Wave equation in cylindrical and spherical coordinates

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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


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \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{align}

This is often written in the more compact form


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \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{align}

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