T. l. La bdsqueda de una esencia dnica del lenguaje a tra—
2.3.5. Orftica al cardcter fundante otorgado a la dé
Standing waves can also be produced within pipes which can have open or closed ends. Consider fi rst a pipe of length L that is open at both ends, i.e. the end conditions are antinode–antinode (Figure 4.65). A fl ute is an example of this. A travelling wave sent down the pipe will refl ect from the ends (even though they are open) and we again have the condition for the formation of a standing wave.
The top diagram in Figure 4.65a represents the fi rst harmonic in a pipe with open ends. The dots represent molecules of air in the pipe. The double-headed arrows show how far these molecules oscillate back and forth (the amplitude of the oscillations). We see that the molecules at the ends oscillate the most: they are at antinodes. The molecules in the middle of the pipe do not oscillate at all: they are at a node. We have antinodes at the open ends and there is a node in the middle. The lower diagram is how we normally represent the standing wave in the pipe – you must understand that it represents what the top diagram shows. Figure 4.65b represents the second harmonic.
Exam tip
You must be able to explain how molecules move in a longitudinal standing wave such as those in pipes.
a
b
a
b a
b
Figure 4.65 a A pipe with both ends open has two antinodes at the open ends and a node in the middle. b The second harmonic in an open pipe.
Note that these diagrams also give the harmonics for the unrealistic case of a string with both ends free.
The case of a pipe with both ends closed (which is not very useful) is similar to that of a string with ends fi xed: Figure 4.65 shows the fi rst and second harmonics.
The wavelength for pipes with both ends closed or both ends open is:
λn = 2L
n , n = 1, 2, 3, 4, …
We consider fi nally the case of a pipe with one closed and one open end (i.e. end conditions node–antinode). This could apply to some organ pipes.
The closed end will be a node and the open end an antinode.
a
a
b a
b
Figure 4.66 The fi rst and second harmonics for a pipe with both ends closed.
Figure 4.67 shows the fi rst two harmonics. The distance between a node and an antinode is a quarter of a wavelength and so the wavelength of fi rst harmonic (the fundamental wavelength) is given by:
1 × λ4 = L1
⇒ λ1 = 4L = 4L 1
The wavelength of the next harmonic is:
3 × λ4 = L3
⇒ λ3 = 4L 3
Notice that there only ‘odd’ harmonics present. In general, the allowed wavelengths are:
λn = 4L
n , n = 1, 3, 5, …
This formula also gives the wavelength in the unrealistic case of a string with one fi xed and one free end.
Table 4.2 summarises the relationships for standing waves in strings and pipes.
String of length L
Both ends fi xed or both free: λn =2L
n n = 1, 2, 3, 4, … One end fi xed, the other free: λn = 4L
n n = 1, 3, 5, … Pipe of length L Both ends open or both closed: λn =2L
n n = 1, 2, 3, 4, … One end closed, the other open: λn = 4L
n n = 1, 3, 5, … Table 4.2 Wavelengths for allowed harmonics for standing waves in strings and pipes.
N A N A N A
a b
Figure 4.67 The fi rst two harmonics in a pipe with one open and one closed end.
Worked examples
4.16 A standing wave is set up on a string with both ends fi xed. The frequency of the fi rst harmonic is 150 Hz.
Calculate:
a the length of the string
b the wavelength of the sound produced.
(The speed of the wave on the string is 240 m s−1 and the speed of sound in air is 340 m s−1.) a The wavelength is given by:
λ1 = 240
150 = 1.6 m
The wavelength of the fi rst harmonic is 2L and so L = 0.80 m.
b The sound will have the same frequency as that of the standing wave, i.e. 150 Hz. The wavelength of the sound is thus:
λ = 340 150 ≈ 2.3 m
4.17 A pipe has one open and one closed end. Determine the ratio of the frequency of the fi rst harmonic to that of the next harmonic.
