The radial coordinate wave equation close to the finger is not like the one-dimensional standard wave equation of a string. The circular waves close to the finger will partially reflect back. A solution is a standing wave whose radial profile is a Bessel function and which does synchronized oscillation up and down.
Far from the finger, the radial wave equation is close to the equation for a string, and the waves will spread with little reflection back.
If we throw a stone into water, the center will keep oscillating up and down for quite some time while waves who got far enough from the center will start spreading somewhat like plane waves.
If the medium is not uniform, then we can describe the propagation of a plane wave as the main wave plus perturbation sources caused by the nonuniformity of the medium. The plane wave may gradually decay into a various more or less random disturbances of the field.
Does the space around an event horizon heat up?
How does a wave which is approaching an event horizon behave?
If we believe in the equivalence principle, we may try studying the light wave in a freely falling laboratory. There are tidal forces in the lab, but at least we get rid of the strong gravitational pull.
We may assume that a static observer drops the lab, so that the lab can for a while study a light wave propagating downward.
The frequency of the light will drop because of the Doppler effect. That is, the input to the laboratory is a down chirp.
A wave of a form
ψ = exp(-i ω(t, x) (t - x))
does not usually satisfy the standard wave equation
d^2/dx^2 ψ = d^2/dt^2 ψ,
if ω changes with time t and position x.
A simple formula for a chirp would be
ω = ω_0 + ε (t - x).
Maybe the down chirp cannot be transmitted through the laboratory without producing a perturbation source?
Since a wave will cycle through an infinite number of wavelengths before reaching the event horizon, it must travel through an infinite number of laboratories. If a perturbation source is produced in each laboratory, then the wave may decay entirely into perturbations before it reaches the horizon. If this is the case, then it is hard for anything to fall through the horizon because diffusion of heat is slow compared to the propagation of a plane wave.
A wave in an infinitely long swimming pool and fishing floats
Suppose that we have an infinitely long pool of water of some fixed width. We can make a wave which travels along the pool without any reflection or disturbance.
Suppose then that we add a fishing float at every 25 meters of the pool. The float will produce a minuscule perturbation wave each time the crest of our water wave passes it. Since the pool is infinitely long, eventually the floats will absorb all the energy of the wave we made, and convert it to more or less random waves zigzagging the pool.
This example shows that a minor modification to a wave equation may cause all energy to be reflected.
If a light wave which approaches an event horizon does not obey the standard wave equation exactly under some choices of coordinates, then most probably all of its energy will eventually be reflected.
We need to write the wave equation close to the horizon in a way which contains the standard wave equation plus a small source term (= perturbation). Then we can study what is the impact of the source term.
In the Schwarzschild coordinates, where x is a short distance from the horizon, the speed of light is proportional to x.
This corresponds to a wave equation of the form
An ingoing gaussian pulse
Graham Reid calculated in the Schwarzschild metric the fate of an ingoing spherically symmetric gaussian pulse for various widths of the pulse. The result is that the black hole absorbs between 0% and 100% of the pulse, less for narrow pulses.
Our conjecture was a 0% absorption. We need to check the assumptions in the calculation.
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