Topic 4 – Progressive Waves

Waves are caused by oscillations.  Oscillations are complete to-and-fro movements, of which vibrations are one example.  Another example is the oscillation of electrons, which cause radio waves. We have studied oscillations in more detail with simple harmonic motion.

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.  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.

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.

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.

Terms Used with Waves

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:

f = 1/T

Wave velocity, code v, units metres per second (m/s), tells us the speed of propagation of the wave, i.e. how fast it travels.  For water waves this is a few cm/s.  In air, sound waves propagate at 340 m/s.  For light the speed is 3 x 10⁸ m/s.  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.

Now have a go at the Wave Definitions Exercise to see how much you have taken in.

The Wave Equation

The frequency, speed, and wavelength of any wave can be linked by the simple equation:

where v is the speed of the wave in m/s, 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

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 ´ 10⁸ m/s.  So the equation is written c = fl.

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 180o 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.

Transverse and Longitudinal Waves

A transverse wave is one in which the displacement of the particles is at 90^o to the direction of travel.  In a water wave, the particles move up and down while the wave travels horizontally.  All electromagnetic waves are transverse.

We can show the features of a transverse wave in the diagram below:

In longitudinal waves, the displacement is parallel to the direction of travel of the wave.  There are regions of high pressure, compression, and regions of low pressure, rarefaction.  In a sound wave the air molecules move forwards and backwards; where they are squashed together, a compression results, where they are forced further apart, there is a rarefaction.  Like all mechanical waves, a medium or material is required.  The speed of sound in air is 336 m/s, in water 1400 m/s, in steel it is 6000 m/s.  Other examples of longitudinal waves include some kinds of earthquake waves (the pressure or P-wave).  We can see the features of a longitudinal wave in the diagram below.

Now go on to Graphical Representation of Waves