The fi rst harmonic has wavelength λ1 = 4L and the next harmonic has wavelength λ3 = 4L
3 . Hence:
f1
f3 = 4L/3 4L = 1
3
4.18 A source of sound of frequency 2100 Hz is placed at the open end of a tube. The other end of the tube is closed. Powder is sprinkled inside the tube. When the source is turned on it is observed that the powder collects in heaps a distance of 8.0 cm apart.
a Explain this observation.
b Use this information to estimate the speed of sound.
a A standing wave is established inside the tube since the travelling waves from the source superpose with the refl ected waves from the closed end. At the antinodes air oscillates the most and pushes the powder right and left. The powder collects at the nodes where the air does not move.
b The heaps collect at the nodes and the distance between nodes is half a wavelength. So the wavelength is 16 cm.
The speed of sound is then:
v = f λ = 2100 × 0.16 = 336 ≈ 340 m s−1
4.19 A tube with both ends open is placed inside a container of water. When a tuning fork above the tube is sounded a loud sound comes out of the tube. The shortest length of the column of air for which this happens is L. The frequency of the tuning fork is 486 Hz and the speed of sound is 340 m s−1.
a Determine the length L.
b Predict the least distance by which the tube must be raised for another loud sound to be heard from the tube when the same tuning fork is sounding.
Figure 4.68
a The wave in the tube must be the fi rst harmonic whose wavelength is 4L. The wavelength is given by:
λ = 340
486 = 0.6996 ≈ 0.70 m and so:
L = 0.6996
4 = 0.1749 ≈ 0.17 m
b The length of the air column in the tube must be increased so that the next harmonic can fi t. This means that the distance by which the tube must be raised is a half wavelength, i.e. 0.35 m.
L
Exam tip
Draw the standing wave in part a. It is the fi rst harmonic. Now raise the tube and draw the next harmonic. What is the connection between the distance the tube was raised and the wavelength?
Nature of science
Physics is universal
The universality of physics is evident almost everywhere including in the theory of standing waves. From the time of Pythagoras onwards philosophers and scientists have used mathematics to model the formation of standing waves on strings and in pipes. The theory that we have
developed here applies to simple vibrating strings and air columns, but it can be used to give detailed accounts of the formation of musical sound in instruments as well as the stability of buildings shaken by earthquakes.
41 A glass tube with one end open and the other closed is used in an experiment to determine the speed of sound. A tuning fork of frequency 427 Hz is used and a loud sound is heard when the air column has length equal to 20.0 cm.
a Calculate the speed of sound.
b Predict the next length of air column when a loud sound will again be heard.
42 A pipe with both ends open has two consecutive harmonics of frequency 300 Hz and 360 Hz.
a Suggest which harmonics are excited in the pipe.
b Determine the length of the pipe.
(Take the speed of sound to be 340 m s−1.) 43 A pipe X with both ends open and a pipe Y with
one open and one closed end have the same frequency in the fi rst harmonic. Calculate the ratio of the length of pipe X to that of pipe Y.
44 If you walk at one step a second holding a cup of water (diameter 8 cm) the water will spill out of the cup. Use this information to estimate the speed of the waves in water.
45 Consider a string with both ends fi xed. A standing wave in the second harmonic mode is established on the string, as shown in the diagram. The speed of the wave is 180 m s−1.
a Explain the meaning of wave speed in the context of standing waves.
b Consider the vibrations of two points on the string, P and Q. The displacement of point P is given by the equation y = 5.0 cos (45πt), where y is in mm and t is in seconds. Calculate the length of the string.
c State the phase diff erence between the oscillation of point P and that of point Q.
Hence write down the equation giving the displacement of point Q.
Test yourself
P
Q
32 Describe what is meant by a standing wave. List the ways in which a standing wave diff ers from a travelling wave.
33 Outline how a standing wave is formed.
34 In the context of standing waves describe what is meant by:
a node b antinode c wave speed.
35 a Describe how you would arrange for a string that is kept under tension, with both ends fi xed, to vibrate in its second harmonic mode.
b Draw the shape of the string when it is vibrating in its second harmonic mode.
