Difference between revisions of "Dictionary:Wave equation"

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{{#category_index:W|wave equation}}
 
An equation that relates the spatial and time dependence of a disturbance which can propagate as a wave. In rectangular coordinates ''x'', ''y'', ''z'', it is  
 
An equation that relates the spatial and time dependence of a disturbance which can propagate as a wave. In rectangular coordinates ''x'', ''y'', ''z'', it is  
  
<center>''<sup>2</sup>&#x03C8;=(&#x2202;<sup>2</sup>&#x03C8;)/(&#x2202;x<sup>2</sup>)+(&#x2202;<sup>2</sup>&#x03C8;)/(&#x2202;y<sup>2</sup>)+(&#x2202;<sup>2</sup>&#x03C8;)/(&#x2202;z<sup>2</sup>)=((1)/(V<sup>2</sup>))(&#x2202;<sup>2</sup>&#x03C8;)/(&#x2202;t<sup>2</sup>),</center>
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<center><math>\nabla^2 \psi = \frac {\partial^2 \psi}{\partial x^2 } + \frac {\partial^2 \psi}{\partial y^2} + \frac {\partial^2 \psi}{\partial z^2} =\left( \frac{1}{V^2}\right)\frac {\partial^2 \psi}{\partial t^2} </math>,</center>
  
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where <math>\psi</math> represents wave displacement (pressure, rotation, dilatation, etc.) and ''V'' the velocity of the wave. Functions <math>f(lx + my + nz \pm Vt)</math> are solutions to this equation.
  
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In spherical coordinates where ''r'' is the radius, <math>\theta </math> the colatitude, and <math>\phi </math>the longitude, the wave equation becomes:
  
where &#x03C8; represents wave displacement (pressure, rotation, dilatation, etc.) and ''V'' the velocity of the wave. Functions ''f''(&#x2113;''x''+''my''+''nz''&#x00B1;''Vt'') are solutions to this equation. In spherical coordinates where ''r'' is the radius, &#x03B8; the colatitude, and '' the longitude, the wave equation becomes:
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<center> <math> \left(\frac{1}{V^2}\right) \frac{\partial^2 \Psi}{\partial t^2}= \left(\frac{1}{r^2}\right)\left[ \left(\frac{\partial}{\partial r} \right) \left( r^2 \frac{\partial \Psi}{\partial r}\right) + \left(\frac{1}{sin{\theta}}\right)\left(\frac{\partial}{\partial \theta}\right)\left(sin{\theta}\frac{\partial \Psi}{\partial \theta}\right) + \left( \frac{1}{sin^2{\theta}}\right) \frac{\partial^2 \Psi}{\partial \phi^2} \right]</math> </center>
  
<center>((1)/(V<sup>2</sup>))(&#x2202;<sup>2</sup>&#x03A8;)/(&#x2202;t<sup>2</sup>)=((1)/(r<sup>2</sup>))&#x005B;((&#x2202;)/(&#x2202;r))(r<sup>2</sup>(&#x2202;&#x03A8;)/(&#x2202;r))+((1)/(sin&#x03B8;))((&#x2202;)/(&#x2202;&#x03B8;))(sin&#x03B8;(&#x2202;&#x03A8;)/(&#x2202;&#x03B8;))+((1)/(sin<sup>2</sup>&#x03B8;))(&#x2202;<sup>2</sup>&#x03A8;)/(&#x2202;''<sup>2</sup>)&#x005D;</center>
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The foregoing are forms of the <b>scalar wave equation</b> These forms do not provide for the conversion of P-waves to S-waves nor vice-versa.
  
  
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The <b>vector wave equation</b> is more general; for isotropic media it is
  
The foregoing are forms of the <b>scalar wave equation</b> These forms do not provide for the conversion of P-waves to S-waves nor vice-versa. The <b>vector wave equation</b> is more general; it is
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<center><math> \left(2\mu + \lambda \right) \nabla (\nabla \cdot \psi)- \mu \nabla \times (\nabla \times \psi)=\rho \frac{\partial ^2 \psi}{\partial t^2} </math>,</center>
<center>(2&#x03BC;+&#x03BB;)''(''''&#x03C8;)&#x2013;&#x03BC;''&#x00D7;(''&#x00D7;&#x03C8;)=&#x03C1;&#x2202;<sup>2</sup>&#x03C8;/&#x2202;''t''<sup>2</sup>,</center>
 
 
 
  
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which can be written in component form as
  
which can be written in component form as
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<center><math>\mu \nabla ^2 \Psi _x + (\mu + \lambda) \frac{\partial}{\partial x} \left( \frac{\partial \Psi_x}{\partial x} + \frac{\partial \Psi_y}{\partial y} + \frac{\partial \Psi_z}{\partial z}\right) = \rho \frac {\partial^2 \Psi_x}{\partial t^2} </math>.</center>
  
<center>&#x03BC;''<sup>2</sup>&#x03C8;<sub>''x''</sub>+(&#x03BC;+&#x03BB;)(&#x2202;/&#x2202;<b>''''x''''</b>)(&#x2202;&#x03C8;<sub>''x''</sub>/&#x2202;''x''+&#x2202;&#x03C8;<sub>''y''</sub>/&#x2202;''y''+&#x2202;&#x03C8;<sub>''z''</sub>/&#x2202;''z'')=&#x03C1;&#x2202;<sup>2</sup>&#x03C8;/&#x2202;''t''<sup>2</sup>.</center>
 
  
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If <math> \nabla \cdot \Psi =0 </math>, this gives an S-wave; if <math> \nabla \times \Psi =0 </math>, a P-wave. The wave equation in polar anisotropic (transversely isotropic) media is given in Figure [[Special:MyLanguage/Dictionary:Fig_T-13|T-13]].
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If div&#x03C8;=0, this gives an S-wave; if curl &#x03C8;=0, a P-wave. The wave equation in polar anisotropic (transversely isotropic) media is given in Figure [[Dictionary:Fig_T-13|T-13]].
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For a derivation, and the relation of the ''wave equation'' to the ''equation of motion'', see the main page: [[Wave equation]].

Latest revision as of 16:38, 27 September 2020

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An equation that relates the spatial and time dependence of a disturbance which can propagate as a wave. In rectangular coordinates x, y, z, it is

,

where represents wave displacement (pressure, rotation, dilatation, etc.) and V the velocity of the wave. Functions are solutions to this equation.

In spherical coordinates where r is the radius, the colatitude, and the longitude, the wave equation becomes:

The foregoing are forms of the scalar wave equation These forms do not provide for the conversion of P-waves to S-waves nor vice-versa.


The vector wave equation is more general; for isotropic media it is

,

which can be written in component form as

.


If , this gives an S-wave; if , a P-wave. The wave equation in polar anisotropic (transversely isotropic) media is given in Figure T-13.


For a derivation, and the relation of the wave equation to the equation of motion, see the main page: Wave equation.