# What are examples of Solutions of Newton's Laws that have analogs to Solutions of the Wave Equation?

The idea is to give examples of processes that deal with properties of a particle that have clear wave analogues.

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I'm not sure this question is clear. What sort of analogues are you looking for? Why do you expect such solutions to exist? And what exactly are you trying to find out by asking this question (or, what prompted you to ask it)? –  David Z Jun 15 '11 at 20:43
Ideally an analog would be where the description of a wave process is mathematically identical to the description of a particle process...a weaker analog would be where phenomena are similar but have different underlying equations....as for what prompted me--> curiosity –  munali Jun 15 '11 at 20:51
Waves on a tensile rope or string are represented by wave equations the derive from Newton's Laws plus Hooke's Law. Is that what you are looking for? –  dmckee Jun 15 '11 at 21:22
I also don't quite get this question. Are you looking for localized waves that behave like particles (under the action of some force)? Or are you looking for something like derivation of Newton's law from Schroedinger equation? Or derivation of wave equation from Newton laws? Or something like soliton? –  Marek Jul 27 '11 at 6:32

## 1 Answer

The two traditional properties which were considered analogous were refraction/sliding-down-a-potential-step and reflection. But all properties are analogous, because Newton's laws emerge as the small $\hbar$ limit of the Schrodinger equation.

• Refraction: suppose a marble is rolling along a surface and encounters a smooth ramp which takes it down to a lower level. What happens to the particle? You can see that it is refracted--- it moves at a different angle. If the ramp is translation invariant in y, and the particle comes in with $p_x$,$p_y$, then the outgoing momentum is $p_x+\Delta p$,$p_y$, and the matching conditions for refraction are satisfied if the wavenumbers $k_x$ and $k_y$ are proportional to the momentum.
• Reflection: if a particle hits a wall, the angle of incidence is equal to the angle of reflection. If you boost to a moving frame, you can find the condition for the angle of reflection when the wall is moving. The condition again matches momenta as though they were wavenumbers.
• Conservation laws: if a particle is entering a force field, this is like light propagating in a material with a smoothly varying index of refraction. Again, the deflection is consistent if the momentum is the wavenumber. The conservation of energy becomes the conservation of frequency in a time-invariant medium.

These coincidences led Hamilton to suspect that particle motion should be thought of as a small-wavelength approximation to a wave equation.

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