(Are you sure you want to spoil the fun of figuring it out on your own? If not, then go back to part I now !)
If one wave had a negative polarity, the collision would give zero pressure, but the energy cannot have disappeared. What happened?
Acoustical energy can be defined at a point as the sum of kinetic energy (velocity squared times density) plus potential energy (pressure squared times compressibility).
Where the positive pressure pulses collide (left side of the figure), the potential energy is twice as great but the kinetic energy vanishes (by symmetry because the motion of a wave moving positively is opposite to one moving negatively). Similarly, in the second case (right side) the kinetic energy doubles but the potential energy (and amplitude) vanishes.
Newton's law, mass times acceleration equals force, tells us how to compute the velocity from the pressure.
To get velocity from pressure, it is only necessary to differentiate with respect to x and integrate with respect to t. Observe how this equation relates to Figure 4 .
So, the energy at a point is not just the square of the pressure. The velocity must be accounted for as well.
Although the two cases in Eisner's paradox have the same potential energy, they do not have the same kinetic energy. As can be seen in Figure 2 , at the focus in case B it is more like plane waves colliding head-on, where the kinetic energy extinguishes, and at the focus in case A it is more like a single plane wave with all the energy going in one direction.
In the end, the lesson of Eisner's paradox is that ``energy equals amplitude squared'' is not true when more than one wave is involved. We fall into the habit of thinking it is always true because we so often see single plane waves, where it is true.
An advanced exercise is to show how a complex variable can have pressure as the real part and velocity as the imaginary part, and the Hilbert transform envelope as the energy. For a start, note the symmetries present in Figure 4 .