Oscillations & Waves, Reflection & Refraction

WAVES (Part 2) (return to CONTENTS)


PRINCIPLE OF SUPERPOSITION (return to start of page)

Suppose that wave pulses travel in opposite directions along rope:

The individual molecules in the rope can only be in one place at a time, and so have to respond to both pulses simultaneously - so the pulses add together as they pass through each other, but they emerge again in their original form. By 'vector sum' we mean that the direction of the displacement produced by each wave at a point is taken into account when the waves are added. This is indicated in the way the following wave pulses are added to produce their resultants:
Coherent waves Consider two waves with the same wavelength and same velocity, but not in step:
The crests, for example, do not coincide, but their relative positions stay the same, i.e. the phase difference is constant, i.e. the waves are coherent.

Special cases



INTERFERENCE (return to start of page)

In a region where waves overlap, superposition occurs. This produces a steady pattern, called an interference pattern or a system of fringes, if:

a) Water waves

The following represents a top view of a ripple tank, with two ball-ended dippers producing circular ripples:

If the lines represent crests, then between each pair of crests is a trough. Along lines such as those indicated by dots, there are points at which crests coincide with each other and between which troughs coincide with each other. Superposition produces larger crests and troughs along these lines. Between the dotted lines, crests coincide with troughs producing cancellation. The above represents the waves produced by each dipper - these superpose to produce a steady interference pattern.

b) Light waves - Young’s double slit experiment (1801)

The set-up

In Young’s experiment two coherent light sources are produced by splitting a single light source:

'Monochromatic light' means light of a single frequency (or wavelength) or a very narrow band of frequencies - and therefore of a single colour.

The ‘slits’ need not actually be holes. They may be clear lines through which light can pass, on an otherwise black projector slide.

The single slit S diffracts light falling on it. Light from S is then diffracted by slits S1 and S2, which act as two coherent light sources.

In the region of overlap, interference occurs, and on the screen alternate bright and dark bands are produced, called interference fringes. The central band is bright, and then they are alternately dark and bright.

Recall that when two waves are in phase they add constructively, and when they are out of phase they add destructively.

The change from bright to dark is gradual, though this may not be obvious to the naked eye.

Note: A laser emits a monochromatic and coherent beam of light, and can be shone directly onto the double slits, and interference fringes are produced.

Conditions for bright and dark fringes

For simplicity, the sources S1 and S2 will now be represented by dots. Consider light arriving at some point P on the screen:

Light waves leave S1 and S2 in phase but they travel different distances to reach P, and so their phase difference at P depends upon the path difference (= S2P - S1P). Thus, the fringe at P is:

Fringe spacing, y

Let P be the nth and Q be the (n+1)th bright fringes (E.g. 3rd and 4th or 4th and 5th).

Typically (when using a light source), d<1mm while D may be several metres.


This gives the spacing between successive bright fringes, or between successive dark fringes - from a bright fringe to the next dark fringe the distance is half of y.

Example


Note: Interference is considered to be a typically wave-type phenomenon. Young’s double slit experiment can also be performed with sound waves.

Example

Two whistles with the same frequency of 1.50kHz are 4m apart, and are blown simultaneously. An observer moving along a line as indicated below, observes minima of sound at a series of points spaced 1.10m apart. Calculate the speed of sound in air.


DIFFRACTION OF LIGHT (return to start of page)

a) Single slit

Light passing through a sufficiently narrow single slit is diffracted significantly, and can produce a set of bright and dark bands on a screen:

The central band is bright, with alternate dark and bright bands either side of it.
If the slit is made narrower, the central band gets wider. It has is maximum width when the slit is about equal to the wavelength of the light.
b) Multiple slits

Two slits

Suppose that light passes through two, narrow, parallel, close slits (as in Young’s set-up). Each slit produces a similar diffraction pattern in the same direction. However, the diffraction pattern is now seen to be crossed by a number of bands, and these must be produced by interference between light from each slit.

When more slits are used, subsidiary maxima appear between the principal maxima.

Three slits

As the number of slits increases, the sharpness of the principal maxima increases. The number of subsidiary maxima rises, but they get weaker and weaker.

Six slits

c) Diffraction grating

A diffraction grating consists of a piece of glass or plastic with a large number of parallel lines marked on it. The lines scatter light but the thin clear strips between them transmit light and act as slits.

A grating may have hundreds of lines per mm, so following on from the above, only principal maxima are produced, the subsidiary maxima being too weak to be seen.

If the rays diffracted at a certain angle are brought to a focus, they will constructively interfere, producing a bright image, if they are in phase. The rays leave corresponding points, such as A and B, in phase, and they will be bought to focus in phase if the path difference AC equals zero or a whole number of wavelengths.

Thus, constructive interference occurs if:


The value of n gives the ‘order’ of an image:

Note: If a grating is stated as having N lines per mm (per cm or per metre), then d in mm (or cm or metres) can be found simply from:

Maximum value of n


Example

For yellow light of wavelength 600nm, using a grating with 500 lines per mm, what is the highest order image visible?

N = 500 lines per mm = 500*1000 lines per metre. Wavelength 600 nm = 600 * 10-9 m

Thus, the highest order visible is the third (n=3).

