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Line 1: − + − ==One-Dimensional Wave Equation==+ − One type of wave equation is the one-dimensional wave equation. The mathematical relation describes a wave whose parts only [[oscillate]] in ''one dimension''. This wave, however, can ''propagate'' in all three spacial dimensions. An example would be a vibrating rope or string with both ends fixed. − The one-dimensional wave equation can be written<ref>Pain, H.J. ''The Physics of Vibrations and Waves'' 6th edition. Southern Gate, Chichester, West Sussex, England: John Wiley & Sons, 2005</ref>:+ + + − + + + + + + + − where y = y(x,t), the y-direction of motion at a point x that changes with time t,+ − and v = the [[phase velocity]] of the wave. − + − + + + + + + + + + + + + + + + + + + + + + + + + +
Rewrite. Not finished. Will do illustration and a little more detail (not a lot!) about applications.
The '''wave equation''' is an important [[differential equation]] in [[physics]]. It describes how [[waves]] propagate through mediums, whether they be [[transverse wave]]s (e.g. [[electromagnetic radiation]]) or [[longitudinal wave]]s (e.g. [[sound]] waves).
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 derivative]]s.
In one dimension, and denoting the "quantity" as <math>\psi</math> the equation is:
:<math>\frac{\partial^2 \psi}{\partial t^2} = v^2\ \frac{\partial^2 \psi}{\partial x^2}</math>
for some constant <math>v</math>. <math>v</math> is the velocity of the wave.
<math> \frac{\partial^2 y}{dx^2} = \frac{1}{v^2} \frac{\partial^2 y}{dt^2} </math>
Solutions to this equation are abundant. For any function <math>\psi(q)</math> of a single variable <math>q</math>, if we turn it into a function of two variables by substituting <math>q = x - vt</math> (or <math>q = x + vt</math>), then, by the [[chain rule]], we have:
:<math>\frac{\partial \psi}{\partial t} = \psi' \frac{\partial}{\partial t}(x - vt) = - \psi' v</math>
Taking the derivative again, we get:
:<math>\frac{\partial^2 \psi}{\partial t^2} = \psi'' v^2</math>
Similarly:
:<math>\frac{\partial \psi}{\partial x} = \psi' \frac{\partial}{\partial x}(x - vt) = \psi'</math>
:<math>\frac{\partial^2 \psi}{\partial x^2} = \psi''</math>
The figure on the right illustrates this for <math>\psi(q) = e^{-q^2}</math>, or <math>\psi(x, t) = e^{- (x - vt)^2}</math>. One can clearly see the "wave", graphed as a function of <math>x</math>, moving to the right as time progresses.
==References==
The quantity <math>v</math> 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. When dealing with electromagnetic waves, it is common to use <math>c</math>, the standard symbol for the speed of light.
<small><references/></small>
==In Higher Dimensions and the Laplacian==
In two dimensions, and Cartesian coordinates, the wave equation is:
:<math>\frac{\partial^2 \psi}{\partial t^2} = v^2\ \left(\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2}\right)</math>
In three dimensions it is:
:<math>\frac{\partial^2 \psi}{\partial t^2} = v^2\ \left(\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2}\right)</math>
In each case the quantity in parentheses is called the [[Laplacian]] operator, denoted thusly:
:<math>\frac{\partial^2 \psi}{\partial t^2} = v^2\ \nabla^2 \psi</math>
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
:<math>\frac{\partial^2 \psi}{\partial t^2} = v^2\ \nabla^2 \psi</math>
==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 (woodwinf 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
<!--==References==
<small><references/></small>-->
[[category:physics]]
[[category:physics]]