1. 2.9 Beats Source 1 Detector , x Source 2
Consider two sound sources that produce harmonic waves of equal amplitude A but different frequencies f1 and f2. harmonic waves,
Source 1
Source 2
Detector , x
What is the signal recorded by a detector placed a distance x from each source?
Need to use the principle of linear superposition to obtain the answer.

2. Let the displacement of the sound wave from source 1 be
y1(x,t) = Asin(k1x-w1t) and for source 2
y2(x,t) = Asin(k2x-w2t)
From principle of linear superposition we have
yt(x,t) = y1(x,t) + y2(x,t)
yt(x,t) = Asin(k1x-w1t) + Asin(k2x-w2t)
Which becomes

3. For the two sound waves that interact the resulting displacement is
To simplify the problem let us assume that x = 0 and (w1+w2)/2 = wave and (w1-w2) = wbeat
Here we have used sin(-a) = -sin(a) and cos(-a) = cos(a)

4. Thus at x we have two two harmonic waves interacting together one vibrating at the average frequency and the other vibrating at the beat frequency.
Harmonic wave at ave frequency
Harmonic wave at beat frequency
When the ear detects this signal the ear responds to the intensity of the signal.

5. We can use the fact that
To give
So when wbeatt = 2nπ the intensity is a maximum
So when wbeatt = (2n+1)π the intensity is zero
Thus there is a periodic modulation of the intensity and the period is

6. 2.10 How are beats formed?
Need to consider the way in which the phase of the two sound waves varies at the point x.
Initially the two waves are in phase. Hence the displacement caused by each wave is the same. So the particle experiences a maximum displacement.
As time increases the phase of the two waves evolves at different rates because the frequency of each wave is different (w1t ≠ w2t).

7. After some time T the phase of wave 1 has evolved such that it is π out of phase with respect to wave 2. At this point the displacement caused by each wave is equal but opposite resulting in zero net displacement. T
Time increases and the two waves come back into phase resulting in the maximum displacement.

8. 2.11 The Doppler effect So far the wave source has been stationary.
What happens if there is a relative motion between the source and the observer?
When there is a relative motion between a wave source and an observer there is an apparent change in frequency.
This is known as the Doppler Effect.
There are two cases to consider
Source stationary and the observer moving.
Source moving and observer stationary.

9. 2.11(I) The Doppler effect- Source stationary observer moving
Consider a stationary source generating sound waves at a frequency fs and wave speed vs.
The observer moves in a straight line towards the source at a speed vo.

10. 2.11(I) The Doppler effect- Source stationary observer moving
The source generates wave fronts that are spaced by ls.
As the observer moves towards the observer passes the wave fronts with an apparent speed va = vo + vs.
The spacing between the wave fronts does not change and so the observer senses an apparent frequency fa = va/ls. So
But ls = vs/fs
Hence
If the observer was moving away from the source the the apparent frequency shift is

11. 2.11(II) The Doppler effect- Source moving observer stationary
Consider a stationary source generating sound waves at a frequency fs and wave speed vs.
The source moves in a straight line towards the observer at a speed vm.
If the source was stationary then spacing between successive wave fronts would be ls and the observer would sense the wave fronts at their spacing ls.The source moves in a straight line towards the observer at a speed vm.
As the source is moving at a speed vm then the spacing between successive wave fronts is changed.

12. Source stationary wave fronts spaced by ls
Source now moves with speed vm. Successive wave fronts spaced by la. The time taken for successive wave fronts to be emitted is T, the period of the source. In this time the source moves towards the observer a distance vmT. Hence the distance between the new wave font and old wave front is reduced.

13. 2.11(II) The Doppler effect- Source moving and observer stationary
The apparent wavelength is given by la = ls - vmT.
But la = vs/fa and ls = vs/fs and T = 1/fs So
Hence
If source was moving away from the source the the apparent frequency shift is

14. 2.11(II) The Doppler effect- Source moving observer moving
If both the source and the observer are moving then the apparent change in frequency is given by