on a station platform and a train sounding its whistle is approaching at ...... he electron microscope makes use of the
WAVE PHENOMENA
WAVE PHENOMENA (SL Option A2) Standing (Stationary) Waves
11.2
(SL Option A3) Doppler Effect
11.3
(SL Option A4) Diffraction
11.4
(SL Option A5) Resolution
11.5
(SL Option A6) Polarization
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11.1
11.1 (sL OPTiON A.2) sTANdiNg (sTATiONAry) WAVEs 11.1.1 Describe the nature of standing (stationary) waves. 11.1.2 Explain the formation of one-dimensional standing waves. 11.1.3 Discuss the modes of vibration of strings and air in open and in closed pipes. 11.1.4 Compare standing waves and travelling waves.
of course oscillating but the wave is not moving forward. You can actually get the wave to appear to stand still by illuminating it with a strobe light that lashes at the same frequency of vibration as your hand. he fact that the wave is not progressing is the reason why such waves are called standing or stationary waves.
A
B
A
B
L
L
diagram 1
diagram 2
Figures 1101 and 1102 11.1.5 Solve problems involving standing waves. © IBO 2007
11.1.1 (A.2.1) THE NATurE Of sTANdiNg WAVEs
A
very interesting situation arises when considering a travelling wave that is relected and the relected wave interferes with the forward moving wave. It can be demonstrated either with a rubber tube, stretched string or a slinky spring attached to a rigid support. In the diagram the tube is attached at A and you set up a wave by moving your hand back and forth at B. If you get the frequency of your hand movement just right then the tube appears to take the shape as shown in Fig 1101. When you move your hand faster you can get the tube to take the shape shown in Fig 1102. All points other than A and B on the tube are
here are several things to note about standing waves. he fact the wave is not moving forward means that no energy is being propagated. If you increase the amplitude with which you shake your hand, this increase in energy input to the wave will result in a greater maximum displacement of the tube. It can also be seen that points alomg the wave oscillate with diferent amplitudes. In this respect, the amplitude of a standing wave clearly varies along its length. To illustrate this, Fig 1103 shows a standing wave set up in a string of length 30 cm at a particular instant in time. Fig 1104 shows the variation with time t of the amplitude x0 of the string at a point 13 cm along the string.
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11.1.2 (A.2.2)THE fOrMATiON Of
10 8 6
sTANdiNg WAVEs
4
d /cm
2 0 -2
0
5
10
15
20
25
x /cm 30
-4 -6 -8 -10 displacement d of string vs distance x along string
Figure 1103 A standing wave 4 3 2
x0 /m
0 0
0.05
0.1
0.15
-1
0.2
t /s
0.25
-2
A
-3
20
-4
15
amplitude x 0 vs time t
Figure 1104
Variation of amplitude with time
Since energy is not propagated by a standing wave, it doesn’t really make a lot of sense to talk about the speed of a standing wave.
displacement of tube/cm
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A standing wave arises from the interference of two waves moving in opposite directions. To understand this, let us look at the situation of a standing wave in a string or tube as described above. Initially when you start moving the free end of the tube up and down, a wave travels along the tube. When it reaches the ixed end, it is relected and, as described in Topic 4, the relected wave is π out of phase with the forward (incident) wave. he forward wave and relected wave interfere and the resultant displacement of the tube is found from the principle of superposition. his is illustrated in Fig 1105 which shows, at a particular incident of time, the displacement of the tube due to the incident wave, the displacement of the tube due to the relected wave and the resultant displacement due to the interference of the two waves.
5
Also, since at any one time all the particles in a standing wave are either moving “up” or moving “down”, it follows that all the particles are either in phase or in anti-phase with each other.
