Wave equation in cylindrical and spherical coordinates

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	2
Pages	7 - 46
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 2.6a

Show that the wave equation (2.5a) can be written in cylindrical coordinates (see Figure 2.6a) as

${\begin{aligned}{\frac {\partial ^{2}\psi }{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}\psi }{\partial \theta ^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}={\frac {1}{V^{2}}}{\frac {\partial ^{2}\psi }{\partial t^{2}}}.\end{aligned}}$

(2.6a)

Solution

Figure 2.6a.  Cylindrical coordinates.

We shall solve by direct substitution. We have $x=r\cos {\mathrm {\theta } }$, $y=r\sin {\mathrm {\theta } }$, $z=z$, and $r^{2}=x^{2}+y^{2}$, ${\mathrm {\theta } }=\tan ^{-1}\left(y/x\right)$. The following solution is lengthy, so we use subscripts to denote partial derivatives and write

${\begin{aligned}a=\sin {\mathrm {\theta } }=y/r,{\mathrm {\quad } }b=\cos {\mathrm {\theta } }=x/r.\end{aligned}}$

We shall require the derivatives:

${\begin{aligned}{\mathrm {\partial } }r/{\mathrm {\partial } }x=r_{x}=x/r=\cos {\mathrm {\theta } }=b,\\{\mathrm {\partial } }r/{\mathrm {\partial } }y=r_{y}=y/r=\sin {\mathrm {\theta } }=a;\\{\mathrm {\partial } }\theta /{\mathrm {\partial } }x=\theta _{x}=\left[{\frac {\mathrm {\partial } }{{\mathrm {\partial } }x}}\left(y/x\right)\right]/[1+\left(y/x\right)^{2}]=-{\frac {y}{x^{2}}}\left({\frac {1}{1+({\frac {y}{x}})^{2}}}\right)\\=-\left(\sin \theta \right)/r=-a/r,\\{\mathrm {\partial } }\theta /{\mathrm {\partial } }y=\theta _{y}=\left[{\frac {\mathrm {\partial } }{{\mathrm {\partial } }y}}\left(y/x\right)\right]/[1+\left(y/x\right)^{2}]={\frac {1}{x}}\left({\frac {1}{1+({\frac {y}{x}})^{2}}}\right)\\=\left(\cos \theta \right)/r=b/r.\end{aligned}}$

To replace ${\mathrm {\psi } }_{xx}$ and ${\mathrm {\psi } }_{yy}$ with derivatives with respect to $r$ and ${\theta }$, we write:

${\begin{aligned}\psi _{x}&=\psi _{r}r_{x}+\psi _{\theta }\theta _{x}=\psi _{r}b-\psi _{\theta }a/r,\\\psi _{y}&=\psi _{r}r_{y}+\psi _{\theta }\theta _{y}=\psi _{r}a+\psi _{\theta }b/r.\end{aligned}}$

