Difference between revisions of "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.
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A place where nobody dared to go
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The love that we came to know
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They call it Xanadu
  
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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.
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And now
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Open your eyes and see
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What we have made is real
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We are in Xanadu
  
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In one dimension, and denoting the "quantity" as <math>\psi</math>, the equation is:
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A million lights are dancing
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:<math>\frac{\partial^2 \psi}{\partial t^2} = c^2\ \frac{\partial^2 \psi}{\partial x^2}</math>
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And there you are
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for some constant <math>c</math>.  <math>c</math> is the velocity of the wave.  It is common to use this symbol, even for waves other than light waves.
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A shooting star
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An everlasting world
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And you're here with me
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Eternally
  
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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:
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Xanadu, Xanadu,  
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:<math>\frac{\partial \psi}{\partial t} = \psi' \frac{\partial}{\partial t}(x - ct) = - c\ \psi'</math>
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(now we are here)  
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Taking the derivative again, we get:
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In Xanadu
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:<math>\frac{\partial^2 \psi}{\partial t^2} = c^2\ \psi''</math>
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Xanadu, Xanadu,
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Similarly:
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(now we are here)
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:<math>\frac{\partial \psi}{\partial x} = \psi' \frac{\partial}{\partial x}(x - ct) = \psi'</math>
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In Xanadu
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:<math>\frac{\partial^2 \psi}{\partial x^2} = \psi''</math>
 
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{{clear}}
 
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[[File:Wave.jpg|thumb|right|450px|Graphs of the function <math>e^{-(x-t)^2}</math> for t=0 (blue), t=1 (red), t=2 (green) and t=3 (magenta)]]
 
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The figure on the right illustrates this for <math>\psi(q) = e^{-q^2}</math>, or <math>\psi(x, t) = e^{- (x - ct)^2}</math>.  One can clearly see the "wave", graphed as a function of <math>x</math>, moving to the right as time progresses.
 
  
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The quantity <math>c</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.
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Xanadu, your neon lights will shine
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For you, Xanadu
  
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==In Higher Dimensions, and the Laplacian==
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The love
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In two dimensions, and Cartesian coordinates, the wave equation is:
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The echoes of long ago
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:<math>\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)</math>
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You needed the world to know
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In three dimensions it is:
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They are in Xanadu
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:<math>\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)</math>
 
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In each case the quantity in parentheses is called the [[Laplacian]] operator, denoted thusly:
 
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:<math>\frac{\partial^2 \psi}{\partial t^2} = c^2\ \nabla^2 \psi</math>
 
  
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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
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The dream
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:<math>\frac{\partial^2 \psi}{\partial t^2} = c^2\ \nabla^2 \psi</math>
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That came through a million years
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That lived on through all the tears
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It came to Xanadu
  
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==Applications==
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A million lights are dancing
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There are many problems in physics that, when analyzed mathematically, turn into the wave equation.  These include:
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And there you are
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*air pressure (hence sound waves)
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A shooting star
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*vibrating strings (hence stringed instruments)
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An everlasting world
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*vibrating columns of air (woodwind and brass instruments)
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And you're here with me
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*vibrating membranes (kettle drums)
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Eternally
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*Maxwell's equations for electrodynamics (electromagnetic waves)
 
  
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==Additional reading==
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Xanadu, Xanadu,  
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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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(now we are here)
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In Xanadu
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Xanadu, Xanadu,
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(now we are here)
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In Xanadu
  
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<!--==References==
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Now that I'm here
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<small><references/></small>-->
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Now that you're near in Xanadu
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Now that I'm here
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[[category:physics]]
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Now that you're near in Xanadu
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Xanadu...

Revision as of 01:07, August 10, 2010

A place where nobody dared to go The love that we came to know They call it Xanadu

And now Open your eyes and see What we have made is real We are in Xanadu

A million lights are dancing And there you are A shooting star An everlasting world And you're here with me Eternally

Xanadu, Xanadu, (now we are here) In Xanadu Xanadu, Xanadu, (now we are here) In Xanadu

Xanadu, your neon lights will shine For you, Xanadu

The love The echoes of long ago You needed the world to know They are in Xanadu

The dream That came through a million years That lived on through all the tears It came to Xanadu

A million lights are dancing And there you are A shooting star An everlasting world And you're here with me Eternally

Xanadu, Xanadu, (now we are here) In Xanadu Xanadu, Xanadu, (now we are here) In Xanadu

Now that I'm here Now that you're near in Xanadu Now that I'm here Now that you're near in Xanadu Xanadu...

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