User talk:Paul Wormer/scratchbook2

In classical physics, the wave equation is the name given to a certain partial differential equation containing second derivatives with respect to spatial coordinates and time. It has the general form

\left[\nabla^2 - \frac{1}{v^2} \frac{\partial^2}{\partial t^2}\right] f(\mathbf{r},t) = 0. $$ The quantity v is a parameter with the dimension of velocity (length over time); &nabla;2 is the Laplace operator' t is the time and r represents the space coordinates x, y, and z. The function f(r,t) is a physical observable; that is, it can be measured and it is a real function. As will be shown, the function f has different meanings in different physical situations.

Motion of a string
The oldest—and simplest—example of a wave equation in classical physics is that governing that the transverse motion of a string under tension and constrained to move in a plane, the xy-plane. Let y(x,t) be the transverse displacement of the string from its equilibrium at point x and time t. The wave equation is

\frac{\partial^2 y(x,t)}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y(x,t)}{\partial t^2}. $$ The velocity v appearing in the wave equation for the string is

v = \sqrt{\frac{T}{\rho}} $$ The tension T as well as the linear mass density &rho; are constant over the string.

A solution of the wave equation is of the form y = f(x&plusmn; vt). Indeed, by the chain rule

\frac{\partial f(x,t)}{\partial x} = \frac{\partial f(x,t)}{\partial (x \pm vt)}\quad \hbox{and}\quad \frac{\partial f(x,t)}{\partial t} = \pm v\frac{\partial f(x,t)}{\partial (x \pm vt)}, $$ and also

\frac{\partial^2 f(x,t)}{\partial x^2} = \frac{\partial^2 f(x,t)}{\partial (x \pm vt)^2}\quad \hbox{and} \quad \frac{\partial^2 f(x,t)}{\partial t^2} = (\pm v)^2\frac{\partial^2 f(x,t)}{\partial (x \pm vt)^2}. $$ Hence

\frac{\partial^2 f(x,t)}{\partial t^2} = v^2\frac{\partial^2 f(x,t)}{\partial x^2}. $$

The initial condition y(x,0) = f(x) specifies the deformation of the string from equilibrium at t = 0. Since f(x&plusmn;vt) differs from f(x) only by a translation of magnitude vt along the x-axis (&minus;vt in positive direction, +vt in negative direction), it is apparent that the deformation of the string moves along the x-axis with velocity v, the wave propagation velocity. Note that, as is shown a few lines earlier,

\frac{\partial y(x,t)}{\partial t} = \pm v\frac{\partial y(x,t)}{\partial x}, $$ which means that v is not the time derivative of the transverse displacement y, i.e., v is not the transverse velocity.