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List of solutions to the wave equation in three dimensions
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Wikipedia describes plane waves and $e^i(omega tpm kr)$ as well as a general integral formula as solutions to the wave equation. As far as I understand, all solutions can be constructed from these by finite and infinite sums. But: are there other "simple" closed form functions known that solve the wave equation?
wave-equation
asked Aug 1 at 20:03
Harald
1968
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Wikipedia describes plane waves and $e^i(omega tpm kr)$ as well as a general integral formula as solutions to the wave equation. As far as I understand, all solutions can be constructed from these by finite and infinite sums. But: are there other "simple" closed form functions known that solve the wave equation?
wave-equation
asked Aug 1 at 20:03
Harald
1968
1
I am not sure whether you consider the functions $sin(omega tpm kr)$ and $cos(omega tpm kr)$ as simple enough or not. But, yes indeed these two solutions can be constructed out of the exponential you gave there. But I do not know if this what you are looking for.
– mrtaurho
Aug 1 at 20:08
Well, surely simple enough and could serve as number two and three of the list, just after $e^i(omega tpm kr)$ despite them being contained in it already. What else goes on the list?
– Harald
Aug 4 at 8:50
I guess one of the main properties of a general solution is the fact that you can construct each single particular solution out of it. Therefore the given solution, $e^i(omega tpm rt)$, is the key to derive every single other solution like for example the trigonometric functions. Therefore the exponential is the whole list of solutions you are asking for I would say.
– mrtaurho
Aug 4 at 21:13
Surely all (most) functions on the list would also be constructible from the exponential. So maybe a similar question would be, if there are "simple" closed form functions that are the sum of a few (or a series of) $e^i(omega tpm rt)$.
– Harald
Aug 5 at 5:03
I have to ask for a detail: are we talking about the general solution in terms of plane waves or in terms of the exponential. Since Wikipedia claims your given exponential, $e^i(omega tpm kr)$, as a part of monochromatic spherical waves. Thus the general solution would be $c_1e^i(omega tpm kr)+c_2e^-i(omega tpm kr)$ from which on I could construct a way more solutions than out of the single exponetial. Furthermore the construction of the sine and cosine functions only works out with the two exponentials.
– mrtaurho
13 hours ago
add a comment |Â
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up vote
0
down vote
favorite
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1
Wikipedia describes plane waves and $e^i(omega tpm kr)$ as well as a general integral formula as solutions to the wave equation. As far as I understand, all solutions can be constructed from these by finite and infinite sums. But: are there other "simple" closed form functions known that solve the wave equation?
wave-equation
asked Aug 1 at 20:03
Harald
1968
Wikipedia describes plane waves and $e^i(omega tpm kr)$ as well as a general integral formula as solutions to the wave equation. As far as I understand, all solutions can be constructed from these by finite and infinite sums. But: are there other "simple" closed form functions known that solve the wave equation?
wave-equation
asked Aug 1 at 20:03
Harald
1968
asked Aug 1 at 20:03
Harald
1968
asked Aug 1 at 20:03
Harald
1968
asked Aug 1 at 20:03
Harald
1968
1968
1
I am not sure whether you consider the functions $sin(omega tpm kr)$ and $cos(omega tpm kr)$ as simple enough or not. But, yes indeed these two solutions can be constructed out of the exponential you gave there. But I do not know if this what you are looking for.
– mrtaurho
Aug 1 at 20:08
Well, surely simple enough and could serve as number two and three of the list, just after $e^i(omega tpm kr)$ despite them being contained in it already. What else goes on the list?
– Harald
Aug 4 at 8:50
I guess one of the main properties of a general solution is the fact that you can construct each single particular solution out of it. Therefore the given solution, $e^i(omega tpm rt)$, is the key to derive every single other solution like for example the trigonometric functions. Therefore the exponential is the whole list of solutions you are asking for I would say.
– mrtaurho
Aug 4 at 21:13
Surely all (most) functions on the list would also be constructible from the exponential. So maybe a similar question would be, if there are "simple" closed form functions that are the sum of a few (or a series of) $e^i(omega tpm rt)$.
– Harald
Aug 5 at 5:03
I have to ask for a detail: are we talking about the general solution in terms of plane waves or in terms of the exponential. Since Wikipedia claims your given exponential, $e^i(omega tpm kr)$, as a part of monochromatic spherical waves. Thus the general solution would be $c_1e^i(omega tpm kr)+c_2e^-i(omega tpm kr)$ from which on I could construct a way more solutions than out of the single exponetial. Furthermore the construction of the sine and cosine functions only works out with the two exponentials.
– mrtaurho
13 hours ago
add a comment |Â
1
I am not sure whether you consider the functions $sin(omega tpm kr)$ and $cos(omega tpm kr)$ as simple enough or not. But, yes indeed these two solutions can be constructed out of the exponential you gave there. But I do not know if this what you are looking for.
– mrtaurho
Aug 1 at 20:08
Well, surely simple enough and could serve as number two and three of the list, just after $e^i(omega tpm kr)$ despite them being contained in it already. What else goes on the list?
– Harald
Aug 4 at 8:50
I guess one of the main properties of a general solution is the fact that you can construct each single particular solution out of it. Therefore the given solution, $e^i(omega tpm rt)$, is the key to derive every single other solution like for example the trigonometric functions. Therefore the exponential is the whole list of solutions you are asking for I would say.