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forward wave
0 -5
0
10
20
30
reflected wave standing wave
N
-10 -15 -20
his speed is the speed of a travelling wave in the string. As we have seen, the speed of a travelling wave is determined by the nature and properties of the material through which it travels. For the string therefore, the speed of the travelling wave in the string determines the frequency with which you have to oscillate the string to produce the standing wave. (From Figure 1104 we see that for rhis situation the frequency of oscillation of the string is f = 1/T = 1/0.25 = 4.0 Hz)
N
10
A
Figure 1105 Nodes and anti-nodes If at the ixed end, due to the incident wave, the tube is moving upwards, then due to the relected wave it will be moving downwards such that the net displacement is always zero. Similarly, at the mid-point of the string, the displacements of the tube due to each wave are always in anti-phase. Hence the net displacement at this point is always zero. Points on a standing wave at which the displacement is always zero, are called nodes or nodal points. hese are labelled N in Fig 1105. he points at which a standing wave reach maximum displacement are called antinodes. In Fig 1105, the antinodes are at points one quarter and three-quarters the length of the tube (7.5 cm and 15 cm) and are labelled A. he amplitude of the forward wave and the relected wave is 10 cm, hence the maximum displacement at an antinode in the tube is 20 cm. he maximum displacement at the antinodes will occur at the times when the forward and relected waves are in phase.
WAVE PHENOMENA Fig 1106 shows an instance in time when the interference of the forward and relected wave produce an the overall displacement of zero in the standing wave. 20 15 10 5
forward wave reflected wave
0 0
5
10
15
20
25
-5
30
standing wave
-10 -15
that the ear will hear above all the others. his is what enables you to sing in tune with the note emitted by the plucked string. If we were to vibrate the stretched string at a frequency equal to the fundamental or to one of its harmonics rather than just pluck it, then we set up a standing wave as earlier described. We have used the phenomenon of resonance to produce a single standing wave. In the case of the stretched string it has an ininite number of natural frequencies of oscillations, each corresponding to a standing wave. Hence, when plucked, we obtain an ininite number of harmonics.
Figure 1106 Forward and reflected waves
Exercise
Sketch the shape of the forward and relected wave in a string at an instant in time that results in the antinodes of the standing wave having maximum displacement.
11.1.3 (A.2.3) sTANdiNg WAVEs iN sTriNgs ANd PiPEs If you take a wire and stretch it between two points and pluck it in the middle then you will actually set up standing waves in the wire as described above. he number of standing waves that you set up will actually be ininite. Fundamental (1st harmonic)
2nd harmonic
3rd harmonic
4th harmonic
Figure 1107
The first four modes of vibration
Fig 1107 shows the irst four modes of vibration i.e.standing waves in the string he modes of vibration are called harmonics.he irst harmonic is called the fundamental. his is the dominant vibration and will in fact be the one
Fig 1107 shows part of what is called a harmonic series. Diferent fundamentals can be obtained by pinching the string along its length and then plucking it or by altering its tension. In a violin for example the four strings are of the same length, but under diferent tensions so that each produces a diferent fundamental. he notes of the harmonic series associated with each fundamental are obtained by holding the string down at diferent places and then bowing it. he harmonics essentially efect the quality of the note that you hear. he presence of harmonics is one reason why diferent types of musical instruments sounding a note of the same frequency actually sound diferent. It is not the only reason that musical instruments have diferent sound qualities. An A string on a guitar sounds diferent from the A string of a violin, because they are also produced in diferent ways and the sound box of each instrument is very diferent in construction. he actual construction of the violin for example distinguishes the quality of the notes produced by a Stradivarius from those produced by a plastic replica. Fig 1107 enables us to derive a relationship between the wavelengths of the harmonics and length of the string. If the length of the string is L then clearly, the wavelength of the fundamental (irst harmonic) is λ = 2L, for the second 2L harmonic λ = L and for the third harmonic λ = . From 3 this sequence, we see that the wavelength λn of the nth harmonic is given by
λn =
2L n
Resonance and standing waves also play their part in the production of sound from pipes. If you take a pipe that is open at one end and blow across the top it will produce a sound. By blowing faster you can produce a diferent sound. In this situation you have used resonance to set up a standing wave in the pipe. It is now the air molecules inside the pipe that are set vibrating. he sound wave that
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CHAPTER 11 you create at the open-end travels to the bottom of the pipe, is relected back and then again relected when it reaches the open end. he waves interfere to produce a standing wave. However, when waves are relected from an “open” boundary they do not undergo a phase change so that there is always an antinode at the open end. he fundamental and the irst three harmonics for a pipe open at one end are shown Fig 1108. A
1st harmonic (Fundamental)
A
3rd harmonic
A
5th harmonic N
N
N
N
N
11.1.4 (A.2.4) COMPArisON Of TrAVELLiNg ANd sTANdiNg WAVEs he Figure 1110 summarises the diferences between travelling and standing waves. Property Energy propagation Amplitude Phase diference
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A pipe that is open at both ends and that has the same dimensions, as the previous pipe, will produce a diferent fundamental note and a diferent harmonic series. his is shown in Fig 1109 A
A
2nd harmonic
A
3rd harmonic N
Single amplitude Variable amplitude All phase diferences Only 0, 2π and π between 0 and 2π phase diference
Since travelling waves have a single amplitude, it follows that there are no nodal or anti-nodal points in a travelling wave.
