# The Sonic Boom

In 1986, a Royal Australian Air Force pilot was fined $100 for disobeying orders and flying faster than the speed of sound over the city of Adelaide, Australia. His plane’s sonic boom caused $512,000 worth of damage to buildings, including $256,000 worth of damage to a single greenhouse by shattering its walls.

What is a sonic boom?

As a plane flies through the air, its nose pushes the air molecules in its path out of the way. This is the same principle as the bow of a boat which pushes away water molecules as it moves forward, creating a wake. The physics here is similar to last week’s discussion about the Doppler effect, where we saw how a difference in relative velocities between a signal source and the observer causes a change in frequency.

Sonic booms occur when planes fly faster than the speed of sound. The speed of sound is in the neighborhood of 760 miles/hour or 340 meters/second but varies with atmospheric conditions. Planes flying through the air generate air-pressure waves that can only propagate at the speed the air medium allows, like 340 meters/second. When the plane exceeds the speed of sound, these pressure waves combine and form shock waves. A plane traveling at supersonic speeds continues to generate shock waves and cause sonic boom along its path. If you’ve taken some physics, this is an example of constructive interference. You can read more about this at https://www.af.mil/About-Us/Fact-Sheets/Display/Article/104540/sonic-boom/.

The shock waves propagate out behind the plane like a sideways ice cream cone. As the radius of their circular cross-sections gets bigger, they eventually intersect with the ground, forming an area defined by two intersecting lines. These are just the sides of the ice cream cone intersecting with the ground. If you’re still having trouble imagining this, take a look at the pictures at https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/17-8-shock-waves/#:~:text=A%20sonic%20boom%20is%20the,boat%20moving%20through%20the%20water.

We commonly refer to the speed of a supersonic plane by its Mach number M, which is just the ratio of the speed of the plane v_{s} to the speed of sound v:

M = v_{s} / v.

It turns out that the sine of one-half of the angle theta between the two intersecting lines ( or one half of the angle measure of the vertex angle of the cone) can be given by the reciprocal of the Mach number, or:

Sin (theta/2) = 1/M

Let’s consider a real example. An F-35 fighter is flying at a speed Mach 1.2, and an observer hears the sonic boom 15 seconds after the plane flies directly overhead. What is the altitude of the plane? Assume that the speed of sound is v = 343 meters/second.

I’ve drawn a picture of the scenario in the attached diagram. The angle theta/2 can be found by the relationship

sin theta/2 = 1/1.2

If I apply the inverse sin function to both sides, I find that

theta/2 = 56.47 degrees

The distance d the plane travels in 15 seconds is

15 seconds * 343 meters/second * 1.2 = 6174 meters

That’s the horizontal leg of the right triangle in our diagram. We want to find the vertical height h, that is, the vertical leg of our right triangle. So using some basic trigonometry, we find

h = d * tan (theta/2) = 6174 * tan 56.47 = 9317.3 meters = 9.31 kilometers

If you want to hear and see some sonic booms for yourself, check out the following video:

https://www.youtube.com/watch?v=gWGLAAYdbbc

Is there anywhere else you might encounter a sonic boom? Yes. The sound of a gun firing and the crack of a whip are also examples of sonic booms. Both bullets and whips can travel at speeds exceeding the speed of sound, causing the same shock wave effects we talked about for supersonic planes.

So now we understand a little bit about sonic booms, and how sound waves propagate through the air. Armed with this knowledge, we’ll move on to another sound-related technology next week. Of great importance to the Navy, sonar relies on the propagation of sound through the water.