Peter's Physics Pages

An Introductory Physics Course with Peter Eyland
Lecture 3 (Waves)

In this lecture the following are introduced:
• Waves moving energy through a medium without moving the whole medium.
• Longitudinal and transverse waves.
• Amplitude, wavelength, period, frequency and phase difference.
• Reflection, refraction, diffraction, interference and polarisation.

Waves transfer energy

Any vibrating body that is connected to its environment will transfer energy to its environment. The vibrations are then transferred though the environment from neighbour to neighbour. This energy transfer is called wave motion. Waves move energy through a medium without moving the whole medium.

Leonardo di Vinci
"waves made in a field of grain by the wind, ... we see the waves running across the field while the grain remains in place."

Types of waves

Longitudinal waves

When waves transfer energy by pushing neighbours in the same direction that the energy moves, the waves are called longitudinal waves. In the simulation below (from Dr. Daniel A. Russell) you can see energy move to the right while individual particles vibrate to the left and right about fixed points. The places when the particles cluster together are volumes of high pressure so these waves are also called pressure waves. Sound waves are an example of pressure waves and they can move through gases, liquids and solids. For sound waves, the denser the medium the faster the speed.
Speed through air (1atm, 200) =344 m.s-1
Speed through sea water = 1531 m.s-1
Speed through iron = 5130 m.s-1

Transverse waves

When waves transfer energy by pulling neighbours sideways to the direction of travel, the waves are called transverse waves. In the simulation below (also from Dr. Daniel A. Russell) you can see energy move to the right while individual particles vibrate up and down about fixed points. Electromagnetic waves (X-rays, light, radio, radar and TV waves) are examples of transverse waves formed by electric and magnetic fields vibrating together at right angles to the wave's motion. They don't need any medium so they can move through a vacuum, (good for us or we wouldn't see the Sun!). They all move at the same speed of 300,000 km.s-1 when they travel through vacuum. They slow down when they travel through a medium (this is an average speed between interactions).

Mechanically twisting or pulling a medium sideways is called shearing so waves formed this way are also called shear waves.

Longitudinal and Transverse waves together

Sometimes longitudinal and transverse waves occur together. Ocean waves are a combination of longitudinal and transverse waves because the surface of the water can be pulled sideways as well as pushed longitudinally. In the simulation below (also from Dr. Daniel A. Russell) you can see energy move to the right while individual particles move clockwise in circles or ellipses. When ocean waves get to a shelving beach the speed of the waves changes relative to each other and circles go to ellipses and then the wave breaks.

Seismic waves are formed when there is a sudden movement (or slip) between layers in the Earth's crust. This may happen anywhere between several km and several 100s km down from the surface. The wave motions that occur through the crust have Pressure ("P") components and Shear ("S") components.
The P waves move at 5 - 14 km. s-1
The S waves move at 3 - 8 km. s-1
When they reach the surface an Earthquake occurs, and the timing between the arrivals of the The S and P waves and their sizes at different places will enable the epicentre to be determined. (Note: seismic waves can also have "surface" waves. Again see Dr. Daniel A. Russell's excellent pages for details.)

Measuring Waves

Waves have peaks (maxima) and troughs (minima). The maxima link to form "wavefronts" which can be lines (like "lines of breakers" at the beach) or more generally areas. Plane (or "Straight") waves have wavefronts which are planes (seen as straight lines in 2 dimensions). The direction that a wave travels (or "propagates") is always at right angles to the wavefronts and usually indicated by arrowed lines or "rays". • The distance between maxima (or minima) is called the Wavelength and is measured in metres. Wavelength is usually given the symbol λ (lambda).
• The Amplitude measures the "size" of the wave. It is half the measurement between a minimum to a maximum and can be a height or a pressure etc.
• The time between successive maxima arriving is called the Period. It is usually given the symbol T, and measured in seconds.
• The Frequency is the number of maxima that pass per second. It is often given the symbol ν (nu). It is the reciprocal of the Period, so ν=1/T. The unit of frequency is s-1, which is given the special name Hertz. Be careful to distinguish ν from velocity (v) - even though they look much the same here in unicode! ν should come to a point at the bottom and both sides curve like an angled ) - see pictures below. When v (velocity) is hand written it is (usually) round at the bottom.

