Equation of the Day #1
Tonight's equation: The wave equation.
This says that the sum of the change in the change in the function, ψ, with respect to the coordinates used to represent it is equal to the inverse square of the speed of the wave,c, modeled by ψ times the change in the change of ψ with respect to time.
This equation is the governing equation for all wave phenomena in our world. Sound waves, light waves, water waves, earthquakes, etc. are governed by this mathematical equation. In one dimension, the wave equation simplifies to
which has the lovely solutions
where A and B are determined by appropriate boundary conditions, and ω/k = c. This equation governs things like vibrations of a string, sound made by an air column in a pipe (like that of an organ, trumpet, or didgeridoo), or even waves created by playing with a slinky. It also governs the resonances of certain optical cavities, such as a laser or Fabry-Perot cavity.
Since waves are one of my favorite physical phenomena, I find it very appropriate to start with this one.