– mrtaurho
Aug 4 at 21:13
Surely all (most) functions on the list would also be constructible from the exponential. So maybe a similar question would be, if there are "simple" closed form functions that are the sum of a few (or a series of) $e^i(omega tpm rt)$.
– Harald
Aug 5 at 5:03
I have to ask for a detail: are we talking about the general solution in terms of plane waves or in terms of the exponential. Since Wikipedia claims your given exponential, $e^i(omega tpm kr)$, as a part of monochromatic spherical waves. Thus the general solution would be $c_1e^i(omega tpm kr)+c_2e^-i(omega tpm kr)$ from which on I could construct a way more solutions than out of the single exponetial. Furthermore the construction of the sine and cosine functions only works out with the two exponentials.
– mrtaurho
13 hours ago
1
1
I am not sure whether you consider the functions $sin(omega tpm kr)$ and $cos(omega tpm kr)$ as simple enough or not. But, yes indeed these two solutions can be constructed out of the exponential you gave there. But I do not know if this what you are looking for.
– mrtaurho
Aug 1 at 20:08
I am not sure whether you consider the functions $sin(omega tpm kr)$ and $cos(omega tpm kr)$ as simple enough or not. But, yes indeed these two solutions can be constructed out of the exponential you gave there. But I do not know if this what you are looking for.
– mrtaurho
Aug 1 at 20:08
Well, surely simple enough and could serve as number two and three of the list, just after $e^i(omega tpm kr)$ despite them being contained in it already. What else goes on the list?
– Harald
Aug 4 at 8:50
Well, surely simple enough and could serve as number two and three of the list, just after $e^i(omega tpm kr)$ despite them being contained in it already. What else goes on the list?
– Harald
Aug 4 at 8:50
I guess one of the main properties of a general solution is the fact that you can construct each single particular solution out of it. Therefore the given solution, $e^i(omega tpm rt)$, is the key to derive every single other solution like for example the trigonometric functions. Therefore the exponential is the whole list of solutions you are asking for I would say.
– mrtaurho
Aug 4 at 21:13
I guess one of the main properties of a general solution is the fact that you can construct each single particular solution out of it. Therefore the given solution, $e^i(omega tpm rt)$, is the key to derive every single other solution like for example the trigonometric functions. Therefore the exponential is the whole list of solutions you are asking for I would say.
– mrtaurho
Aug 4 at 21:13
Surely all (most) functions on the list would also be constructible from the exponential. So maybe a similar question would be, if there are "simple" closed form functions that are the sum of a few (or a series of) $e^i(omega tpm rt)$.
– Harald
Aug 5 at 5:03
Surely all (most) functions on the list would also be constructible from the exponential. So maybe a similar question would be, if there are "simple" closed form functions that are the sum of a few (or a series of) $e^i(omega tpm rt)$.
– Harald
Aug 5 at 5:03
I have to ask for a detail: are we talking about the general solution in terms of plane waves or in terms of the exponential. Since Wikipedia claims your given exponential, $e^i(omega tpm kr)$, as a part of monochromatic spherical waves. Thus the general solution would be $c_1e^i(omega tpm kr)+c_2e^-i(omega tpm kr)$ from which on I could construct a way more solutions than out of the single exponetial. Furthermore the construction of the sine and cosine functions only works out with the two exponentials.
– mrtaurho
13 hours ago
I have to ask for a detail: are we talking about the general solution in terms of plane waves or in terms of the exponential. Since Wikipedia claims your given exponential, $e^i(omega tpm kr)$, as a part of monochromatic spherical waves. Thus the general solution would be $c_1e^i(omega tpm kr)+c_2e^-i(omega tpm kr)$ from which on I could construct a way more solutions than out of the single exponetial. Furthermore the construction of the sine and cosine functions only works out with the two exponentials.
– mrtaurho
13 hours ago
add a comment |Â
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1
I am not sure whether you consider the functions $sin(omega tpm kr)$ and $cos(omega tpm kr)$ as simple enough or not. But, yes indeed these two solutions can be constructed out of the exponential you gave there. But I do not know if this what you are looking for.
– mrtaurho
Aug 1 at 20:08
Well, surely simple enough and could serve as number two and three of the list, just after $e^i(omega tpm kr)$ despite them being contained in it already. What else goes on the list?
– Harald
Aug 4 at 8:50
I guess one of the main properties of a general solution is the fact that you can construct each single particular solution out of it. Therefore the given solution, $e^i(omega tpm rt)$, is the key to derive every single other solution like for example the trigonometric functions. Therefore the exponential is the whole list of solutions you are asking for I would say.
– mrtaurho
Aug 4 at 21:13
Surely all (most) functions on the list would also be constructible from the exponential. So maybe a similar question would be, if there are "simple" closed form functions that are the sum of a few (or a series of) $e^i(omega tpm rt)$.
– Harald
Aug 5 at 5:03
I have to ask for a detail: are we talking about the general solution in terms of plane waves or in terms of the exponential. Since Wikipedia claims your given exponential, $e^i(omega tpm kr)$, as a part of monochromatic spherical waves. Thus the general solution would be $c_1e^i(omega tpm kr)+c_2e^-i(omega tpm kr)$ from which on I could construct a way more solutions than out of the single exponetial. Furthermore the construction of the sine and cosine functions only works out with the two exponentials.
– mrtaurho
13 hours ago