Waves 1.4 Solution
Back to Waves Question Set 1.Question
(a) A point sound source produces sound energy at a rate of 1.0 kilowatt.
(i) Describe (in words) how the intensity varies with distance from the source.
(ii) Calculate the intensity in W.m-2 at a distance of 50 m from the source.
(iii) Calculate the sound level in decibels at this position.
(b) Two musical instruments are playing together. Source 1 has a fundamental frequency of 555 Hz and source 2 has a fundamental frequency of 565 Hz.
(i) Describe what you hear.
Now show this mathematically by answering the following questions.
(ii) Write down expressions for the displacement at some position due to source 1 alone and source 2 alone and add them. Assume that the displacement amplitudes are equal.
(iii) Why can you simply add the displacements?
(iv) Manipulate this result to explain what you hear.
[The following information may prove useful:
standard reference intensity is 10-12 W.m-2
Solution
(a) (i) The intensity varies as 1/r2 with distance r from the source.
(ii) 1.0 kilowatt = 1000 W is the power from the source.
At a distance r from the source this power has spread out over a sphere of radius r. The intensity is given by dividing the power by the area over which it has spread, which is 4r2. In this case, r=50. So the intensity I is given by
(iii) The sound level in decibels is given by:
(b) Take f1 = 555 Hz to be the frequency of source 1 and f2 = 565 Hz to be the frequency of source 2.
(i) We hear a single tone of frequency (f1+f2) / 2 = 560 Hz. This tone beats at a frequency of (f2-f1) / 2 = 5 Hz.
(ii) At some position p where both sources have the same amplitude A the displacement from source 1 is s1 = A sin (2pf1t) and the displacement from source 2 is s2 = A sin (2pf2t).
We find the total displacement to be s = s1 + s2 = A [ sin (2pf1t) + sin (2pf2t) ].
(iii) For amplitudes and ranges in the usual audio range we can assume the ear is a linear system – i.e. the loudness of the sound heard is a linear function of the air displacement at the ear. So we add the two displacements to get the total sound heard by the ear.
(iv) Use the standard trigonometric identity given in the question for addition of sines.
The first term gives the single combined tone at a frequency of (f1+f2) / 2, the second term gives the beat at a frequency (f2-f1) / 2.