Difference between revisions of "Dictionary:Wave equation"

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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  
 
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>
+
<center>[[File:Nabla.gif]]<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>
  
  
  
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:  
+
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 [[File:Fgr.gif]] the longitude, the wave equation becomes:  
  
<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>
+
<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;[[File:Fgr.gif]]<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. The <b>vector wave equation</b> is more general; 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  
  
<center>(2&#x03BC;+&#x03BB;)''(''''&#x03C8;)&#x2013;&#x03BC;''&#x00D7;(''&#x00D7;&#x03C8;)=&#x03C1;&#x2202;<sup>2</sup>&#x03C8;/&#x2202;''t''<sup>2</sup>,</center>
+
<center>(2&#x03BC;+&#x03BB;)[[File:Nabla.gif]]([[File:Nabla.gif]][[File:Middot.gif]]&#x03C8;)&#x2013;&#x03BC;[[File:Nabla.gif]]&#x00D7;([[File:Nabla.gif]]&#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  
  
<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>
+
<center>&#x03BC;[[File:Nabla.gif]]<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>
  
  
  
 
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]].
 
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]].

Revision as of 11:56, 18 January 2012

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

Nabla.gif2ψ=(∂2ψ)/(∂x2)+(∂2ψ)/(∂y2)+(∂2ψ)/(∂z2)=((1)/(V2))(∂2ψ)/(∂t2),


where ψ represents wave displacement (pressure, rotation, dilatation, etc.) and V the velocity of the wave. Functions f(ℓx+my+nz±Vt) are solutions to this equation. In spherical coordinates where r is the radius, θ the colatitude, and Fgr.gif the longitude, the wave equation becomes:

((1)/(V2))(∂2Ψ)/(∂t2)=((1)/(r2))[((∂)/(∂r))(r2(∂Ψ)/(∂r))+((1)/(sinθ))((∂)/(∂θ))(sinθ(∂Ψ)/(∂θ))+((1)/(sin2θ))(∂2Ψ)/(∂Fgr.gif2)]


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; it is

(2μ+λ)Nabla.gif(Nabla.gifMiddot.gifψ)–μNabla.gif×(Nabla.gif×ψ)=ρ∂2ψ/∂t2,


which can be written in component form as

μNabla.gif2ψx+(μ+λ)(∂/∂x)(∂ψx/∂x+∂ψy/∂y+∂ψz/∂z)=ρ∂2ψ/∂t2.


If divψ=0, this gives an S-wave; if curl ψ=0, a P-wave. The wave equation in polar anisotropic (transversely isotropic) media is given in Figure T-13.