Waves are disturbances which propagate (move) through a medium, Light is a special case, it exhibits wave-like properties but does not require a medium through which to Propagate. Waves occur frequently in nature. The most obvious examples are waves in water, on a dam, in the ocean, or in a bucket. Waves can be viewed as transfer energy rather than the movement of a particle. Particles form the medium through which waves propagate but they are not the wave. Waves in water consist of moving peaks and troughs. A peak is a place where the water rises higher than when the water is still and a trough is a place where the water sinks lower than when the water is still. A single peak or trough we call a pulse. A wave consists of a train of pulses.

So waves have peaks and troughs. The following diagram shows the peaks and troughs on a wave. If we look very carefully we notice that the height of the peaks above the level of the still water is the same as the depth of the troughs below the level of the still water. The size of the peaks and troughs is the same.

### Characteristics of Waves

#### Amplitude

The characteristic height of a peak and depth of a trough is called the amplitude of the wave. The vertical distance between the bottom of the trough and the top of the peak is twice the amplitude. We use symbols agreed upon by convention to label the characteristic quantities of the waves. Normally the letter A is used for the amplitude of a wave. The units of amplitude are metres (m). It is the height of the crest or depth of a trough of a wave. It is generally expressed by the letter ‘a’. The amplitude of a wave determines the intensity of the radiation.

#### Wavelength

Look a little closer at the peaks and the troughs. The distance between two adjacent (next to each other) peaks is the same no matter which two adjacent peaks you choose. So there is a fixed distance between the peaks.

Looking closer you'll notice that the distance between two adjacent troughs is the same no matter which two troughs you look at. But, more importantly, it is the same as the distance between the peaks. This distance which is a characteristic of the wave is called the wavelength.
Waves have a characteristic wavelength. The units are metres (m). The wavelength is the distance between any two adjacent points which are in phase. Two points in phase are separate by an integer (0,1,2,3,...) number of complete wave cycles. They don't have to be peaks or trough but they must be separated by a complete number of waves.

The distance between the two consecutive troughs or crests of a wave is known as Wave-length of the wave. It is denoted by λ (Lambda).

#### Period

The time between two adjacent peaks is same and also the time between two adjacent troughs always the same, no matter which two adjacent troughs you pick. The time you have been measuring is the time for one wavelength to pass by. We call this time the period and it is a characteristic of the wave.

Waves have a characteristic time interval which we call the period of the wave and denote with the symbol T. It is the time it takes for any two adjacent points which are in phase to pass a fixed point. The units are seconds (s). #### Frequency There is another way of characterising the time interval of a wave. We timed how long it takes for one wavelength to pass a fixed point to get the period. We could also turn this around and say how many waves go by in 1 second. The number of waves made per second is called Frequency. It is denoted by ν (nu) .

We can easily determine this number, which we call the frequency and denote f. To determine the frequency, how many waves go per second, we work out what fraction of a waves goes by in 1 second by dividing 1 second by the time it takes T. The unit of frequency is hertz (Hz).
Waves have a characteristic frequency. F=1/T

#### Speed Now if you are watching a wave go by you will notice that they move at a constant velocity. The speed is the distance you travel divided by the time you take to travel that distance. This is excellent because we know that the waves travel a distance equal to wavelength in a time T. This means that we can determine the speed.

There are a number of relationships involving the various characteristic quantities of waves.
A simple example of how this would be useful is how to determine the velocity when you have the frequency and the wavelength. We can take the above equation and substitute the relationship between frequency and period to produce an equation for speed of the form.

### Types of Waves We agreed that a wave was a moving set of peaks and troughs and we used water as an example. Moving peaks and troughs, with all the characteristics we described, in any medium constitute a wave. It is possible to have waves where the peaks and troughs are perpendicular to the direction of motion, like in the case of water waves. These waves are called transverse waves.

There is another type of wave called a longitudinal wave and it has the peaks and troughs in the same direction as the wave is moving. The question is how do we construct such a wave?

An example of a longitudinal wave is a pressure wave moving through a gas. The peaks in this wave are places where the pressure reaches a peak and the troughs are places where the pressure is a minimum.

### Properties of Waves

We have discussed some of the simple characteristics of waves that we need to know. Now we can progress onto some more interesting and, perhaps, less intuitive properties of waves.