Then,

${\begin{aligned}{\mathrm {\psi } }_{xx}={\frac {\mathrm {\partial } }{{\mathrm {\partial } }r}}\left({\mathrm {\psi } }_{r}b-{\mathrm {\psi } }_{\mathrm {\mathrm {\theta } } }a/r\right)r_{x}+{\frac {\mathrm {\partial } }{{\mathrm {\partial } }{\mathrm {\theta } }}}\left({\mathrm {\psi } }_{r}b-{\mathrm {\psi } }_{\mathrm {\theta } }a/r\right)\theta _{x}\\=\left({\mathrm {\psi } }_{rr}b-{\mathrm {\psi } }_{r{\mathrm {\theta } }}a/r+{\mathrm {\psi } }_{\mathrm {\mathrm {\theta } } }a/r^{2}\right)b\\{\quad }+\left({\mathrm {\psi } }_{r{\mathrm {\theta } }}b-{\mathrm {\psi } }_{r}a-{\mathrm {\psi } }_{\theta \theta }a/r-{\mathrm {\psi } }_{\mathrm {\theta } }b/r\right)\left(-a/r\right)\\=[{\mathrm {\psi } }_{rr}b^{2}-{\mathrm {\psi } }_{r{\mathrm {\theta } }}\left(2ab/r\right)+{\mathrm {\psi } }_{\mathrm {\theta } }\left(2ab/r^{2}\right)+{\mathrm {\psi } }_{r}\left(a^{2}/r\right)+{\mathrm {\psi } }_{\theta \theta }\left(a/r)^{2}\right],\\{\mathrm {\psi } }_{yy}={\frac {\mathrm {\partial } }{{\mathrm {\partial } }r}}\left({\mathrm {\psi } }_{r}a+{\mathrm {\psi } }_{\mathrm {\theta } }b/r\right)r_{y}+{\frac {\mathrm {\partial } }{{\mathrm {\partial } }{\mathrm {\theta } }}}\left({\mathrm {\psi } }_{r}a+{\mathrm {\psi } }_{\mathrm {\theta } }b/r\right)\theta _{y}\\=\left({\mathrm {\psi } }_{rr}a+{\mathrm {\psi } }_{r{\mathrm {\theta } }}b/r-{\mathrm {\psi } }_{\mathrm {\mathrm {\theta } } },b/r^{2}\right)a\\{\quad }+\left({\mathrm {\psi } }_{r{\mathrm {\theta } }}a+\psi _{r}b+\psi _{\theta \theta }b/r-\psi _{\mathrm {\theta } }a/r\right)\left(b/r\right)\\=\left({\mathrm {\psi } }_{rr}a^{2}+\psi _{r{\mathrm {\theta } }}2ab/r-{\mathrm {\psi } }_{\mathrm {\theta } }2ab/r^{2}\right)+\left({\mathrm {\psi } }_{r}b^{2}/r+{\mathrm {\psi } }_{\theta \theta }b^{2}r^{2}\right).\end{aligned}}$

Thus

${\begin{aligned}{\nabla }^{2}{\mathrm {\psi } }={\mathrm {\psi } }_{rr}\left(a^{2}+b^{2}\right)+{\mathrm {\psi } }_{r}\left(a^{2}+b^{2}\right)/r+{\mathrm {\psi } }_{\theta \theta }\left(a^{2}+b^{2}\right)/r^{2}+{\mathrm {\psi } }_{zz},\end{aligned}}$

so

${\frac {{\mathrm {\partial } }^{2}{\mathrm {\psi } }}{{\mathrm {\partial } }r^{2}}}+{\frac {1}{r}}{\frac {{\mathrm {\partial } }{\mathrm {\psi } }}{{\mathrm {\partial } }r}}+{\frac {1}{r^{2}}}{\frac {{\mathrm {\partial } }^{2}{\mathrm {\psi } }}{{\mathrm {\partial } }{\mathrm {\theta } }^{2}}}+{\frac {{\mathrm {\partial } }^{2}{\mathrm {\psi } }}{{\mathrm {\partial } }z^{2}}}={\frac {1}{V^{2}}}{\frac {{\mathrm {\partial } }^{2}{\mathrm {\psi } }}{{\mathrm {\partial } }t^{2}}}.$

Problem 2.6b

Transform the wave equation into spherical coordinates (see Figure 2.6b), showing that it becomes

${\begin{aligned}{\frac {1}{r^{2}}}\left[{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial \psi }{\partial r}}\right)+{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial \psi }{\partial \theta }}\right)+{\frac {1}{{\sin }^{2}\theta }}{\frac {\partial ^{2}\psi }{\partial \phi ^{2}}}\right]={\frac {1}{V^{2}}}{\frac {\partial ^{2}\psi }{\partial t^{2}}}.\end{aligned}}$

(2.6b)

Solution

Spherical coordinates $\left(r,\;\theta ,\;\phi \right)$ and rectangular coordinates are related as follows (see Figure 2.6b):

${\begin{aligned}x=r\sin \theta \cos \phi ,\\y=r\sin \theta \sin \phi ,\\z=r\cos \theta ,\\r=(x^{2}+y^{2}+z^{2})^{1/2},\\\theta ={\rm {cos}}^{-1}\left(z/r\right),\\\phi =\tan ^{-1}\left(y/x\right).\end{aligned}}$

We continue to use subscripts to denote derivatives and letters to denote sines and cosines:

${\begin{aligned}{\begin{array}{l}a=\sin \theta ,\\m=\sin \phi ,\\a_{\mathrm {\theta } }=b,\\m_{\phi }=n,\end{array}}{\quad }{\begin{array}{l}b=\cos \theta ,\\n=\cos \phi ,\\b_{\mathrm {\theta } }=-a,\\n_{\phi }=-m.\end{array}}\end{aligned}}$

