Hero and Fermat on Receding Mirrors
The law of reflection for a simple plane surface (first annunicated by Hero of Alexandria) states that the angle of incidence equals the angle of reflection as illustrated below.
If the source and receiver are at the same height above the mirror, the point of reflection will be directly below the mid-point between them - assuming the source, mirror, and receiver are mutually stationary. However, if the mirror has some downward velocity v, the point of reflection will not lie along the centerline. Instead, it will be slightly shifted toward the source.
There are several ways of determining the optical rays for the case when the mirror has some downward speed. The simplest approach is to work in the rest frame of the mirror, with respect to which the source and receiver are both moving upward at the speed v. This makes it clear that a light pulse emitted from the source at a given height above the mirror at time t arrives at the receiver at a later time t + Dt, when the receiver is at a slightly greater height above the mirror. Hence the ray traces out a path as indicated in the figure below.
Throughout this discussion we will express both spatial and temporal magnitudes in units so that c = 1 and the distance from the center line to the source and receiver is 1. Now, by similar triangles we have
Solving this for Dt gives
Also, letting r1 and r2 denote the lengths of the incident and reflected rays respectively, we have
and therefore
Substituting the previous expression for Dt into this equation and solving for v gives
It follows that atanh(dv/dx) = asinh(h). Also, solving for x as a function of v and h, we have
Incidentally, this function is singular when v = h (the numerator and denominator both go to zero), but the singularity is removable, and the function goes to x = (1+v2)/2 at this point.
Since v = dh/dt, we can choose the origin such that h = vt, and substitute into the above formula to give the lateral position of the point of reflection as a function of time
A plot of x(t) for several values of v is shown below.
Notice that for any fixed velocity v, the value of x approaches v in the limit as t approaches infinity. On the other hand, as v approaches zero, the function goes to
Another approach to the problem is to apply Fermat's Principle of Least Time, taking into account the motion of the mirror. As before, the horizontal distance from the centerline to the source and receiver is unity, and the vertical distance between source and mirror at the instant a pulse of light is emitted from the source is h. If the pulse takes Dt to reach the mirror, the mirror will have moved vDt since the emission event, so the time from the emission event to the instant of reflection satisfies the relation
From here the light is reflected up to the receiver, and the time required for this second leg of the journey is
Combining these two equations, we see that
The total duration of the light transit is T = t1 + t2, so we have
Solving equation (2) for t1 and inserting into this equation gives
To find the value of x that minimizes this transit time in accord with Fermat's principle we then must solve the equation dT/dx = 0 for x. We might expect this to give the same result as equation (1), but if we set h = 1 and plot the two profiles of x as a function of v we get slightly different results, as shown below.
What has gone wrong? Does this demonstrate that Hero's law of reflection is incompatible with Fermat's Principle in this situation? No, because the comparison has not been made on a consistent basis. The parameter h was defined in both cases as the vertical distance between the source and the mirror at the instant when the pulse of light was emitted, but this was evaluated with respect to two different frames of references, first in the frame of the mirror, and then in the frame of the light source. These two objects are moving apart with a mutual speed v, so they have different loci of simultaneity (when working in the respective systems of inertial coordinates, in which the speed of light is unity). In order to place the results of the two analyses on an equal basis, we must account for this relativistic effect. The situation is shown in the space-time diagram below.
(The vertical axis actually represents the worldline of the point on the mirror directly beneath the emitter, so the light ray and reflection event shown in this drawing are actually projections along the y axis, merely to indicate the relative orientations.) The value of h that we used when applying Hero's law of reflection was evaluated with respect to the rest frame of the mirror, whereas we applied Fermat's Principle with respect to the rest frame of the emitter, so in the preceding expression for T we really need to replace h with
Making this substitution for h into the preceding expression for T, and clearing fractions, we arrive at
We need to find the value of x that minimizes T for any fixed v and h, so we can just as well minimize T times the square root of 1-v2. If we let Q denote the square root of h2 + (1-x)2, we can set to zero the derivative of the above expression with respect to x, and multiply through by Q, to give
Multiplying through by the denominator of the right hand side, and squaring both sides, we find that the reflection point x is a solution of
Expanding both sides, replacing every Q2 by x2 + (1-x)2, and collecting the coefficients of Q, we have
Clearing the common factors, squaring both sides, and bringing the right side over to the left, we get
The left side factors as
The relevant root of the left hand factor is the same as equation (1), and hence the results of Hero's law and Fermat's Principle are in exact agreement, as shown below.