11.1.5 (A.2.5) PrObLEMs iNVOLViNg sTANdiNg WAVEs
N
N
Example
N N A
A
A
The harmonics of a pipe open at both ends
We see that, whereas a pipe open at both ends produces all the odd and even harmonics of the fundamental λ = 2L, 2L (λn = ), a pipe closed at one end can produce only the n odd harmonics of the fundamental λ = 4L. With open and close pipes we are essentially looking at the way in which organs, brass instruments and woodwind instruments produce musical sounds.
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Not propagated
Figure 1110 Comparing travelling and standing waves
N
Figure 1109
Propagated
Standing wave
N
Figure 1108 The harmonics of a pipe open at one end
1st harmonic (Fundamental)
Travelling wave
he speed v of a wave travelling in a string is given by the expression
v=
T µ
where T is the tension in the string and µ is the mass per unit length of the string. (a)
Deduce an expression for the frequency f of the fundamental in a string of length L.
(b)
Use your answer to (a) to estimate the tension in the ‘A string of a violin (frequency = 440 Hz).
WAVE PHENOMENA
11.2 (sL OPTiON A.3) dOPPLEr EffECT
Solution
he wavelength of the fundamental is 2L
Using f =
(b)
1 T v we have f = 2L µ λ
As this is an estimate, we are not looking for an exact value but we do need to make some sensible estimates of L and µ.
L = 0.5 m say and 2.0 × 10-3 kg m-1 ie. About 2 g per metre. From f =
1 T we have that T = 4 L2 µ f 2 = 2L µ
4 × 0.25 × 2.0 × 10-3 × (440)2 ≈ 400 N
Exercise
1.
Calculate (i)
(b)
11.2.2 Explain the Doppler effect by reference to wavefront diagrams for moving-detector and moving-source situations. 11.2.3 Apply the Doppler effect equations for sound. 11.2.4 Solve problems on the Doppler effect for sound. Problems will not include situations where both source and detector are moving 11.2.5 Solve problems on the Doppler effect for electromagnetic waves using the v approximation ∆f = f
An organ pipe is closed at one end and produces a fundamental note of frequency 128 Hz. (a)
11.2.1 Describe what is meant by the Doppler effect.
the frequencies of the next two harmonics in the harmonic series of the pipe.
(ii)
the frequencies of the corresponding harmonics for an open pipe whose fundamental is 128 Hz
(iii)
the ratio of the length of the closed pipe to that of the open pipe.
