Name: ___________________

Math 122 Lab 5: Breaking the Sound Barrier

Introduction

You have heard of a sonic boom, but have you ever seen one? In the remarkable photograph below, water vapor outlines a shock wave as an F-18 jet breaks the sound barrier.

Why does the shock wave have a conical shape? We will show in this assignment that this conical shape is predicted by the physics of shock waves.

What to Turn In

Turn in this worksheet with everything filled in. This will be graded for accuracy and understanding, so write good explanations when you are asked to explain something. If you need more space, turn in additional pages typed or neatly written.

Sound Waves

The starting point for the analysis of a shock wave is an understanding of how sound propagates. Imagine a firecracker exploding. The sound will reach everyone standing 10 feet away at the same time. That is, in two dimensions, sound propagates in expanding concentric circles. Suppose a sound wave is emitted from the origin at time 0. Define 1 “unit” to be the distance traveled by sound in 1 second. After r seconds (r > 0), the position in units of the sound wave is modeled by x = r cosq and y = r sinq, where the parameter q has range 0 £ q £ 2p. Find parametric equations for the position after 5 seconds of a sound wave emitted from the point (2,3). Write the answer here.

Now, generalize this. That is, write out parametric equations for the position after r seconds of a sound wave emitted from the point (a,b). Write the answer here. You will use this general result several times in what follows for different values of a and r.

A Subsonic Flight

Suppose that a jet has speed 0.8 units per second (i.e., Mach 0.8) with position function x(t) = 0.8t and y(t) = 0. Graph the position of the jet and briefly describe which direction it is moving.

The jet is noisy, so that sound waves are constantly being emitted by the jet. You will graph the position at time t = 5 seconds of various sound waves emitted by the jet. At time 0, the jet is at position (0,0) and emits a sound wave. Give parametric equations for the position of this sound wave at time 5. That is, the wave starts at (0,0) and travels for 5 seconds. What is its position?

Next, we look at where the jet is at time 1. Its position is (0.8,0). Another sound wave is emitted. Give parametric equations for the position of this sound wave at time 5. That is, the wave starts at (0.8,0) and travels for 4 seconds. What is its position?

Continue on to get the position of the sound wave emitted by the jet at time 2. (Hint: you need the position of the jet at this time and how many seconds the wave travels.)

Find the position of the waves emitted at times 3 and 4.

You now have 5 sets of parametric equations. Use the calculator to graph all of them on the same axes. Copy this below and mark the position of the jet at time 5 on this graph. How does the jet’s position compare to the sound waves?

The Sound Barrier

The next step is to redo the above graph for a jet going exactly the speed of sound. This is Mach 1, so in our units the speed is 1. The sound wave emitted at time 0 is now a wave that starts at (0,0) and travels for 5 seconds. Write down its equations. The sound wave emitted at time 1 starts at (1,0) and travels for 4 seconds. Write down its equations.

Similarly, write down equations for the sound waves emitted at times 2, 3 and 4.

You now have 5 more sets of parametric equations. Use the calculator to graph all of them on the same axes. Copy this below and mark the position of the jet at time 5 on this graph. How does the jet’s position compare to the sound waves? Briefly describe what the “sound barrier” is that a jet must break through to go supersonic.

Supersonic Flight

Now, repeat the above for a jet flying at Mach 1.4. That is, write down equations for the sound waves emitted at times 0, 1, 2, 3 and 4 and graph all of them together.

Sonic Booms

In this last set of graphs, you should notice sound waves intersecting each other. The intersections form the “shock wave” that we hear as a sonic boom. Based on your graphs, does the sonic boom only occur right as the jet passes through the sound barrier or does it occur for any jet flying faster than the speed of sound?

Shock Waves

Theoretically, the angle q between the shock wave and the x-axis satisfies the equation

sin q = 1/m

where m is the Mach speed of the jet. Show that for m=1.4, the theoretical shock wave is formed by the lines x(t) = 7 - t, y(t) = t and x(t) = 7 - t, y(t) = - t. Superimpose these lines onto your last graph.

We’ve seen in two dimensions that the shock wave of a jet flying at Mach 1.4 is modeled by two lines. In three dimensions, the shock wave has circular cross sections. Use this to describe the three-dimensional figure formed by the shock wave.