion.

• The maximum displacement from equilibrium is referred to as the amplitude.
• The time required to complete one complete cycle is reffered to as the period.
• The frequency is the number of cycles per second.

For example if the period of a pendulum is 2.0 sec, its frequency is 1/2 sec^-1. The pendulum completes a half of a cycle per second.

What are Waves?

Waves are rhythmic disturbances that are transmitted through a medium (ie. air, slinky, water). They are created by something that vibrates at regular intervals. The energy from the vibrations is transmitted through the medium. The particles in the medium vibrate, but undergo no net displacement.

There are two kinds of waves.

Transverse waves are waves whose vibrations oscillate perpendicular to the direction of transmission of the wave. The best example of this kind of a wave is waves you observe when you throw a pebble into a pond. The ripples you see are the transverse waves propogating through the water. When you observe closely, you would notice a buoyant object such as a fishing "bobber" moves up and down in the direction of the vibrations but exhibits no movement in the direction of the wave's transmission.
 

Transverse Wave	Longitudinal Wave

Longitudinal waves are waves whose vibrations oscillate parallel to the direction of transmission of the wave. Sound waves are a good example of longitudinal waves. The tines of a tuning fork vibrate back and forth alternately compressing the air molecules into a compression, and pulling them further apart into a rarefaction. As with any wave, there is no net displacement of the particles in the medium. They vibrate back and forth, but do not "go anywhere".

Measurements Associated With Waves

As with simple harmonic motion, waves have a frequency and period. Also in transverse waves, amplitude can be measured by measuring the displacement from equilibrium to crest or equilibrium to trough. In longitudinal waves, the amplitude can be visualized by how "squished together" the compressions are or how "stretched out" the rarefactions are.

The distance between two corresponding points on a wave is the wavelength. This can be easily measured from crest to crest in a transverse wave and compression to compression in a longitudinal wave.

The SI units of wavelength and frequency are meters and sec^-1 respectively. We can relate both to the speed of the wave by using the universal wave equation:

                                                                                                                       v = l * f

Phenomena Associated With Waves

Reflection

When waves strike a barrier, they bounce off, not unlike how a racquetball bounces off a wall. When a wave reflects off of a barrier, its velocity, wavelength and frequency remain unchanged. If the barrier absorbs some energy from the wave, the reflected wave may have a somewhat smaller amplitude. When a wave is incident on a barrier, the reflected wave's direction can be predicted by the law of reflection. All waves reflect so that the angle of incidence (the angle the incoming wave makes with the normal) is equal to the angle of reflection (the angle the reflecting wave makes with the normal). The normal is an imaginary line drawn perpendicular to the barrier at the point that the wave strikes it.
 

Reflection off of a Straight Barrier	Reflection off of a Curved Barrier

When a waves reflect off of a curved surface, the law of reflection still applies, but because of the curvature, each wave strikes with a different angle of incidence and therefore reflects with a different angle of reflection. If the incident waves are parallel to each other, they reflect back onto a focal point.

Refraction

When a wave passes from one medium to another, it usually changes its speed. For example, light slows down as it passes from air to water. Sound speed up when it travels from air to water. Refraction occurs when a wave passes from one medium to another.
If the wave strikes the boundary between the 2 media "head on" (<i = 0), the wave passes straight through without bending. However, if it strikes the boundary at an angle, the wave's direction will change. We can predict what direction will refract if we know the speed of the wave in each medium.

When the wave passes from a fast medium to a slow medium, it will bend so that the angle of refraction will be less than the angle of incidence. In this case the wave bends towards the normal. If the wave passes from a slow medium to a fast medium, the angle of refraction will be larger than the angle of incidence. The wave bends away from the normal.
 

Fast to Slow Wave bends towards the normal	Slow to Fast Wave bends away from the normal
<I   > <R	<I  < <R

Diffraction

When you thow a pebble into a pond you see the ripples originating from the source. As you watch the waves traveling away from the center point, you notice the waves' amplitude gradually diminishes until you can no longer see them. The energy associated with waves  spreads itself out or dissipates. When a wave meets a barrier, the part of the wave that strikes the barrier reflects back. The other part of the wave travels by the barrier and bends around the corner. This can also be seen when a wave passes through an opening. The bending of waves around a corner is referred to as diffraction. Maximum diffraction occurs at larger wavelengths and smaller opening widths.
 

Waves Diffracting Around a  Corner	Waves Passing Through  an Opening and Diffracting

Superposition

Unlike matter, waves can occupy the same position at the same time. As waves move through each other, an observer sees that  the amplitude of the resultant wave is the sum of the amplitudes of each wave that is superimposed. This is the Law of Superposition.

Constructive Interference is when two or more waves' amplitudes add up to give a resultant wave with a larger amplitude than the individual waves. Destructive Interference occurs when the resultant amplitude is smaller than amplitude of the individual waves.
 

Constructive Interference	Destructive Interference

Standing Waves

When two or more waves have the same amplitude, speed and wavelength, the result is a standing wave. A standing wave looks like it is standing still with the alternating crests and troughs. Selected points where there is no movement of the medium are called nodes. The distance between successive nodes is half of a wavelength. Loops or antinodes are areas where crests alternate with troughs.

Return to Physics Homepage