The derivatives of $r$, ${\theta }$, and $\phi$ now become:

${\begin{aligned}r_{x}&=x/r=an,\\r_{y}&=y/r=am,\\r_{z}&=z/r=b;\\\theta _{x}&={\frac {\partial }{\partial x}}[{\rm {cos}}^{-1}\left(z/r\right)]=-{\frac {\partial }{\partial r}}\left(z/r\right)[1-(z/r)^{2}]^{-1/2}={\frac {Z}{r^{2}}}r_{x}\left(1/a\right)=bn/r,\\\theta _{y}&=-{\frac {\partial }{\partial y}}\left(z/r\right)[1-(z/r)^{2}]^{-1/2}=-{\frac {Z}{r^{2}}}r_{y}\left(1/a\right)=bm/r,\\\theta _{z}&=-{\frac {\partial }{\partial z}}\left(z/r\right)[1-(z/r)^{2}]^{-1/2}=-\left(1/r-z/r^{2}r_{z}\right)[1-(z/r)^{2}]^{-1/2}\\&=\left(-1/r\right)\left(1-zb/r\right)\left(1/a\right)=\left(-1/r\right)\left(1-b^{2}\right)\left(1/a\right)=-a/r,\\\phi _{x}&={\frac {\partial }{\partial x}}[\tan ^{-1}\left(y/x\right)]={\frac {\partial }{\partial x}}\left(y/x\right)[1+(y/x)^{2}]^{-1}=-\left(y/x^{2}\right)[1+(y/x)^{2}]^{-1}\\&=-\left(y/x\right)\left(1/x\right)[1+(y/x)^{2}]^{-1}=-\left(m/n\right)\left(1/ran\right)(1/n^{2})^{-1}=-m/ar,\\\phi _{y}&={\frac {\partial }{\partial y}}\left(y/x\right)[1+(y/x)^{2}]^{-1}=\left(1/x\right)[1+(y/x)^{2}]^{-1}=\left(1/ran\right)(1/n^{2})^{-1}=n/ar,\\\phi _{z}&=0\ {{\text{(because }}\phi {\text{ is not a function of }}z)}.\end{aligned}}$

Figure 2.6b  Spherical coordinates.

Summarizing these results, we have

${\begin{aligned}{\begin{array}{l}r_{x}=an,\\\theta _{x}=bn/r,\\\phi _{x}=-m/ar,\end{array}}\quad {\begin{array}{l}r_{y}=am,\\\theta _{y}=bm/r,\\\phi _{y}=n/ar,\end{array}}\quad {\begin{array}{l}r_{z}=b,\\\theta _{z}=-a/r,\\\phi _{z}=0.\end{array}}\end{aligned}}$

We now calculate the derivatives $\phi _{xx}$, etc.:

