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13 Wave Representations

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Introduction We see waves on the surface of water. They travel across the surface of the water transferring energy:Molecules of the water move up and down.A wave is a periodic disturbance of the water.

Wave travels horizontally

Molecules vibrate up and down (approximately)

The diagram represents the wave as an idealised sine wave. This idea can be used as a model for other phenomena

Sound waves travel through air (or any other medium)

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Wave travelling horizontally

Particles vibrate back and forth

compression

rarefactionrarefaction

Although we may represent sound waves as a sine curve, the particles move back and forth, not up and down.

Light (and other electromagnetic waves) do not require a medium.

They are periodic disturbances of the electric and magnetic fields through which they are travelling.

These fields vary at right angles to the direction of travel.

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Transverse and longitudinal wavesTransverse waves can be made to travel along a stretched rope, by moving one end up and down (or from side to side)Both transverse and longitudinal waves can be demonstrated using a long spring (slinky).For longitudinal waves, the end of the slinky must be pushed back and forth.But it is simplest to represent both types of waves as sine waves.

Longitudinal wave

V V Transverse wave

Vibrations are perpendicular to direction of travel

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Polarisation Light and other transverse waves can be polarised.In un-polarised light, the electric and magnetic fields vibrate in all directions perpendicular to the direction of travel.After passing through a piece of Polaroid, each vibrates only in one direction.Only transverse waves can be polarised.

Wave fronts and raysThe ripple tank shows another way to represent waves.We draw wave fronts as though we are looking down on the ripples from above.Rays can be added: these are always perpendicular to the wave frony

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Wave fronts

rays

Circular waves spreading out from a point source

Note that the separation of the wave fronts is constant

All waves can be reflected and refracted.

When a wave enters a medium where it travels more slowly, its wavelength decreases, but its frequency remains constant.

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Questions 1. Classify the following as transverse or

longitudinal: light, sound, water, infrared waves.2. A guitarist plucks a string. A wave travels along

the string. Is this longitudinal or transverse?3. Draw a ray diagram to show a single ray being

reflected by a mirror at 45o to its path. Add wave fronts to show how these are reflected by the mirror.

4. Copy and complete the diagram to show what happens when waves enter a medium where they travel more slowly. The boundary is parallel to the wave fronts.

slowerfaster

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Wave quantitiesSeveral quantities are needed to fully describe a wave: amplitude, wavelength, frequency, phase.Take care not to confuse them.

Wavelength and amplitudey

x/t

Horizontal axis = distance or timeAmplitude is the height of a crest measured from the horizontal axis

T

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Wavelength and amplitude

The displacement y is the distance moved by any particle from its undisturbed position.The wavelength λ of a wave is the distance between adjacent crests (or troughs), or between any two adjacent points which are at the same point in the cycle.(i.e. which are in phase with each other)The amplitude a of a wave is the maximum displacement of any particle.

Period and frequencyThe period T is the time for one complete cycle of the wave.This is related to the wave’s frequency f: T = 1 / f (or f = 1 / T)

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Frequency is measured in hertz (Hz) 1 Hz = 1 wave/s = 1 s-11 kHz = 103 Hz 1 MHz = 106 Hz 1 GHz = 109 Hz

Think of it like this: the frequency is the number of waves per second; the period is the number of seconds per wave.

Phase differenceTwo waves may have the same wavelength but may be out of phase (out of step)

Phase difference is expressed as a fraction of a cycle, or in radians (rad) or degrees (o)

1 cycle = 1 complete wave = 2π rad = 360o

½ cycle = π rad = 180o ¼ cycle = π/2 rad = 90o

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Measuring frequencyTo find the frequency of a sound wave, plug a microphone into an oscilloscope (c.r.o) and use it to display the sound.Step 1 adjust the time-base setting to give two or three complete waves on the screen

Time-base setting = 0.02 s div-1 (time-base settings may be given in divisions or centimetres)

Step 2 measure the width of a number of complete waves

Two waves occupy 5.0 divisionsStep 3 calculate the time represented by this number of divisions.

Time = 5.0 div x 0.02 s div-1 = 0.10 sStep 4 calculate the frequency = number of waves / time Frequency = 2 waves / 0.10 s = 20 Hz

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Questions 1. Calculate the period for waves of the following

frequencies: 2 Hz, 2 kHz, 0.5 MHz.2. On a displacement-time axes, sketch two waves

with a phase difference of π radians; one wave has twice the amplitude of the other.

3. An oscilloscope is set with its time-base at 5 ms cm-1. An alternating signal gives foue complete waves across the 6 cm screen. What is the frequency of this signal.