Waves

Simple harmonic motion

For SHM a mass is displaced from a central position, where it is in equilibrium. A restoring force acts in the oposite direction to the displacement; for SHM the force must be proportional to the displacement. This gives the mass an acceleration, towards the central position, that is proportional to the displacement.

SHM occurs when the acceleration of a mass is directed towards a fixed pointand is proportional to its displacement form that point.

For a pendulum, the energy of teh oscillating mass is transformed back and fourth between kinetic and potential forms.

• Midpoint: maximum velocity, therefore maximum kinetic energy and zero potential energy.
• Endpoints: zero velocity, therefore zero kinetic energy and maximum potential energy.
• The total energy (kinetik + potential) is constant.

For a mass oscillating vertically on a spring, the potential energy is partly gravitational potential energy and partly elastic strain energy. At all times the total energy (potential + kinetic) is constant. this is true for all freely osscilating systems.

If an oscillating mass loses no energy it will oscillate for ever with the same amplitude. However, if it loses energy, we say that the oscillations are damped. Their amplitude decreases.

• Light damping: the amplitude decreases gradually as the mass oscillates.
• Critical damping: the damping is heavy enough for teh displacement just to decrease to zero (the equilibrium position) without oscillation. With a little less damping, the mass overshoots the equilibrium position.

Damping is caused by frictonal forces, e.g. drag of the air or viscous drag in oil.

It may be possible to force a mass to oscillate at any frequency by applying a periodic force. This is called a forced oscillation or forced vibration. If the forcing frequency matches the natural frequency then the amplitude increases to a large value this is called resonance.

Light(and other transverse waves) can be polarised. In unpolarised light, theelectric and magnetic filds vibrate in all directions perpindicular to the direction of travel. After passing through a sheet of polaroid, each vibrates in only one direction or plane. The light is said to be plane polarised.

A second sheet of polaroid will let through nearly all the light if its plain of polarisation is lined up with that of teh first sheet. If it is 'crossed'(at 90degrees) with the first, all the plain polarised vibrations will be absorbed and no light will pass through.

Only transverse waves can be polarised. If a wave can be polarised, it must b transverse- that's hwo we know electromagnetic waves are transverse.

Wavelenth and amplitude

• The displacement is the distance the distance moved by a particle from its undisturbed pattern.
• The wavelength of a wave is the distance between adjacent crests, or between any two adjacent points that are at the same point in the cycle(i.e. in phase with one another)
• The amplitude of a wave is the maximum displacement of any particle.
• The time period is the time for one complete cycle of the wave.
• This is related to the waves frequency f=1/T
• The frequency is the number of cycles of the wave per second.

Light shows interference. To produce two waves, light is shone through a pair of parallel slits. Where the light falls on a screen beyond the slits, light and dark interference 'fringes' are seen.

• The light from the slits must be coherent. The light leaving the double slit is placed in front of an illuminated single slit. Alternatively, a laser can be shone directly on the double slit.
• As light passes through each slit, it spreads out into the space beyond.
• The fringe seperation can be measured using a traveling microscope. If a laser is used, the fringe seperation can be measured fairly accurately by marking the fringes on graph paper stuck on a screen a few metres away.
• Increase the slit-screen distance makes the fringes wider but dimmer.

When two or more waves cross at a point, the result is found by the principle of superposition. At any instant, the resultant displacement is simply the sum of the displacements of teh individual waves. Constructive and destructive interference are obvious examples of this idea. It also explains the formation of stationary waves.

Stationary waves on a stretched string

The ******** sends waves along the string. They are reflected at each end. The outgoing and reflected waves then interfere. At certain frequencies, a stationary wave pattern of loops is formed.

• At certain points-nodes- the two waves interfere destructively. There is no vibration. There are nodes at the ends of the string.
• Half-way between the nodes are antinodes. The string vibrates with a large amplitude.
• Changing the frequency slightly causes the stationary wave to disappear. Changing the length, tension or thicknessof the string causes the stationary waves to appear at different frequencies.
• The wavelength of the wave is twice the distance from one wave to another.

Conditions for a stationary wave

Two identical but oppositely traveling waves interfere with each other to form a stationary wave. Often, one wave is the reflection of the other.

When the frequency of the loudspeaker is changed, a point is reached where the note becomes much louder. Sound waves are reflected by the water and a stationary wave has formed in the air column inside teh cylinder. There is a node at the foot of teh air column and an antinode at the top.

At the lowest frequency at which this occurs, the length of teh air column is one quarter of a wavelength of the sound. A stationary wave is formed again at three times this frequency, with three quarters of a wave fitting in the column.