- PRINCIPLE OF SUPERPOSITION
- INTERFERENCE
- DIFFRACTION OF LIGHT
- STATIONARY WAVES ON STRETCHED STRINGS
- STATIONARY WAVES IN AIR COLUMNS

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.*The principle of superposition states that when two waves meet, the total displacement at a point is the vector sum of the displacements due to each wave at that point*

*Two waves are coherent if they have a constant phase difference between them*

The crests, for example, do not coincide, but their

**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:**

*the sources are coherent*(i.e. they have a constant phase difference between them)*the waves have approximately equal amplitude*

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 S_{1} and S_{2}, 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 S_{1} and S_{2} will now be represented by dots. Consider
light arriving at some point P on the screen:

Light waves leave S

- If S
_{2}P - S_{1}P equals zero or a whole number of wavelengths, then the waves arrive at P still in phase, and constructive interference occurs producing a bright fringe - If S
_{2}P - S_{1}P equals an odd number of half wavelengths, then the waves arrive at P out of phase, and destructive interference occurs producing a dark fringe

**Fringe spacing, y**

Let P be the n

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.

**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**

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:

**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:

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^{-}^{7}m; wavelength of violet light = 4 *
10^{-}^{7}m)

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 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):- Energy is not transmitted, it is trapped in the loops
- Particles in one loop are in phase with each other, and so have their maximum displacements simultaneously
- The amplitude of oscillation varies from zero (at points called nodes, N) to maxima (at points called antinodes, A)

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 f _{0}**

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.

To represent the above graphically we, in effect, turn all the
arrows though 90^{0} 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:

- There is always a node at the closed end, since the air cannot vibrate freely
- There is always an antinode at the open end, since the air is free to vibrate
- Nodes and antinodes alternate

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.

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:

- There is always an antinode at both open ends
- Nodes and antinodes alternate

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.

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