rate at which that displacement's speed is itself changing its acceleration. represents space or position. is a term for how the displacement is varying at the point in one of the dimensions (like one of the axes on a graph). It's not the rate at which the displacement is changing across space, but in fact the rate at which the change itself is changing across space its second derivative. In other words, this term shows how the displacement's changes are squashed up in a tiny surrounding area. The equation states that at any given instance, at any given point, the way the displacement accelerates is proportional to the way the displacement's changes are squashed up in the surrounding area. Or, in even simpler terms, the way displacements get pushed around is proportional to how pointy the displacement is, and conversely. Using the notations of Newtonian mechanics and vector calculus, the wave equation can be written more compactly as where the double dot on denotes double time derivative of , is the nabla operator, and is the (spatial) Laplacian operator (not vector Laplacian): An even more compact notation sometimes used in physics reads simply where all operators are combined into the d'Alembert operator (denoted by a box): A solution of this (two-way) wave equation can be quite complicated, but it can be analyzed as a linear combination of simple solutions that are sinusoidal plane waves with various directions of propagation and wavelengths but all with the same propagation speed . This analysis is possible because the wave equation is linear and homogeneous, so that any multiple of a solution is also a solution, and the sum of any two solutions is again a solution. This property is called the superposition principle in physics. The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments. The two-way wave equation describing a standing wave field is the simplest example of a second-order hyperbolic differential equation. It and its modifications play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity, geophysics, and many other scientific and technical disciplines.