We consider the times and angles you perceive when a source changes position faster than the speed of sound. The same analysis applies when a source changes position faster than the speed of light, which actually can happen, as discussed below.

*   Contents

• 1 Setup and Notation
• 2 Key Relationships
• 3 Solutions
• 4 Special Cases
• 5 Scaling

1 Setup and Notation

The time t is measured relative to the point of closest approach, so for the situation shown in the diagram, t is less than zero.

Figure 1: Mach Angle Construction

2 Key Relationships

At all times:

See also the scaled versions in section 5.

3 Solutions

It is now straightforward to solve for the arrival time a and angle θ (both which you can observe directly) as functions of emission time t (which you can’t). The results are shown in figure 2. For this example, we assume y=1, c=1, and v=2.

Figure 2: Mach Arrival Time and Angle versus Emission Time

• Signals emitted at the point of closest approach take unit time to arrive, at zero angle. Signals emitted slightly to the left of there arrive early, because they get a head start (to first order) and the distance of travel is the same (to first order).
• Signals emitted at early times take a long time to arrive, because even though they got an early start, they have a long ways to travel. They arrive at an angle of nearly 90^∘ left.
• Meanwhile, signals emitted at late times times take an extra-long time to arrive, because they got a late start and have a long ways to travel. They arrive at an angle of nearly 90^∘ right.

Here’s the graph of what you actually observe: Arrival angles as functions of arrival time. For each time there are two angles.

Figure 3: Mach Arrival Angle versus Arrival Time

When there are two signals arriving at the same time, it is tricky to perceive the angle of either of them using just your two ears. It may be possible to untangle them using frequency information, due to the Doppler shift and/or dispersion. You could easily untangle them using a phased-array passive sonar receiver.

Things are even more interesting when you look at a source that changes its position faster than the speed of light. Your eyes contain millions of photoreceptors, so it is possible to perceive both of the arriving signals. You can’t have a source that physically moves faster than the speed of light, but you can have a long array of sub-sources that light up in turn, by pre-arrangement, creating a pattern that moves faster than the speed of light. As a particularly simple example, you can use to grids to create a moireé pattern that moves faster than the speed of light, even though neither grid is moving very fast, as shown in figure 4.

Figure 4: Moiré Pattern

4 Special Cases

At the time of onset, when the first signal arrives, [e x b] is a right triangle, and β = θ. We know it’s a right triangle because the line [x b] is tangent to the wavefront, as suggested by the circular arc in the diagram. This can be used as a check on other ways of calculating the minimum time of flight f. Also it tells us that

which is the famous formula for the Mach angle.

Also, the emission time of the onset signal is:

In the special case where v/c = 2, at onset the triangle [e x b] is a 30-60-90 triangle. This is the case shown in the diagram.

At some later time, we might be tempted to consider the triangle [e′ x b′], but (1) it doesn’t tell us anything we don’t already know, (2) it bears no relationship to the physics of the Mach angle, and (3) it is not a right triangle.

5 Scaling

It is good practice to thing about how a problem scales. For the Mach arrivals problem, all the speeds scale in proportion to c, and all the times scale in proportion to y/c. If we replace v/c by m (the Mach number) and replace y/c by τ (the time of flight from the point of closest approach), then c drops out of the equations entirely. This reduces the number of variables involved. All the legs in figure 5 have dimensions of time (not distance).

Figure 5: Scaled Mach Angle Construction

The key equations become: