Wikipedia describes plane waves and $e^{i(\omega t\pm kr)}$ 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?
Asked
Active
Viewed 508 times
0
-
1I am not sure whether you consider the functions $\sin(\omega t\pm kr)$ and $\cos(\omega t\pm kr)$ 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 01 '18 at 20:08
-
Well, surely simple enough and could serve as number two and three of the list, just after $e^{i(\omega t\pm kr)}$ despite them being contained in it already. What else goes on the list? – Harald Aug 04 '18 at 08:50
-
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, $e^{i(\omega t\pm rt)}$, 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 04 '18 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) $e^{i(\omega t\pm rt)}$. – Harald Aug 05 '18 at 05:03
-
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, $e^{i(\omega t\pm kr)}$, as a part of monochromatic spherical waves. Thus the general solution would be $c_1e^{i(\omega t\pm kr)}+c_2e^{-i(\omega t\pm kr)}$ 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 Aug 07 '18 at 20:15
-
Whatever, any solution. And I don't csre how they are constructed, in fact I would rather appreciate if a solution has a nice closed form but is an infinite seties of exponentials. I am not interestet in the general solution as series of exponentials, but in particular, remarkable solutions, if any. – Harald Aug 08 '18 at 19:49
1 Answers
0
Suppose the general solution for the wave equation in three dimensions as the following
$$y(x)=c_1e^{i(\omega t\pm kr)}+c_2e^{-i(\omega t\pm kr)}$$
with $c_1,c_2\in\mathbb{C}$. From hereon we can construct some other functions, which are closely related to the exponential
$$\begin{align} &y_{1,2}(x)~=~ce^{\pm i(\omega t\pm kr)}\tag{I}\\ &y_{3,4}(x)~=~c\lambda^{\pm i(\omega t\pm kr)}\tag{II}\\ &y_{5,6}(x)~=~c_1\sin(\omega t\pm kr)+c_2\cos(\omega t\pm kr)\tag{III}\\ &y_{5,6}(x)~=~c_1\sin(\lambda_1(\omega t\pm kr))+c_2\cos(\lambda_2(\omega t\pm kr)\tag{IV}\\ &y_{7,8}(x)~=~c_1\sinh(i(\omega t\pm kr))+c_2\cosh(i(\omega t\pm kr))\tag{V}\\ &y_{9,10}(x)~=~c_1\sinh(\lambda_1i(\omega t\pm kr))+c_2\cosh(\lambda_2i(\omega t\pm kr)\tag{VI}\\ \end{align}$$
We can go even further and say the Generalized Hypergeometric Function $_0F_0(;;i(\omega t\pm kr))$ is a solution of the equation since it is just a more general way to write down the exponential.
The main problem with your question is that you can in fact construct many other closed functions out of the exponential. But hence they are just a construction out of the general solution mathematician agreed on just considering the general solution - or in this case the fundamental set of solutions - as the one solution to the differential equation so that they have not to write down a list with about $10$ entries every single time they solve an equation.
Howsoever I hope that you are satisfied with my collocation of possible functions which can be constructed.
mrtaurho
- 16,103