Waves
are caused by oscillations. Oscillations are complete toandfro movements, of which
vibrations are one example. Another
example is the oscillation of electrons, which cause radio waves. We will study oscillations in more detail with simple harmonic motion.
Write down three different examples of oscillations. 
Waves occur when a disturbance at the source of the wave causes particles to oscillate about a fixed central point. There is a maximum displacement from the central point, which is called the equilibrium position, or average level. When particles reach that maximum displacement, they start to move towards the central point. They pass through the central point as they move to the maximum displacement on the other side.
What are the features of wave motion? 
We can show this on a water wave. The particles of water oscillate up and down from the equilibrium position. The wave is travelling from left to right. P is going down, Q is at the maximum displacement, and R is going up.
The wave is called a progressive wave because it is moving in a particular direction. It is transferring energy from the point of disturbance, but the particles are not travelling with the wave, merely going up and down.
How is energy transferred if particles do not travel with the wave? 
Ripples in solid materials like sand are NOT waves. They are caused by the piling up of sand in the wind. They do not oscillate.

Waves can be considered to travel either as plane wavefronts, from a plane source or as circular wavefronts from a point source:
In 3
dimensions, the waves would propagate spherically from a point
source.
Displacement of a particle is the distance at any given moment
from the central or equilibrium position, i.e. the undisturbed position.
It is given the Physics Code
s
or
x, and the SI unit is metre (m).
The displacement decreases the further the wave progresses from its
source.
Intensity of waves at a point is the
power per unit area at that point.
The energy of a wave increases as the square of its amplitude.
However the energy decreases as the square of the distance from the
source, which is known as the inverse square law. The
physics code for intensity is
I
and the units are watts per square metre (W m^{2}).
Amplitude of a wave, code
A or
r, units metres (m),
is the maximum displacement of a
particle from its equilibrium position.
In
other words it is the height of the wave from the average level.
It is NOT the height from crest to trough.
(NB: Be careful of the code.
Here amplitude is given the code
A,
but in many texts you will see
a.
This could be confused with acceleration.)
Wavelength is defined as the distance between any two points
on adjacent cycles that are in phase,
in other words the distance between adjacent peaks or troughs.
The code for wavelength is
l
(lambda, a Greek letter ‘l’).
The units for wavelength are metre (m).
Frequency, code
f, has the unit hertz (Hz), and is the number of
waves passing a given point every second.
Period is the time taken for one complete oscillation. The code
is
T and the units seconds (s).
Frequency is the reciprocal of period
and is related to period by the simple equation:
Wave velocity, code
v,
units metres per second (m s^{1}), tells us the speed of
propagation of the wave, i.e. how fast it travels.
For water waves this is a few cm s^{1}.
In air, sound waves propagate at 340 m s^{1}.
For light the speed is 3 x 10^{8} m s^{1}.
The speed of light is given the code
c.
Mechanical waves are produced by a disturbance in a
material, or a medium, and can be
longitudinal
or transverse. Mechanical
waves need a medium or material to
travel in. In
electromagnetic waves the disturbances are in the form of
oscillating electrical and magnetic fields. They are always transverse.
Electromagnetic waves can travel in a vacuum.
The
phase of a particle is the fraction of the cycle a particle has
passed through relative to a given starting point. We describe the difference in the motion of particles in
terms of the phase difference.
This is the fraction of a wavelength by which their motions are
different.
The
path difference between two waves is the number of cycles
difference there is in the distance they have to travel.
Graphical Representation of Waves
We can show the features of waves in two ways:
A displacement  displacement graph which shows how the wave varies in space.
A displacement  time graph which shows how the wave varies with time.
The displacement  displacement graph shows the physical features of a wave:
The displacement  time graph shows how the displacement varies with time and it is like this:
You need to note these points:
The displacement time graph does not show the physical features of the graph.
The wavelength should not be confused with the period.
It is not correct to call the positive amplitude a crest, not the negative amplitude a trough.
Alternating currents vary sinusoidally like the wave above.