${\begin{aligned}\psi _{x}&=\psi _{r}r_{x}+\psi _{\theta }\theta _{x}+\psi _{\phi }\phi _{x}\\&=\psi _{r}an+\psi _{\theta }bn/r-\psi _{\phi }m/ar;\\\psi _{xx}&={\frac {\partial }{\partial r}}\left(\psi _{r}an+\psi _{\theta }bn/r-\psi _{\phi }m/ar\right)\left(an\right)\\&\quad +{\frac {\partial }{\partial \theta }}\left(\psi _{r}an+\psi _{\theta }bn/r-\psi _{\phi }m/ar\right)\left(bn/r\right)\\&\quad +{\frac {\partial }{\partial \theta }}\left(\psi _{r}an+\psi _{\theta }bn/r-\psi _{\phi }m/ar\right)\left(-m/ar\right),\\&=\left(\psi _{rr}an+\psi _{r\theta }bn/r-\psi _{\theta }bn/r^{2}-\psi _{\gamma \phi }m/ar+\psi _{\phi }m/ar^{2}\right)an\\&\quad +(\psi _{r\theta }an+\psi _{r}bn+\psi _{\theta \theta }bn/r-\psi _{\theta }an/r-\psi _{\theta \phi }m/ar\\&\quad +\psi _{\phi }mb/a^{2}r)\left(bn/r\right)+(\psi _{\gamma \phi }an-\psi _{r}am+\psi _{\theta \phi }bn/r\\&\quad -\psi _{\theta }bm/r-\psi _{\phi \phi }m/ar-\psi _{\phi }n/ar)\left(-m/ar\right)\\&=\psi _{rr}a^{2}n^{2}+\psi _{r\theta }\left(2abn^{2}/r\right)-\psi _{\gamma \phi }\left(2mn/r\right)+\psi _{r}\left({\frac {b^{2}n^{2}+m^{2}}{r}}\right)\\&\quad +\psi _{\theta \theta }\;\left({\frac {bn}{r}}\right)^{2}+\psi _{\theta \phi }\left({\frac {b}{a}}\right)\left({\frac {2mn}{r}}\right)+\psi _{\theta }\;\left({\frac {bm^{2}}{ar^{2}}}-{\frac {2abn}{r^{2}}}\right)\\&\quad +\psi _{\phi \phi }\left({\frac {m}{ar}}\right)^{2}+\psi _{\phi }\left({\frac {2mn}{a^{2}r^{2}}}\right);\\\psi _{y}&=\psi _{r}r_{y}+\psi _{\theta }\theta _{y}+\psi _{\phi }\phi _{y}\\&=\psi _{r}am+\psi _{\theta }bm/r+\psi _{\phi }n/ar;\\\psi _{yy}&={\frac {\partial }{\partial r}}\left(\psi _{r}am+\psi _{\theta }bm/r+\psi _{\phi }n/ar\right)\left(am\right)\\&\quad +{\frac {\partial }{\partial \theta }}\left(\psi _{r}am+\psi _{\theta }bm/r+\psi _{\phi }n/ar\right)\left(bm/r\right)\\&\quad +{\frac {\partial }{\partial \phi }}\left(\psi _{r}am+\psi _{\theta }bm/r+\psi _{\phi }n/ar\right)\left(n/ar\right)\\&=\left(\psi _{rr}am+\psi _{r\theta }bm/r-\psi _{\theta }bm/r^{2}+\psi _{\gamma \phi }n/ar-\psi _{\phi }n/ar^{2}\right)am\\&\quad +(\psi _{r\theta }am+\psi _{r}bm+\psi _{\theta \theta }bm/r-\psi _{\theta }am/r+\psi _{\theta \phi }n/ar\\&\quad -\psi _{\phi }bn/a^{2}r)\left(bm/r\right)\\&\quad +(\psi _{\gamma \phi }am+\psi _{r}an+\psi _{\theta \phi }bm/r+\psi _{\theta }bn/r+\psi _{\phi \phi }n/ar\\&\quad -\psi _{\phi }m/ar)\left(n/ar\right)\\&=\psi _{rr}a^{2}m^{2}+\psi _{r\theta }\;\left({\frac {2abm}{r}}\right)+\psi _{\gamma \phi }\left({\frac {2mn}{r}}\right)+\psi _{r}\left({\frac {1-a^{2}m^{2}}{r}}\right)\\&\quad +\psi _{\theta \theta }\;\left({\frac {bm}{r}}\right)^{2}+\psi _{\theta \phi }\left({\frac {b}{a}}\right)\left({\frac {2mn}{r^{2}}}\right)+\psi _{\theta }\;\left(-{\frac {2abm^{2}}{r}}+{\frac {bn^{2}}{ar^{2}}}\right)\\&\quad +\psi _{\phi \phi }\left({\frac {n}{ar}}\right)^{2_{-\psi }}\phi \left({\frac {mn}{r^{2}}}\right)\left(1+{\frac {b^{2}}{a^{2}}}+{\frac {1}{a^{2}}}\right);\\\psi _{z}&=\psi _{r}r_{z}+\psi _{\theta }\theta _{z}+\psi _{\phi }\phi _{z}=\psi _{r}b-\psi _{\theta }a/r;\\\psi _{zz}&=\left(\psi _{rr}b-\psi _{r\theta }a/r+\psi _{\theta }a/r^{2}\right)b\\&\quad +\left(\psi _{r\theta }b-\psi _{r}a-\psi _{\theta \theta }a/r-\psi _{\theta }b/r\right)\left(-a/r\right)\\&=[\psi _{rr}b^{2}-\psi _{r\theta }\left(2ab/r\right)+\psi _{r}\left(a^{2}/r\right)+\psi _{\theta \theta }\left(a/r)^{2}+\psi _{\theta }\left(2ab/r^{2}\right)\right].\end{aligned}}$