The speed of a wave is measured by the time for one wavelength to pass by. Example 3.1
Sound waves travel at 1530 m.s-1 below the surface of the ocean, with a wavelength of 1.02m. Find the frequency of the waves. Example 3.2
Seismic Shear waves travel at 4000 m.s-1 and have a period of 0.12s. Find the wavelength of these waves. Example 3.3

An ocean wave has a period of 15s and wavelength 345m. Find the wave speed. Phase difference

The moon has phases, which go from new, to 1st quarter, to full, to 3rd quarter etc. The phase of something tells us how far through a cycle it is. A full cycle is called 3600 or 2π radians, (that's 2 x Pi). Radians are the preferred unit. To understand radians, think of a circle with two radii drawn from the centre outwards to the circular curve. The angle at the centre between those two radii is measured by the arc length around the circular curve divided by the radius. For a full circle the arc length is the circumference (2πr) divided by the radius, i.e. 2π. For a wave, each point along it will have a phase angle associated with it. Example 3.4
A wave has a wavelength of 1.4m. Find the phase difference between a point 0.3m from the peak of a wave and another point 0.7m further along from the same peak. Waves meeting objects

Reflection

Reflection occurs when a surface throws back the heat, light, sound etc that falls on it. When light falls on mirror surfaces, images are formed and "magicians" can dazzle their viewers.

For waves, the "glancing" or "grazing" angle is the angle between the incident wavefront and the surface it meets.
The "incident" angle is the angle between the normal to the incident wavefront and the normal to the surface. The "reflected" angle is the angle between the reflected "ray" and the normal to the surface. When a plane wave meets a plane reflector the incident angle equals the reflected angle. When plane waves meet a concave spherical reflector the reflected wavefronts converge to a point. This point is called the focal point. When spherical waves from a point source in front of a plane reflector meet the reflector, then the reflected waves are spherical and appear to come from a source which is the same distance behind the reflector as the point source was in front of it. When circular waves originate from the focal point of a concave spherical reflector and move towards the reflector, then the reflected waves have plane wavefronts. Refraction

Refraction gives us rainbows, corrective lenses for our eyes, telescopes, movies and mirages. Refraction is a change in the direction of a wave, that is caused by a change in the wave's speed as the medium that it is passing through changes.

Diagram from A.Ryer It may be an actual change from one medium to another or, simply a change in the properties of the same medium.

Diffraction

When waves pass a gap, or an edge which is of a comparable size to the wavelength, the wavefront bends near the edge. This means the waves spread out and the phenomenon is called "diffraction". Note that the medium does not change as in refraction. Interference

When waves pass through each other at a point in space, the result at that point is simply the sum (or "superposition") of the wave heights. If two peaks occur at the same spot the result is a higher peak. This is called constructive interference. If two troughs occur at the same spot the result is a deeper trough. This is also constructive. If a peak and a trough occur at the same spot there is a small result (zero - if they are the same amplitude). This is called destructive interference because the effect of the waves is cancelled out. Stable patterns of constructive and destructive interference are called interference patterns.

Polarisation

Polarisation gives us blue skies, red sunsets, enables bees to find their way home, and sunglasses that cut out reflected glare. Normally waves come bundled together in overlapping groups of limited length.
Overlapping transverse waves usually have their particles vibrating with random orientations so that the resulting vibration direction of the group is continually changing. This is said to be an "unpolarised" wave.
When transverse waves have their particles always vibrating in the same plane, the wave is said to be "plane polarised". Longitudinal waves can't be polarised because their particles vibrate in the same direction that the wave travels.
A "polariser" forces the transverse waves which emerge from it, to be vibrating only in one plane. A second polariser can be rotated in the emergent waves to check on the polarisation. When two polarisers have their polarising directions at right angles, no waves emerge from the second polariser.
For more on polarisation see Polarized Light in Nature and Technology

Summarising:

Waves move energy through a medium without moving the whole medium.
Longitudinal waves transfer energy by pushing neighbours in the same direction that the energy moves.
Transverse waves transfer energy by pulling neighbours sideways to the direction that the energy moves.
The amplitude is half the measurement between a minimum to a maximum.
The wavelength is the distance between successive maxima.
The period is the time between successive maxima.
The frequency is the number of maxima that pass per second.
The phase tells us how far through a cycle something is.
Reflection occurs when a surface throws back the heat, light, sound etc that falls on it.
Refraction is the change of direction that occurs when the medium changes.
Diffraction occurs when a wavefront bends near a gap or an edge in the same medium.
Interference occurs when waves pass through each other at a point in space and the result at that point is the sum of the wave heights.
Polarisation occurs for transverse waves when the vibration directions are not randomly oriented. email Write me a note if you found this useful