The
frequency, speed, and wavelength of any wave can be linked by the simple
equation:
v = fl
where
v
is the speed of the wave in m s^{1},
f is the frequency in Hz,
l is the wavelength in m.
When
we use this equation, we do not always have to use SI units, but it is important
to be consistent.
Worked
Example What is the frequency of water waves of wavelength 4 cm travelling at a speed of 1.6 m s^{1}? 
Answer Formula first: v = fl 
Rearrange: f = v/l 
f
= 1.60 m s^{1}
÷
0.04 m = 40 Hz (Note that the 4 cm was converted to 0.04 m to keep the units consistent. The speed (1.6 m s^{1}) could have been changed to 160 cm s^{1}) 
The wave equation is used for longitudinal and transverse waves.
Note that when we use the equation for light or radio waves, we use the code c for the speed of light  c = 3.0 ´ 10^{8} m s^{1}. So the equation is written c = fl.
(a)
What is the frequency of the waves?
(b)
What is the wavelength of the waves? (c) What is the wave speed? 
Energy in a Wave
If there is a bad storm, the news reports show spectacular waves smashing against the coastline. Often there are shots of the damage that has been done to sea walls and other buildings. This is because the wind disturbs the water to make high amplitude waves.
The energy is related to the amplitude of a mechanical wave:
This can be converted into an equation:
Where:
E  energy (J);
k  some constant (J m^{2});
A  amplitude (m).
Therefore, when the amplitude doubles, the energy carried by the wave goes up by four times.
A more detailed treatment is studied at university level, but the energy in waves along a string can be found HERE.
The physics code A can stand for amplitude or area. Be careful that you know the context in which it's being used.
The photon energy in an electromagnetic wave is given by E = hf. Amplitude is not involved. 
Intensity
Intensity is defined as:
Energy per unit area
The physics code is I and the units are joule per square metre (J m^{2}). The equation is:
The energy has a maximum value, often written I_{0}, at the source. As waves propagate, they spread out. For each doubling of radius, the intensity goes down by 4 times:
This is called the Inverse Square Law. The equation for the inverse square law is:
This is true for all waves.
Phase
When a wave is travelling, all the particles are in continuous motion. The different particles have different displacements, velocities and directions. Indeed this is true even of adjacent particles. The phase of a particle is the fraction of the cycle a particle has passed through relative to a given starting point. We describe the difference in the motion of particles in terms of the phase difference. This is the fraction of a wavelength by which their motions are different.
Consider the two particles
X
and
Y.
X
is at the trough of a wave, whereas
Y
is at the crest. Their directions are upwards and downwards respectively.
They are half a wavelength (l/2)
out
of phase. By linking oscillation to rotary movement, we can also
describe X and Y as being 180^{o} or
p
radians out of phase. We say that these particles are in
antiphase.
W and Z are one wavelength, 360^{o} or 2p radians apart. They are both at the starting point of a cycle. Their motion, including displacement, velocity and direction, is identical. We can therefore say that they are in phase. Particles can be any amount out of phase.
If we have two waves, we can measure their path difference. If the waves have a path difference of 1 wavelength (or any other whole number of wavelengths), they are in step or in phase. Waves with a path difference of 1/4 wavelength have a phase difference of 1/4 of a cycle (90^{o} or p/2 rad). If they have a path difference of 1/2 a wavelength, the waves are in antiphase. Path difference is important when we analyse how waves superpose.

The phase can be calculated for any points on the wave that are d metres apart, using the relationship:
The term d is the distance between the two points. The curious symbol f is ‘phi’, a Greek letter ‘f’, used as Physics code for phase angle.
The angle in this relationship is in radians, where 2p rad = 360 ^{o}. Hence 1 rad ≈ 57^{ o}.
When you use radians, you must ensure that your calculator is set to radians. The easiest way to lose marks is put your angle in radians while you are set to degrees…
It is up to you to make sure how to work your calculator. 
Radians is a dimensionless unit, and some people leave it out. In these notes I will always use the shorthand rad.
The graph is a summary of phase relationships:
Modelling Waves (Extension)
Questions that use this concept are not likely to be asked in AS exams. It is entirely possible that they will come up in Alevel.
The simplest type of wave is called a sine wave. This is because the displacement varies with the sine of the time. The equation is shown below:
The terms are:
x  the displacement (m);
A  the amplitude (m) which is the maximum displacement;
w  the angular velocity (rad s^{1});
t  the time (s).
Note that x is often used for the displacement in waves. The code s can be used as well. The symbol w is omega, a Greek lower case letter long 'o' (ō). It represents the frequency of the wave, and is linked to the frequency by the equation:
So we can also write:
Be careful about how you input
wt
into your calculator:

We can use an Excel spreadsheet (other spreadsheets are available) to model the sine waves. You can try it for yourself if you are a dab hand at spreadsheets.
Here is a screen shot with the formulae:
This spreadsheet model has a time interval of 0.01 s for a time period of 1.0 s. The sine function in Excel works with angles in radians. It does not work with degrees. Here is the sine wave produced with the model:
Velocity of the particles that form the wave can be worked out using the gradient of the sine wave. In calculus notation, we can write:
Maths Window The derivatives of trigonometrical functions are as follows:
And
We won't look at the tangent functions here.
If there is a constant being processed by the function, then the derivative is multiplied by the constant. In this case, we'll make the constant B. Therefore:
And:

For acceleration, we can write:
The minus sign tells us that the acceleration is towards the average level (zero displacement).
A wave has an amplitude of 0.25 m and a frequency of 6.0 Hz. Calculate: a. The angular velocity; b. The displacement at 1.1 s; c. The velocity at 1.1 s; d. The acceleration at 1.1 s. 
Sine waves are very closely related to:
Circular motion;
Simple harmonic motion.
Both of these are in the second year A level and
will be discussed in Further Mechanics.