# The Physics Philes, lesson 117: Sonic…BOOOOOOOM

Well, well, well…we’re about to wrap up this section on waves. But before we do let’s take a peak at one of the spectacular things sound waves can do, sonic booms. We can use the equations we learned about previously to show why this happens.

Think about an airplane moving relative to the air at a speed vs. (It’s just the speed, so it’s always positive.) The motion of the plane will produce a sound, but as long as the speed of the airplane is less than the speed of sound v, the waves will be squished together in front of the airplane, given by this equation:

Ooooo but what if the speed of the airplane matches the speed of sound? Then the wavelength equals zero. That means that the wave crests stack up on top of each other and the airplane needs to exert a large force to compress the air. (And, by Newton’s third law, the air has to exert an equally large force back.) That means that the air resistance increases as the airplane approaches the speed of sound and this air resistance is called the sound barrier.

Now, when the speed of the airplane vs exceeds the speed of sound we say the sound is supersonic. When sound is supersonic, the equations we learned for Doppler shift no longer hold. Here’s a really confusing drawing I made that might help illustrate what’s happening:

Whoa, colorful! But what does it mean? The airplane (denoted by the red line because I’m not even going to try to draw an airplane) is moving through the air. As it moves, it displaces the air surrounding it and produces sound. Wave crests are emitted from the nose of the airplane and spreads out in a circle that is centered on the position of the airplane when the crest was emitted. So the airplane was at S1 when it emitted the wave represented by the left-most circle. After a certain time t, the crest of the circle centered at S1 will have expanded to a radius of distance vt. The airplane, meanwhile, is moving faster than sound so the distance it traveled is vst at S2. The waves undergo constructive interference along the blue lines. These lines make an angle α with the airplane velocity. This leads to a wave crest with a very large amplitude along the lines. This is called a shock wave.

You might notice that we have a right triangle. We can use that to find the angle:

The ratio of vs/v is called the Mach number and is greater than 1 for supersonic speeds. As you can see, the sine of the angle is the reciprocal of the Mach number.

The drawing above is two-dimensional, but actual shock waves exist in 3D. The shock wave actually forms a cone around the direction of the source’s motion. If the source – in our example, the airplane – moves with a constant velocity, the angle is constant and the shock wave cone will move along with the source. It’s the arrival of the shock wave that causes a sonic boom. In front of the cone, there is no sound, and inside the cone a stationary listener will hear the Doppler-shifted sound of the airplane moving away.

This is pretty neat in itself, but it has actual practical applications. For example, shock waves can be used to break up kidney stones or gallstones using a technique called extracorporeal shock-wave lithotripsy. Basically a shock wave is produced outside a body that stresses the stone so much that it breaks into small pieces that can be more easily eliminated.

It’s worth noting that shock waves are produced continuously, not just at the instant that the source breaks the sound barrier. The sound waves that combine to form the shock wave are created by the motion of the object, not any sound the object makes. So loud airplane engines don’t have anything to do with the sonic boom that results when the plane exceeds the speed of sound. Not convinced?

Thus ends the section on waves! We’ve learned so much. It was a good time. Don’t worry, though. There are more interesting physics topics to come. Next up: thermodynamics.

*Featured image credit: NASA Goddard Space Flight Center*

## No Comment