Sonic booms occur when the source travels at, or faster than, the speed of sound. Why is this?

Assuming a stationary observer and a source moving at the speed of sound, the Doppler equation predicts an infinite frequency. Why is this?

If the source is traveling at the speed of sound, the waves pile up and move along with the source. All the peaks are at the same place, so the wavelength is zero and the frequency is infinite. This overlay of all the waves produces a shock wave known as a sonic boom.

When the source moves faster than the wave speed the source outruns the wave. The equation can give negative frequency values, but -500 Hz is pretty much the same as +500 Hz as far as an observer is concerned.

Now the waves pile up along a particular angle, again producing a shock wave (sonic boom). The angle at which the shock wave moves away from the path of the source depends on the speed of the source relative to the speed of sound.

Does the pattern look familiar? It looks a lot like the wake left behind by a boat moving through water. What does this tell you about how a boat's speed compares to the speed of water waves?