Suggest why organ pipes that emit notes at the lower end of the organ’s frequency range are usually open pipes.
c
11.2.6 Outline an example in which the Doppler effect is used to measure speed. © IBO 2007
11.2.1 (A.3.1) THE dOPPLEr EffECT Consider two observers A and B at rest with respect to a sound source that emits a sound of constant frequency f. Clearly both observers will hear a sound of the same frequency. However, suppose that the source now moves at constant speed towards A. A will now hear a sound of frequency fA that is greater than f and B will hear a sound of frequency fB that is less than f. his phenomenon is known as the Doppler Efect or Doppler Principle ater C. J. Doppler (1803-1853). he same efect arises for an observer who is either moving towards or away from a stationary source.
11.2.2 (A.3.2) ExPLAiNiNg THE dOPPLEr EffECT Fig 1111 shows the waves spreading out from a stationary source that emits a sound of constant frequency f. he observers A and B hear a sound of the same frequency.
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(a)
CHAPTER 11
A
S
B
wavefront
Figure 1111 Sound waves from a stationary source Suppose now that the source moves towards A with constant speed v. Fig 1112 shows a snapshot of the new wave pattern.
A
B
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source
smaller wavelength
Figure 1112
larger wavelength
11.2.3 (A.3.3) THE dOPPLEr EquATiONs fOr sOuNd
Sound waves from a moving source
he wavefronts are now crowded together in the direction of travel of the source and stretched out in the opposite direction. his is why now the two observers will now hear notes of diferent frequencies. How much the waves bunch together and how much they stretch out will depend on c c the speed v. Essentially, f A = λ and f B = λ where λA < λB B A and v is the speed of sound. If the source is stationary and A is moving towards it, then the waves from the source incident on A will be bunched up. If A is moving away from the stationary source then the waves from the source incident on A will be stretched out. Doppler (1803–1853) actually applied the principle (incorrectly as it happens) to try and explain the colour of stars. However, the Doppler efect does apply to light as well as to sound. If a light source emits a light of frequency f then if it is moving away from an observer the observer will measure the light emitted as having a lower frequency than f. Since the sensation of colour vision is related to the frequency of light (blue light is of a higher frequency than red light), light emitted by objects moving way from an observer is oten referred to as being red-shited whereas if the object is moving toward the observer it is referred to as blue-shited. his idea is used in Option E (Chapter 16). We do not need to consider here the situations where either the source or the observer are accelerating. In a situation for example where an accelerating source is approaching a stationary observer, then the observer will hear a sound
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of ever increasing frequency. his sometimes leads to confusion in describing what is heard when a source approaches, passes and then recedes from a stationary observer. Suppose for example that you are standing on a station platform and a train sounding its whistle is approaching at constant speed. What will you hear as the train approaches and then passes through the station? As the train approaches you will hear a sound of constant pitch but increasing loudness. he pitch of the sound will be greater than if the train were stationary. As the train passes through the station you will hear the pitch change at the moment the train passes you, to a sound, again of constant pitch. he pitch of this sound will be lower than the sound of the approaching train and its intensity will decrease as the train recedes from you. What you do not hear is a sound of increasing pitch and then decreasing pitch.
Although you will not be expected in an IB examination to derive the equations associated with aspects of the Doppler efect, you will be expected to apply them. For completeness therefore, the derivation of the equations associated with the Doppler efect as outlined above is given here. In Figure 1113 the observer O is at rest with respect to a source of sound S is moving with constant speed Vs directly towards O. he source is emitting a note of constant frequency f and the speed of the emitted sound is v. S/ shows the position of the source ∆t later. When the source is at rest, then in a time ∆t the observer will receive f∆t waves and these waves will occupy a distance v∆t . i.e
λ=
v ∆t v = ft∆ f
(Because of the motion of the source this number of waves will now occupy a distance (v∆t – vs∆t). he ‘new’ wavelength is therefore
λ/ =
v∆t − vs ∆ t v − vs = ft∆ f
If f/ is the frequency heard by O then
f/=
v v v − vs λ/ = / = / or λ f f
WAVE PHENOMENA From which
and when the source is moving away from the observer, v equation 11.2 becomes f / − f = ∆f = − f c
v f = f v − vs /
Dividing through by v gives
1 f =f v 1− s v
Provided that v
