Wave equation

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The wave equation is among the most well known, elegant, and important equations in all of mathematical physics. A great many physical problems, usually relating to wave motion or vibration, turn into the wave equation when analyzed mathematically. Some of these applications will be discussed below.

The wave equation concerns some "quantity" that is a function of both space and time. That "quantity" might be air pressure, a magnetic field, the displacement of a string or membrane, or the abstract "wave function" of quantum mechanics. It is a partial differential equation, since it involves partial derivatives.

In one dimension, and denoting the "quantity" as ψ, the equation is:

\frac{\partial^2 \psi}{\partial t^2} = c^2\ \frac{\partial^2 \psi}{\partial x^2}

for some constant c. c is the velocity of the wave. It is common to use this symbol, even for waves other than light waves.

Solutions to this equation are abundant. For any function ψ(q) of a single variable q, if we turn it into a function of two variables by substituting q = xvt (or q = x + vt), then, by the chain rule, we have:

\frac{\partial \psi}{\partial t} = \psi' \frac{\partial}{\partial t}(x - ct) = - c\ \psi'

Taking the derivative again, we get:

\frac{\partial^2 \psi}{\partial t^2} = c^2\ \psi''

Similarly:

\frac{\partial \psi}{\partial x} = \psi' \frac{\partial}{\partial x}(x - ct) = \psi'
\frac{\partial^2 \psi}{\partial x^2} = \psi''
Graphs of the function e^{-(x-t)^2} for t=0 (blue), t=1 (red), t=2 (green) and t=3 (magenta)

The figure on the right illustrates this for \psi(q) = e^{-q^2}, or \psi(x, t) = e^{- (x - ct)^2}. One can clearly see the "wave", graphed as a function of x, moving to the right as time progresses.

The quantity c in the wave equation is the speed of propagation of the wave. Dimensional analysis of the derivatives shows that it has the dimensions of velocity.

In Higher Dimensions, and the Laplacian

In two dimensions, and Cartesian coordinates, the wave equation is:

\frac{\partial^2 \psi}{\partial t^2} = c^2\ \left(\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2}\right)

In three dimensions it is:

\frac{\partial^2 \psi}{\partial t^2} = c^2\ \left(\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2}\right)

In each case the quantity in parentheses is called the Laplacian operator, denoted thusly:

\frac{\partial^2 \psi}{\partial t^2} = c^2\ \nabla^2 \psi

The Laplacian operator is defined in arbitrary coordinate systems (e.g. cylindrical or spherical) to be equivalent to the Cartesian quantity shown above. Therefore, to obtain the wave equation in arbitrary coordinates, one simply looks up the definition of the Laplacian in that coordinate system and substitutes it into

\frac{\partial^2 \psi}{\partial t^2} = c^2\ \nabla^2 \psi

Applications

There are many problems in physics that, when analyzed mathematically, turn into the wave equation. These include:

  • air pressure (hence sound waves)
  • vibrating strings (hence stringed instruments)
  • vibrating columns of air (woodwind and brass instruments)
  • vibrating membranes (kettle drums)
  • Maxwell's equations for electrodynamics (electromagnetic waves)

Additional reading

Pain, H.J. The Physics of Vibrations and Waves 6th edition. Southern Gate, Chichester, West Sussex, England: John Wiley & Sons, 2005

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