#### Reflection

When waves strike a barrier they are reflected. This means that waves bounce off things. Sound waves bounce off walls, light waves bounce off mirrors, radar waves bounce off planes and it can explain how bats can fly at night and avoid things as small as telephone wires. The property of reflection is a very important and useful one.

When waves are reflected, the process of reflection has certain properties. If a wave hits an obstacle at a right angle to the surface then the wave is reflected directly backwards. If the wave strikes the obstacle at some other angle then it is not reflected directly backwards. The angle that the wave arrives at is the same as the angle that the reflected wave leaves at. The angle that waves arrives at or is incident at equals the angle the waves leaves at or is reflected at. #### Refraction

Sometimes waves move from one medium to another. The medium is the substance that is carrying the waves. In our first example this was the water. When the medium properties change it can affect the wave.

Let us start with the simple case of a water wave moving from one depth to another. The speed of the wave depends on the depth. If the wave moves directly from the one medium to the other then we should look closely at the boundary. When a peak arrives at the boundary and moves across it must remain a peak on the other side of the boundary. This means that the peaks pass by at the same time intervals on either side of the boundary. The period and frequency remain the same! But we said the speed of the wave changes, which means that the distance it travels in one time interval is different i.e. the wavelength has changed. Going from one medium to another the period or frequency does not change only the wavelength can change.

Now if we consider a water wave moving at an angle of incidence not 90 degrees towards a change in medium then we immediately know that not the whole wave front will arrive at once. So if a part of the wave arrives and slows down while the rest is still moving faster before it arrives the angle of the wave front is going to change. This is known as refraction. When a wave bends or changes its direction when it goes from one medium to the next. If it slows down it turns towards the perpendicular.
If the wave speeds up in the new medium it turns away from the perpendicular to the medium surface.
When you look at a stick that emerges from water it looks like it is bent. This is because the light from below the surface of the water bends when it leaves the water. Your eyes project the light back in a straight line and so the object looks like it is a different place.

#### Interference

If two waves meet interesting things can happen. Waves are basically collective motion of particles. So when two waves meet they both try to impose their collective motion on the particles. This can have quite different results.

If two identical (same wavelength, amplitude and frequency) waves are both trying to form a peak then they are able to achieve the sum of their efforts. The resulting motion will be a peak which has a height which is the sum of the heights of the two waves. If two waves are both trying to form a trough in the same place then a deeper trough is formed, the depth of which is the sum of the depths of the two waves. Now in this case the two waves have been trying to do the same thing and so add together constructively. This is called constructive interference.
If one wave is trying to form a peak and the other is trying to form a trough then they are competing to do different things. In this case they can cancel out. The amplitude of the resulting wave will depend on the amplitudes of the two waves that are interfering. If the depth of the trough is the same as the height of the peak nothing will happen. If the height of the peak is bigger than the depth of the trough a smaller peak will appear and if the trough is deeper then a less deep trough will appear. This is destructive interference.

#### Standing Waves

When two waves move in opposite directions, through each other, interference takes place. If the two waves have the same frequency and wavelength then a specific type of constructive interference can occur: standing waves can form.

Standing waves are disturbances which don't appear to move, they look like they stay in the same place even though the waves that from them are moving.

#### Diffraction

One of the most interesting, and also very useful, properties of waves is diffraction. When a wave strikes a barrier with a hole only part of the wave can move through the hole. If the hole is similar in size to the wavelength of the wave diffractions occurs. The waves that comes through the hole no longer looks like a straight wave front. It bends around the edges of the hole. If the hole is small enough it acts like a point source of circular waves. This bending around the edges of the hole is called diffraction. To illustrate this behavior we start by with Huygen's principle.

#### Huygen's Principle

Huygen's principle states that each point on a wave front acts like a point source or circular waves. The waves emitted from each point interfere to form another wave front on which each point forms a point source. A long straight line of points emitting waves of the same frequency leads to a straight wave front moving away.

#### Dispersion

Dispersion is a property of waves where the speed of the wave through a medium depends on the frequency. So if two waves enter the same dispersive medium and have different frequencies they will have different speeds in that medium even if they both entered with the same speed.