- #1
solanojedi
- 34
- 0
Hi everyone,
I'm reading about the solution of the wave equation in free space on Stratton - Electromagnetic Theory and Snider - PDE and I got a little confused. The wave equation in 3D (plus time) is the following $$\frac{\partial^{2} \Psi (x,y,z,t)} {\partial t^{2}}=\nabla ^{2}\Psi (x,y,z,t)$$ but the doubt can be carried out also from the "1D" (plus time) wave equation $$\frac{\partial^{2} \Psi (x,t)} {\partial t^{2}}=\nabla ^{2}\Psi (x,t).$$ My intent is to solve the equation in free space, hence ##-\infty<x<+\infty##, but also on positive and negative time. If we apply the separation of variables to the last equation, from ##\Psi(x,t)=X(x)T(t)## we get two separated equation in space and time, i.e. $$\frac{X''}{X}=\lambda,~~~ -\infty<x<+\infty \\ \frac{T''}{T}=\lambda,~~~ -\infty<t<+\infty \\$$ The two equations are identical and are two singular Sturm-Liouville problem, so, as I can see on Snider pagg.474 for the same type of equations, the solution should be the infinite superposition of complex exponentials ##\int_{-\infty}^{+\infty} A(\omega) e^{i \omega x} e^{i \omega t} d\omega = \int_{-\infty}^{+\infty} A(\omega) e^{i\omega(x+t)} d\omega ##, if ##\omega=\sqrt{\lambda}##. However, with this approach, I don't find the two traveling packet waves. Instead, the correct solution is the usual two-traveling-wave equation $$\Psi (x,t)=\int_{-\infty}^{+\infty} [A(\omega) e^{i \omega x}+B(\omega) e^{-i \omega x}] e^{-i\omega t} d\omega.$$ Also Snider's book, when discussing the wave equation in 3D + time, for the spatial variables keeps the singular Sturm Liouvilee approach using complex exponentials, while for the time equation, that has the same form, insert sa ##\sin(\omega t)## as a solution, obtaining $$\Psi (x,y,z,t)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} A(\omega_1, \omega_1, \omega_3) e^{i\omega_1 x}e^{i\omega_2 x}e^{i\omega_3 x} \sin(t \sqrt{\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3}^{2}}) d\omega_{1} d\omega_{2} d\omega_{3}$$
Of course I get that this is the correct solution, but what is wrong with my approach? Thank you in advance.
I'm reading about the solution of the wave equation in free space on Stratton - Electromagnetic Theory and Snider - PDE and I got a little confused. The wave equation in 3D (plus time) is the following $$\frac{\partial^{2} \Psi (x,y,z,t)} {\partial t^{2}}=\nabla ^{2}\Psi (x,y,z,t)$$ but the doubt can be carried out also from the "1D" (plus time) wave equation $$\frac{\partial^{2} \Psi (x,t)} {\partial t^{2}}=\nabla ^{2}\Psi (x,t).$$ My intent is to solve the equation in free space, hence ##-\infty<x<+\infty##, but also on positive and negative time. If we apply the separation of variables to the last equation, from ##\Psi(x,t)=X(x)T(t)## we get two separated equation in space and time, i.e. $$\frac{X''}{X}=\lambda,~~~ -\infty<x<+\infty \\ \frac{T''}{T}=\lambda,~~~ -\infty<t<+\infty \\$$ The two equations are identical and are two singular Sturm-Liouville problem, so, as I can see on Snider pagg.474 for the same type of equations, the solution should be the infinite superposition of complex exponentials ##\int_{-\infty}^{+\infty} A(\omega) e^{i \omega x} e^{i \omega t} d\omega = \int_{-\infty}^{+\infty} A(\omega) e^{i\omega(x+t)} d\omega ##, if ##\omega=\sqrt{\lambda}##. However, with this approach, I don't find the two traveling packet waves. Instead, the correct solution is the usual two-traveling-wave equation $$\Psi (x,t)=\int_{-\infty}^{+\infty} [A(\omega) e^{i \omega x}+B(\omega) e^{-i \omega x}] e^{-i\omega t} d\omega.$$ Also Snider's book, when discussing the wave equation in 3D + time, for the spatial variables keeps the singular Sturm Liouvilee approach using complex exponentials, while for the time equation, that has the same form, insert sa ##\sin(\omega t)## as a solution, obtaining $$\Psi (x,y,z,t)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} A(\omega_1, \omega_1, \omega_3) e^{i\omega_1 x}e^{i\omega_2 x}e^{i\omega_3 x} \sin(t \sqrt{\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3}^{2}}) d\omega_{1} d\omega_{2} d\omega_{3}$$
Of course I get that this is the correct solution, but what is wrong with my approach? Thank you in advance.