Waves 3.3 Solution
Back to Waves Question Set 3.Question
The Sun exhibits radial oscillations, which are somewhat similar to the standing waves in an organ pipe. For these oscillations the Sun's surface is a displacement anti-node.
Shown below is part of the measured standing wave spectrum of the Sun, showing the velocity amplitude at its surface.
(a) Is the centre of the Sun a displacement node or anti-node? Explain your reasoning.
(b)Given that the radius of the Sun is about 700,000 km and using the spectrum shown, give a rough estimate for the velocity of sound in the Sun.
(c)Is your answer in part (c) smaller or larger than the speed of sound in air in the atmosphere on Earth? Comment on this finding.
(a) Solution
The oscillations are radial. The centre of the sun cannot oscillate in all directions at once, hence it is stationary and must be a node.
(a) Actual Student Answers (maximum 3 marks)
Centre of Sun must be a node (it is analogous to a rope tied to a fixed end.) because if it were an antinode, amplitude would be maximum there, but in a 3D standing wave, this would mean the mass which is in the centre of the sun must all move radially in and out in a certain pattern, but this would then leave no matter there (after matter have radially oscillated out). Clearly this does not happen, hence it is a node at the centre
3 marks. Another good explanation of why the centre cannot oscillate.
The centre is a node. For standing waves in an organ pipe, the opened end is an anti-node and the closed end is a node as shown:
Given that the radial oscillations of the Sun is similar to this, and that the surface is an anti-node (open end), then the centre is a ‘closed end’ and is a node.
0 marks. Many students assumed that because the oscillation was likened to an organ pipe, it had to have one node and one antinode. This is not a good explanation – a pipe could have two open ends, making the solar centre an antinode. To explain why the centre of the Sun is in fact a node one must explain why the centre of the sun cannot undergo radial oscillations.
(b) Solution
On the graph we see regularly spaced peaks where the oscillations are much stronger, corresponding to the modes of the oscillation. Consider two adjacent modes, say the nth and (n+1)th modes, with frequencies fn and fn+1 respectively. For any oscillation similar to an organ pipe, adjacent mode frequencies are related to the sound speed v and the length of the oscillating medium L by:
Now from the graph adjacent modes differ in frequency by approximately 0.133 mHz.
So fn+1 – fn = 1.33 x 10-4 Hz, while L = 700 000 km = 7 x 108 m. So the sound speed in the Sun is
(b) Actual Student Answers (maximum 4 marks)
Frequency difference between 2 lines is about 0.13 mHz (millihertz)
Therefore fundamental frequency of the Sun is approx. 0.065 mHz = 6.5 x 10-5 Hz
= 2 800 000 km = 2.8 x 109 m
4 marks. This is a good alternative approach, using the specific relationship between the mode frequency spacing and the fundamental mode frequency for a standing wave with one node and one antinode. The slight variation in the answer is due to measuring a different separation off the graph.
The first harmonic has a frequency of approximately 3.1 mHz (the central, highest amplitude resonant frequency corresponds to the resonance with the least loss due to reflection – the first harmonic).
2 marks. Half marks were awarded for assuming that one of the modes on the graph is the first harmonic and correctly calculating the sound speed from the wavelength and frequency of this mode. Many students gave similar answers, assuming that one of the modes visible on the graph (often the lowest frequency or the strongest mode) was the first harmonic.
However this assumption is wrong - there is nothing to say that any of the prominent modes on the graph is the first harmonic. The modes shown could be the first through to the 13th, the 5th through to 17th or the 80th through to the 92nd modes – the question gives no information on this. What we do know is the spacing between modes and this information should be used to answer the question, either directly (as in the solution) or by working out the fundamental frequency from the spacing (as in the actual student answer given above).
Resonance occurs at point of maximum amplitude on the graph
3 largest resonances v ~ 0.25 ms-1. Therefore speed of sound in the sun ~ 0.25 ms-1.
1 mark. This was another common misconception. The velocity amplitude is the speed at which the surface of the sun is moving during the oscillations, which measures how strong the oscillation is. However it does not measure the speed at which the oscillation propagates through the sun – the wave speed
.
(c) Solution
The sound speed on the earth is 343 ms-1, so the sound speed on the Sun is greater by a factor of ~500. The sun is much hotter than the earth’s atmosphere, so the molecules move more rapidly on the sun. Sound speed is roughly proportional to the molecular speed, so we expect a much higher sound speed on the sun.
(c) Actual Student Answers (3 marks maximum)
Speed of sound in air = 343 ms-1 (on earth)
Above speed of sound in air is much greater than on earth.
Velocity in a medium is given by
B = Bulk modulus, = density
Density is much greater in the sun than in the atmosphere so this would mean velocity would be less in the sun. But Bulk modulus (incompressibility, measure of) is much more greater than density is greater. This results in a much higher velocity.
3 marks. This is an alternative approach, using the formal equation for sound velocity in terms of the density and the bulk modulus. This type of answer is also acceptable, although it does entail making some assumptions about the bulk modulus of the sun. A temperature argument, as given in the solution, relies only on the reasonable assumption of the sun being hotter than the earth.
It is less, due to the different materials and temperature.
2 marks. This is correct, but more detail is needed for full marks.
The speed of sound in air on Earth is much less than the speed of sound in the Sun’s atmosphere. This is because the velocity of sound in the very dense atmosphere of the sun is much greater than that in air.
1 mark. This answer shows an awareness that the density of the medium affects the sound speed, but receives few marks as no knowledge of the details is shown.