We reviewed beats.

We considered standing waves in tubes. When the tube was open at one end and closed at the other, we used the approximation that the wave had to have an anti-node at the first end and a node at the second end. The fundamental then had a quarter of a wavelength inside the tube. The next resonant mode was 3/4 of a wavelength, then 5/4, etc. Finding the corresponding frequencies led to:

Open-Closed: f = v (2n+1) / (4 L)

where L is the length of the tube, v is the speed of sound, and n is an integer n=0,1,2, ....

If the ube has open-open boundary conditions (both ends are open, then both ends must be anti-nodes and the fundamental has half of a wavelength in the tube. Arguments like those above yield

Open-Open: f = v (2n) / (4 L)

where this time n=1,2, ...

We started looking into the Doppler Effect. We argued that when a source of sound moves the effective wavelength is changed. After the source goes through its maximum displacement from equilibrium, that crest begins travelling at the speed of sound toward the observer. After a period T, the source emits another crest. In that time, the first crest traveled v_soundT while the source traveled v_sourceT. Thus the effective wavelength is (v_sound - v_source)T, assuming the source is traveling toward the observer. The corresponding effective frequency is then

f_observed = v_sound f_original / (v_sound - v_source)

We mentioned the case when the speed of the source approaches the speed of sound, mentioning terms like breaking the sound barrier, mach 1, and sonic boom. We still have to consider what happens when the observer moves.