The waves 1 and 2 are alternatively in and out of phase at i, ii, ii and iv. Notice also in the above diagram that the amplitude or maximum displacement of the result of adding the two waves,is the sum of the two amplitudesand– constructive interference – and the minimum is zero – destructive interference.

The beat period is

]]>Oscillatory systems can also display chaotic motion if subject to a forcing term of certain frequencies. For example a pendulum which normally swings backwards and forwards at it's natural frequency, but if the fixed point is made to move up or down or backwards at certain frequencies, chaotic motion results. The pendulum moves around in a small space determined by movement of the fixed point and the length of the pendulum, but the pendulum never follows the same path twice. This is illustrated below.

]]>If light is composed of two plane waves of equal amplitude differing in phase by 90°, then the light is said to be circularly polarized. If you could see the tip of the electric field vector, it would appear to be moving in a circle as it approached you. If while looking at the source, the electric vector of the light coming toward you appears to be rotating counterclockwise, the light is said to be right-circularly polarized. If clockwise, then left-circularly polarized light. The electric field vector makes one complete revolution as the light advances one wavelength toward you. Another way of saying it is that if the thumb of your right hand were pointing in the direction of propagation of the light, the electric vector would be rotating in the direction of your fingers.

Circularly polarized light may be produced by passing linearly polarized light through a quarter-wave plate at an angle of 45° to the optic axis of the plate.

]]>Damping of this sort is often desirable and can be designed into a structure or machine. Large structures such as bridges experience vibrations which, if not damped, can shorten the life of the bridge. The shock absorbers on a car are designed to damp an oscillation very quickly – the vibrations may be dissipated by oil filled cylinders. At any instant the damping force usually increases with increasing velocity. A vibration may be lightly, heavily or critically damped.

]]>If the light incident on a diffraction grating is made up of several different colours or wavelengths, then for each value of n, there will be a bright fringe at a different value of

Atthere is a white maximum, because all the different wavelengths of light are in phase forOn either side of this maximum, white light is diffracted into fringes of different colours, with red light – having the biggest wavelength – having larger values offor eachaccording to (1) so being diffracted the most. Conversely, blue light has the smallest wavelength so is diffracted the least.

Notice also that the width of the fringe – the distance between the blue and red ends – increases as we move away from the central fringe. This is because while the angular width of each fringe might be the same or very similar, the images are formed on a flat screen, which tends to increase the width of the fringes corresponding toThe amount of light reaching the screen decreases for increasing m. This is due to the same effect as in a single fringe diffraction patter.

]]>orall imply simple harmonic motion.

Ifthen we can find the velocityand the accelerationby differentiation:

This is illustrated in the diagram below.

Notice thatandhave opposite signs, andislarger in magnitude.

The displacement leads the velocity byof a cycle and the velocity leads the acceleration byof a cycle.

]]>There are different equations for the Doppler shift for light and sound waves. The equation for sound waves is given and illustrated below.

The frequency of sound heard by the observer is

The frequency of sound heard by the observer is

**Doppler Shift for Electromagnetic Radiation **

(1). Hereis the speed of the observer relative to the source, negative if they approach and positive if they recede from each other. We can write an approximate expression for (1) ifis a small fraction of the speed of light.then , the actual Doppler shift, given by

]]>Originally {jatex options:inline}T= 2 \pi \sqrt{ \frac{L}{g}}{/jatex}, and then {jatex options:inline}2T= 2 \pi \sqrt{ \frac{L+0.1}{g}}{/jatex}.

Dividing the second equation by the first gives {jatex options:inline}2= \sqrt{ \frac{L+0.1}{L}}{/jatex}.

Then {jatex options:inline}4= \frac{L+0.1}{L} \rightarrow 4L=L+0.1 \rightarrow 3L=0.1 \rightarrow L=\frac{0.1}{3}= \frac{1}{30}{/jatex}m.]]>

For the maximum in the first diagram above,and for the second vibration,

Subtracting these gives

Hence

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