Boundary conditions and decomposition on spherical harmonics
$begingroup$
Say I have a function $f(x,y,z)$ satisfying some Laplace-type equation on a domain. This domain is an hemisphere with boundary defined by $x=0$ in Cartesian coordinates. I impose either Dirichlet or Neumann boundary conditions on the plane $x=0$. Now I want to switch to spherical coordinates $(r,theta,varphi)$, where $0<theta<pi$ and normally $0levarphile2pi$.
First, am I allowed to defined the boundary $x=0$ by $varphi=0,pi$, so that $0levarphilepi$ in spherical coordinates?
Assuming so, next I want to expand my function $f(r,theta,varphi)$ on angular modes such that
begin{equation}
f(r,theta,varphi)=sum_{lm}u_{lm}(theta,varphi),f_{lm}(r),,
end{equation}
where $u_{lm}(theta,varphi)$ are related to the spherical harmonics. However, I still want to impose some boundary conditions at $varphi=0,pi$.
For Dirichlet boundary conditions at $varphi=0,pi$, then my guess would be to use only the "sine" part of the real spherical harmonics, that is $u_{lm}(theta,varphi)=sqrt{2}(-1)^m,{rm Im}[Y_{l|m|}]$, with $-lle m<0$, while for Neumann I would use the "cosine" part, i.e. $u_{lm}(theta,varphi)=sqrt{2}(-1)^m,{rm Re}[Y_{lm}]$, with $0le m<l$.
Is that correct? If so, does that carry through to higher dimensions? If not, what should be done instead to specify such boundary conditions?
boundary-value-problem spherical-harmonics
asked Dec 10 '18 at 12:00
KaioKaio
112
$endgroup$
add a comment |
$begingroup$
Say I have a function $f(x,y,z)$ satisfying some Laplace-type equation on a domain. This domain is an hemisphere with boundary defined by $x=0$ in Cartesian coordinates. I impose either Dirichlet or Neumann boundary conditions on the plane $x=0$. Now I want to switch to spherical coordinates $(r,theta,varphi)$, where $0<theta<pi$ and normally $0levarphile2pi$.
First, am I allowed to defined the boundary $x=0$ by $varphi=0,pi$, so that $0levarphilepi$ in spherical coordinates?
Assuming so, next I want to expand my function $f(r,theta,varphi)$ on angular modes such that
begin{equation}
f(r,theta,varphi)=sum_{lm}u_{lm}(theta,varphi),f_{lm}(r),,
end{equation}
where $u_{lm}(theta,varphi)$ are related to the spherical harmonics. However, I still want to impose some boundary conditions at $varphi=0,pi$.
For Dirichlet boundary conditions at $varphi=0,pi$, then my guess would be to use only the "sine" part of the real spherical harmonics, that is $u_{lm}(theta,varphi)=sqrt{2}(-1)^m,{rm Im}[Y_{l|m|}]$, with $-lle m<0$, while for Neumann I would use the "cosine" part, i.e. $u_{lm}(theta,varphi)=sqrt{2}(-1)^m,{rm Re}[Y_{lm}]$, with $0le m<l$.
Is that correct? If so, does that carry through to higher dimensions? If not, what should be done instead to specify such boundary conditions?
boundary-value-problem spherical-harmonics
asked Dec 10 '18 at 12:00
KaioKaio
112
$endgroup$
add a comment |
$begingroup$
Say I have a function $f(x,y,z)$ satisfying some Laplace-type equation on a domain. This domain is an hemisphere with boundary defined by $x=0$ in Cartesian coordinates. I impose either Dirichlet or Neumann boundary conditions on the plane $x=0$. Now I want to switch to spherical coordinates $(r,theta,varphi)$, where $0<theta<pi$ and normally $0levarphile2pi$.
First, am I allowed to defined the boundary $x=0$ by $varphi=0,pi$, so that $0levarphilepi$ in spherical coordinates?
Assuming so, next I want to expand my function $f(r,theta,varphi)$ on angular modes such that
begin{equation}
f(r,theta,varphi)=sum_{lm}u_{lm}(theta,varphi),f_{lm}(r),,
end{equation}
where $u_{lm}(theta,varphi)$ are related to the spherical harmonics. However, I still want to impose some boundary conditions at $varphi=0,pi$.
For Dirichlet boundary conditions at $varphi=0,pi$, then my guess would be to use only the "sine" part of the real spherical harmonics, that is $u_{lm}(theta,varphi)=sqrt{2}(-1)^m,{rm Im}[Y_{l|m|}]$, with $-lle m<0$, while for Neumann I would use the "cosine" part, i.e. $u_{lm}(theta,varphi)=sqrt{2}(-1)^m,{rm Re}[Y_{lm}]$, with $0le m<l$.
Is that correct? If so, does that carry through to higher dimensions? If not, what should be done instead to specify such boundary conditions?
boundary-value-problem spherical-harmonics
asked Dec 10 '18 at 12:00
KaioKaio
112
$endgroup$
Say I have a function $f(x,y,z)$ satisfying some Laplace-type equation on a domain. This domain is an hemisphere with boundary defined by $x=0$ in Cartesian coordinates. I impose either Dirichlet or Neumann boundary conditions on the plane $x=0$. Now I want to switch to spherical coordinates $(r,theta,varphi)$, where $0<theta<pi$ and normally $0levarphile2pi$.
First, am I allowed to defined the boundary $x=0$ by $varphi=0,pi$, so that $0levarphilepi$ in spherical coordinates?
Assuming so, next I want to expand my function $f(r,theta,varphi)$ on angular modes such that
begin{equation}
f(r,theta,varphi)=sum_{lm}u_{lm}(theta,varphi),f_{lm}(r),,
end{equation}
where $u_{lm}(theta,varphi)$ are related to the spherical harmonics. However, I still want to impose some boundary conditions at $varphi=0,pi$.
For Dirichlet boundary conditions at $varphi=0,pi$, then my guess would be to use only the "sine" part of the real spherical harmonics, that is $u_{lm}(theta,varphi)=sqrt{2}(-1)^m,{rm Im}[Y_{l|m|}]$, with $-lle m<0$, while for Neumann I would use the "cosine" part, i.e. $u_{lm}(theta,varphi)=sqrt{2}(-1)^m,{rm Re}[Y_{lm}]$, with $0le m<l$.
Is that correct? If so, does that carry through to higher dimensions? If not, what should be done instead to specify such boundary conditions?
boundary-value-problem spherical-harmonics
boundary-value-problem spherical-harmonics
asked Dec 10 '18 at 12:00
KaioKaio
112
asked Dec 10 '18 at 12:00
KaioKaio
112
asked Dec 10 '18 at 12:00
KaioKaio
112
asked Dec 10 '18 at 12:00
KaioKaio
112
asked Dec 10 '18 at 12:00
KaioKaio
112
112
add a comment |
add a comment |
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