Waves ① force→wave / Waves ② force=wave → Waves ③ wave→force, and "a force that looks like a wave"
After Waves ① "a wave is born from force" and Waves ② "force itself was a wave," we finally close the loop on the relationship between wave and force. First the naive direction ── a wave pushes objects. Light and sound both have pressure, and sunlight pushes a sail to drive a spacecraft forward (the reverse of Waves ②). And deeper still lies the strangest "force" in this whole series ── in the quantum world, the particle itself is a wave, and force appears as "a shift in the wave's phase." Even though locally no force is acting anywhere, the interference fringes shift. The gauge of Episode 9 (the connection of phase) shows its face here as a visible wave pattern. This is the true nature of "a force that looks like a wave."
In Waves ② we saw that "a shaken field becomes a wave and flies." A flying wave carries momentum. So when it hits an object and is absorbed or reflected, it hands over that momentum and pushes. This is the radiation pressure of light. Usually it's far too small to feel, but a solar sail (Japan's "IKAROS," and others) really does move by this, and optical tweezers pinch tiny particles with light. Sound is the same (sound pressure). If Waves ② was "force → wave," this is "wave → force" ── wave and force go back and forth.
From here it gets deep. In quantum mechanics, a particle itself, such as an electron, behaves as a wave (de Broglie). Being a wave, it passes through two paths and interferes (the double slit). And a wave has a phase (the clock hand of Episode 9), and the interference fringes are set by the phase difference of the two paths. What happens when a force acts here?
Because a particle = a wave, force (more precisely, the gauge field \(A\) of Episode 9) acts in a way that shifts the wave's phase.
Even if no force (push/pull) is acting locally anywhere, if the phase shifts along the path ── the interference fringes move sideways. Force becomes visible as a shift in the wave pattern.
What shows this vividly is the Aharonov–Bohm effect. Outside a thin solenoid the magnetic field is zero (= zero local force), yet the phase of the electron wave passing around it is shifted by the \(A\) of Episode 9 (the vector potential = the connection), and the interference fringes move. Classically it's the mystery that "there's an effect even though no force is acting." What we said in Episode 9 ── "the true body of force is the connection \(A\)" ── takes the visible form here of a shift in the wave's interference fringes. In the figure below, change the phase shift (= the effect of the gauge \(A\)) and watch the interference fringes slide.
The conclusion of the Waves trilogy. Force and wave were the front and back of the same thing from the start. ① Line up a restoring force and inertia and a wave is born (force → wave); ② the field that carries that force becomes a wave when shaken (force = wave); ③ a wave pushes objects with momentum (wave → force), and in the quantum world force appears as a shift in the wave's phase (force = phase). What we saw in Episode 9 ── "force = the connection of local symmetry (the connection of phase)" ── takes, in the language of waves, the visible form of "the phase shift that moves the interference fringes." "A force that looks like a wave" means that force was, in its origin, a relationship residing in a wave's phase.
The Aharonov–Bohm effect is real, confirmed by the precision experiments of Akira Tonomura and others. But the reason an effect appears even where "the local force is zero" is that the gauge field \(A\) (the connection) acts along the path, and what is observable is the path difference = relative phase (the loop around = the curvature/magnetic flux of Episode 12). The absolute value of the phase itself is not visible (still the backbone of Episode 9). Radiation pressure is real but is minuscule in everyday life; it is made apparent through devices like solar sails and optical tweezers.
The figure is a schematic reproducing only the key point that the interference fringes slide with the phase difference; it is not a rigorous calculation of the actual intensity distribution of a double-slit / AB configuration.
A wave carries momentum, so it pushes objects (radiation pressure, solar sails, sound pressure) ── this is "wave → force." Further, in the quantum world a particle is a wave, and force (the gauge field \(A\)) acts in a way that shifts the wave's phase. Even with zero local force, if the phase shifts the interference fringes move (the Aharonov–Bohm effect). Episode 9's "force = the connection of phase \(A\)" became here a visible shift of the wave pattern.
What became visible across the Waves trilogy is ── force and wave were the front and back of the same thing from the start. A wave is born from force (①), the field that carries force becomes a wave (②), a wave exerts a force and force resides in a wave's phase (③). "A force that looks like a wave" means that force was, in its origin, the relationship called a wave's phase. The main series' backbone, "force is a relationship," reached, in the language of waves, all the way to the deepest level. Next up is the bonus cluster "The Speed of Force" ── just how fast does that force (or wave) actually travel?
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider lets you see the interference fringes slide as you change the phase shift (= the connection A). "See the answer" opens each solution.