36 A string is held under tension, with both ends fi xed, and has a fi rst harmonic frequency of 250 Hz. The tension in the string is changed so that the speed increases by √2. Predict the new frequency of the fi rst harmonic.
37 A string has both ends fi xed. Determine the ratio of the frequency of the second to that of the fi rst harmonic.
38 The wave velocity of a transverse wave on a string of length 0.500 m is 225 m s−1.
a Determine the frequency of the fi rst harmonic of a standing wave on this string when both ends are kept fi xed.
b Calculate the wavelength of the sound produced in air by the oscillating string in a. (Take the speed of sound in air to be 340 m s−1.)
39 Draw the standing wave representing the third harmonic standing wave in a tube with one closed and one open end.
40 A glass tube is closed at one end. The air column it contains has a length that can be varied between 0.50 m and 1.50 m. A tuning fork of frequency 306 Hz is sounded at the top of the tube. Predict the lengths of the air column at which loud sounds would be heard from the tube. (Take the speed of sound to be 340 m s−1.)
46 A horizontal aluminium rod of length 1.2 m is hit sharply with a hammer. The hammer rebounds from the rod 0.18 ms later.
a Explain why the hammer rebounds.
b Calculate the speed of sound in aluminium.
c The hammer created a longitudinal standing wave in the rod. Estimate the frequency of the sound wave by assuming that the rod vibrates in the fi rst harmonic.
Exam-style questions
1 The diagram shows a point P on a string at a particular instant of time. A transverse wave is travelling along the string from left to right.
P
Which is correct about the direction and the magnitude of the velocity of point P at this instant?
Direction Magnitude
A up maximum
B up minimum
C down maximum
D down minimum
2 A tight horizontal rope with one end tied to a vertical wall is shaken with frequency f so that a travelling wave of wavelength λ is created on the rope. The rope is now shaken with a frequency 2f. Which gives the new wavelength and speed of the wave?
Wavelength Speed
A λ f λ
B λ 2fλ
C λ
2 f λ
D λ
2 2fλ
0
Displacement
Distance
B A
D C
3 The graph shows the displacement of a medium when a longitudinal wave travels through the medium from left to right. Positive displacements correspond to motion to the right. Which point corresponds to the centre of a compression?
4 The diagram shows wavefronts of a wave entering a medium in which the wave speed decreases. Which diagram is correct?
A B C D
0
Displacement
Time
0
Displacement
Time
A
Displacement
Time
B
Displacement Displacement
0
0 0
0
Displacement
Time
A
Displacement
Time
B
Displacement Displacement
0
0 0
5 The graph shows the variation with time of the displacement of a particle in a medium when a wave of intensity I travels through the medium.
The intensity of the wave is halved. Which graph now represents the variation of displacement with time? (The scale on all graphs is the same.)
6 Which of the following does not apply to longitudinal waves?
A superposition
B formation of standing waves C interference
D polarisation
7 Interference is observed with two identical coherent sources. The intensity of the waves at a point of constructive interference is I. What is the intensity when one source is removed?
A 0 B I C I
2 D I
4
8 Unpolarised light of intensity I0 is incident on two polarisers, one behind the other, with parallel transmission axes. The fi rst polariser is rotated by 30° clockwise and the second 30° counter-clockwise. What is the intensity transmitted?
A I0
2 B I0
4 C I0
8 D I0
16
9 A pipe of length 8.0 m is open at one end and closed at the other. The speed of sound is 320 m s−1. Which is the lowest frequency of a standing wave that can be established within this pipe?
A 5.0 Hz B 10 Hz C 15 Hz D 30 Hz
10 Travelling waves of wavelength 32 cm are created in a closed–open pipe X of length 40 cm and an open–open pipe Y of length 50 cm.
X
40 cm
Y
50 cm
0 2 4
Displacement/mm
Distance/m –2
0.1 0.2 0.3 0.4 0.5
–4
In which pipe or pipes will a standing wave be formed?
A X only B Y only C neither X nor Y D both X and Y
11 A longitudinal wave is travelling through a medium. The displacement of the wave at t = 10 s is shown below.
Positive displacements are directed to the right.