2D AND 3D WAVEFRONTS

Contents for this page	Related topics
Principle of superposition of waves Diffraction Interference Standing waves Interference effects with light Interference of light by a double slit Polarization Shock waves Additional questions	Doppler effect with sound Doppler effect with light Light and colour Wave nature of matter	Data Glossary
Learning Outcomes
After studying this section, you will (a) understand the concepts of "constructive" and "destructive interference", (b) the phenomena of diffraction and polarisation and (b) be familiar with the experimental evidence for the wave nature of light.

Principle of superposition of waves:

Consider a wave (or pulse) of amplitude a travelling from left to right in a medium:

Now consider a wave of amplitude a travelling  in the opposite direction in the same medium:

Let us now consider two waves A and B, such that A moves to the right, and B will move to the left. At any point, the displacement is the sum of the separate displacements due to each wave. This is called the principle of superposition.

Observe what happens as the two waves move towards each other. Note that the maximum displacement of a particle in the medium is the sum of the amplitudes of the waves. If the displacement is in the same direction, as is the case here, we have a superposition called CONSTRUCTIVE INTERFERENCE.

We will now repeat the process with wave B having an amplitude equal to (-amplitude of A). Observe what happens as the two waves move towards each other.

In this case the particles of the medium are displaced in opposite directions in each of the waves, and when they superpose, DESTRUCTIVE INTERFERENCE occurs.

Interference can also take place when two waves travel in the same direction

The figure on the right shows two waves travelling at the same speed from left to right. They produce a resulting wave where at any point, the value of the amplitude is the algebraic sum of the amplitudes of each wave at that point. In places, the resulting amplitude is greater than the amplitude of either wave. This is constructive interference. Where the amplitude is less than the amplitude of either wave, we have destructive interference:
Consider a special case where two waves have the same amplitude and wavelength (you only see one because the one obscures the other) travelling at the same speed from left to right. The two waves are in phase, since their maxima and minima coincide exactly. The resulting wave is simply one with the same wavelength but double the amplitude:
In the figure on the right, the two waves are out of phase by 90º (wave 2 is ¼ of a wavelength ahead of wave 1). Again we see a single wave, whose wavelength is the same as the two contributing waves, but whose amplitude is greater than the amplitudes of the contributing waves:
When the waves are 180º degrees out of phase (the position of their maxima are ½ wavelength apart), destructive interference occurs and the resultant wave has zero amplitude:

Standing waves:

When two waves of the same frequency and amplitude travel in opposite directions in the same medium, STANDING WAVES are set up.

Standing waves are characterized by a set of points at which the particles of the medium do not move. These points are called NODES. Nodes are spaced at a distance of λ/2.

Away fom the nodes the particles vibrate with varying amplitudes. The maximum amplitude occurs at the ANTINODES. The antinodes are separated by λ/2.

In the case of standing waves in a vibrating string which has both ends fixed, nodes occur at both ends. Thus, the only possible patterns include a whole number of half-wavelengths.

In the case shown above, λ = L/3, where L is the distance between the fixed ends of the vibrating string.

In general, (N - 1) λ/2 = L

or, λ = 2L/(N - 1)

where N is the number of nodes, and L is the length of the string.

Diffraction:

When a plane wave encounters an obstacle or an aperture through a barrier, the wave spreads out. This phenomenon is called DIFFRACTION.

Diffraction can readily be observed in a ripple tank, for example,

The animations () are simulations of waves in a ripple tank. In the left-hand animation, a wave front impinges on an obstacle that has a slit. The slit then acts as a source of further wave fronts. In the right-hand animations, we see that as a result of diffraction, wave fronts can propagate around a corner.

The diffraction will increase as the wavelength increases, and decreases with increasing aperture width.

Interference:

When two or more waves of the same frequency overlap, the phenomenon of INTERFERENCE occurs. The amplitude of the resultant wave at any point can be calculated by using the principle of superposition.

For example, interference between the waves resulting from two point sources X and Y can be demonstrated in a ripple tank:

Observe that there are lines along which the troughs and the crests intersect. These are called nodal lines, along which the resultant disturbance of the medium is zero.

There are also lines along which the occurence of crests intersecting with crests alternate with troughs intersecting with troughs. Along these lines of reinforcement the resulting disturbance is the greatest.

The number of nodal lines between the sources can be increased by increasing the distance between the sources, as shown above, or decreasing the wavelength of the waves as shown below (this may be achieved by increasing the frequency of oscillation of the sources).

A constant interference pattern will only be set up if the point sources are in phase or have a constant phase difference. Such sources are said to be coherent.

Interference effects with light:

The interference effects described above occur only as a result of the superposition of waves which have the same velocity, frequency, wavelength, amplitude and have a constant phase difference.

Light shows analogous effects. For example, after passing through a narrow slit, a homogeneous monochromatic beam of light is found to diverge from the slit and displays a structure of alternating light and dark bands.

Light can therefore be diffracted, the extent of the diffraction being proportional to λ, the wavelength of the light, and inversely proportional to the width of the slit, w. It can be shown that the angle Θ, at which the first dark line occurs is given by

Interference of light by a double slit:

If we place an obstacle with two slits of an appropriate size in the path of light wave fronts, diffraction occurs at each slit. The secondary waves that are so produced will interact with each other, as shown in the animation () on the right:

If monochromatic light passes through a pair of narrow slits that are an appropriate distance apart, then a pattern of alternating light and dark bands is produced:
	The effects observed in this experiment are in every way analogous to those observed with two oscillating sources in the ripple tank.

The slits are analogous to the point sources. The dark lines are analogous to the projections of the nodal lines.

The number of dark lines may be increased by decreasing the separation of the slits or by decreasing the wavelength of the light. This experiment is proof that light has a wave nature.

What is meant by "polarized light"? (Click here for a discussion)

Shock waves:

Shock waves occur when an object moves through a medium at a speed greater than sound is able to travel through that medium. They are readily produced by explosions and bursting balloons (where gas molecules are propelled outwards at supersonic speeds), or fast moving aircraft and projectiles.

The picture above on the left shows shock waves arising from different parts of an aircraft-line body moving at supersonic speed in air. On the right, an F/A-18 Hornet strike fighter jet at the moment of breaking the sound barrier (pictures in the public domain). The shock waves from aircraft have high energy intensities. An observer on the ground will experience a sudden high intensity sound, the so-called "sonic boom", not unlike a clap of thunder. The diagram on the right, which makes use of Huygens' construction, shows the situation. The wavefront is contained in a cone, the MACH CONE, with semi-vertical angle φ, such that (sinφ)-1 = vs/c, where Vs is the speed of the moving object, and c the speed of sound under the prevailing conditions of temperature, pressure and humidity. The ratio vs/c is called the MACH NUMBER.

Construction of Mach cone:

Consider an object (the source) moving at a speed vs > c, the speed of sound under the prevailing conditions. In a time, T, the source will have generated a wavelet of radius cT, but will have moved to a position vsT.
At a time 2T, the source will produce another wavelet, of radius cT, but will have moved to a new position 2vsT. The original wavelet will now have a radius 2cT.
At a time 3T, the source will produce another wavelet, of radius cT, but will have moved to a new position 3vsT. The previous wavelet will now have a radius 2cT, while the original wavelet will now have a radius 3cT.
At a time 4T, the source will produce another wavelet, of radius cT, but will have moved to a new position 4vsT. All previous wavelets will incease their radii by ct.
The resulting wavefront at this time, 4T, will be the envelope of the wavelets, which will intersect at the source position, 4vsT
See an animation of the above sequence. Observe how the shock wave travels.