Dispersion by a grating


Note: sin-1 stands for ‘inverse sine’. To apply the above equation, the value of nNl is worked out on the calculator and then sin-1 (or the equivalent) is pressed to give the value of the angle.

White light is a mixture of colours from red to violet, and:


This means that the grating 'disperses' or separates the different wavelengths in the light. The visible effect is that the different colours in a light source become visible.

Looking through a diffraction grating at a white light source in an otherwise dark room, we see something like:

Example

Calculate the angular separation of the red and violet rays in the second order spectrum when white light is incident on a diffraction grating with 6000 lines per cm.
(wavelength of red light = 7 * 10-7m; wavelength of violet light = 4 * 10-7m)

N = 6000 lines per cm = 6000 * 100 lines per metre. Second order spectrum means that n = 2.


STATIONARY WAVES ON STRETCHED STRINGS (return to start of page)

The above arrangement allows us to control the frequency of the vibration, the length of the string and the tension in it. The stroboscope provides a flashing light of controllable frequency.

The vibrator produces progressive transverse waves which travel along the string and are reflected at the fixed end, at the pulley. Thus, there are similar waves travelling in opposite directions along the string. At certain frequencies this produces stationary (or standing) waves with 1, 2, 3 etc. loops.

The frequency of the stroboscope can be adjusted until the string is seen to be moving slowly or even ‘frozen’. Oscillations in one loop are seen to be in antiphase to oscillations in the next loop (i.e. one is going up while the next is going down).

In a standing wave (unlike a progressive wave):
  1. Energy is not transmitted, it is trapped in the loops
  2. Particles in one loop are in phase with each other, and so have their maximum displacements simultaneously
  3. The amplitude of oscillation varies from zero (at points called nodes, N) to maxima (at points called antinodes, A)
The end at the pulley is a node. The point of attachment to the vibrator is (nearly) a node - the vibrations only have to be big enough to make good any energy dissipated.


It can be seen in the above diagram, and is true for all the following diagrams that:

We can regard standing waves as a resonance phenomenon, i.e. the string has certain natural frequencies and when it is forced to vibrate at one of these, a large amplitude vibration occurs. Hence, the natural frequencies are also called resonance frequencies.

Expression for f0

The velocity v (m/s) of a transverse wave travelling along a stretched string, subject to a tension T (newtons), is given by:


STATIONARY WAVES IN AIR COLUMNS (return to start of page)

1. Pipe closed at one end

An air column in a pipe has certain natural frequencies, depending on its length, and if the frequency of a tuning fork, held next to the open end, equals one of these frequencies, resonance occurs and a loud note is heard.

Fundamental vibration

The vibrations of the air molecules are parallel to the length of the tube. The vibrations are a maximum at the open end, where the air is free to vibrate, so there is an antinode there and the vibrations actually stick out a bit as indicated. At the closed end the air cannot vibrate sideways, so there is a node there.

Representation

To represent the above graphically we, in effect, turn all the arrows though 900 and join their points end to end:

This is a convenient representation, particularly when we consider more complex standing waves.

If we hold a speaker over the open end, emitting a note which is gradually increased in frequency, a sequence of standing waves is set up, with successively higher pitches.

In drawing diagrams representing these, remember that:

  1. There is always a node at the closed end, since the air cannot vibrate freely
  2. There is always an antinode at the open end, since the air is free to vibrate
  3. Nodes and antinodes alternate
And as before:

Notice that only odd numbered multiples of the fundamental frequency occur.

2. Pipe open at both ends

The air in a pipe open at both ends will also resonate when subject to appropriate forcing frequencies.

Fundamental vibration

In this case the air is free to vibrate at both open ends, so these are always antinodes. In between two antinodes there must be a node.

Representation
Again, if we hold a speaker over an open end, emitting a note which is gradually increased in frequency, a sequence of standing waves is set up, with successively higher pitches.

In drawing diagrams representing these, remember that:

  1. There is always an antinode at both open ends
  2. Nodes and antinodes alternate
And as before:

Notice that all whole number multiples of the fundamental frequency occur.

Example

A small loudspeaker is sounded over the open end of a vertical tube 50cm long, closed at the lower end. The frequency of the note is gradually increased from 100 Hz. At what frequencies will the fundamental resonant note and the first two overtones be heard, given that the speed of sound in air is 340ms-1?

For a pipe closed at one end, the fundamental standing wave has a node at the closed end and an antinode at the open end.

\ the fundamental wavelength = 4 * length of pipe = 4 * 50 = 200 cm = 2.0 m, so:


End correction

In practice, an antinode at the open end of a pipe occurs at a distance c beyond the end, called the end-correction.

a) For a pipe closed at one end, the effective length is (l + c):

b) For a pipe open at both ends, the effective length is (l + 2 c)

Determining speed of sound

Placing the resonance tube in a column of water is a convenient way of controlling the length of the air column in the tube.

The resonance tube is almost completely immersed, and then a tuning fork is sounded over the open end, and the tube is slowly raised until a loud note is heard. The fundamental vibration now exists in the tube, as represented in the diagram above.

Thus, if a set of lengths are determined, as described above, for tuning forks of various frequencies, a graph of l (y axis) against 1/f should be a straight line with slope equal to v/4, and so the speed of sound, v = 4 * slope.


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A Level Physics - Copyright © A C Haynes 1999 & 2004