Adding the three derivatives, we get

${\begin{aligned}\nabla ^{2}\psi &=\psi _{rr}\left(a^{2}m^{2}+a^{2}n^{2}+b^{2}\right)+\psi _{r\theta }\;\left[{\frac {2ab\left(m^{2}+n^{2}\right)}{r}}-{\frac {2ab}{r}}\right]\\&\quad +\psi _{r}\left[{\frac {\left(b^{2}n^{2}+m^{2}\right)+\left(1-a^{2}m^{2}\right)+a^{2}}{r}}\right]\\&\quad +\psi _{\theta \theta }\;\left({\frac {b^{2}n^{2}+b^{2}m^{2}+a^{2}}{r^{2}}}\right)\\&\quad +\psi _{\theta }\;\left[{\frac {\left(bm^{2}/a-2abm^{2}\right)+\left(-2abm^{2}+bn^{2}/a+2ab\right)}{r}}\right]\\&\quad +\psi _{\phi \phi }\left[(-m/ar)^{2}+\left(n/ar\right)^{2}\right]\\&=\psi _{rr}+\left({\frac {2}{r}}\right)\psi _{r}+\left({\frac {1}{r^{2}}}\right)\psi _{\theta \theta }\;+\left({\frac {\cot \theta }{r^{2}}}\right)\psi _{\theta }\;+\left({\frac {1}{a^{2}r^{2}}}\right)\psi _{\phi \phi }.\end{aligned}}$

Substituting the values of $a$, $b$, $m$, and $n$, we get for the wave equation

${\begin{aligned}{\frac {{\mathrm {\partial } }^{2}\psi }{{\mathrm {\partial } }r^{2}}}+{\frac {2}{r}}{\frac {{\mathrm {\partial } }\psi }{{\mathrm {\partial } }r}}+{\frac {1}{r^{2}}}{\frac {{\mathrm {\partial } }^{2}\psi }{{\mathrm {\partial } }\theta ^{2}}}+\left({\frac {{\rm {\;cot\;}}\theta }{r^{2}}}\right){\frac {{\mathrm {\partial } }\psi }{{\mathrm {\partial } }\theta }}+\left({\frac {1}{r^{2}{\rm {sin}}^{2}\theta }}\right){\frac {{\mathrm {\partial } }^{2}\psi }{{\mathrm {\partial } }\phi ^{2}}}={\frac {1}{V^{2}}}{\frac {{\mathrm {\partial } }^{2}\psi }{{\mathrm {\partial } }t^{2}}}.\end{aligned}}$

This is often written in the more compact form

${\begin{aligned}{\frac {1}{r^{2}}}\left[{\frac {\mathrm {\partial } }{{\mathrm {\partial } }r}}\left(r^{2}{\frac {{\mathrm {\partial } }^{2}\psi }{{\mathrm {\partial } }r}}\right)+{\frac {1}{\sin \theta }}{\frac {\mathrm {\partial } }{{\mathrm {\partial } }\theta }}\left(\sin \theta {\frac {{\mathrm {\partial } }\psi }{{\mathrm {\partial } }\theta }}\right)+{\frac {1}{{\rm {sin}}^{2}\theta }}{\frac {{\mathrm {\partial } }^{2}\psi }{{\mathrm {\partial } }\phi ^{2}}}\right]={\frac {1}{V^{2}}}{\frac {{\mathrm {\partial } }^{2}\psi }{{\mathrm {\partial } }t^{2}}}.\end{aligned}}$

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• The basic elastic constants
• Interrelationships among elastic constants
• Magnitude of disturbance from a seismic source
• Magnitudes of elastic constants
• General solutions of the wave equation
• Sum of waves of different frequencies and group velocity
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• Potential functions used to solve wave equations
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• Far- and near-field effects for a point source
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• Tube-wave relationships
• Relation between nepers and decibels
• Attenuation calculations
• Diffraction from a half-plane

External links

find literature about Wave equation in cylindrical and spherical coordinates