Mixing
2nd order differential equation $ E = E_o e^-α' z e^j(Ét - kz)$
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Show that $ E = E_o e^-α' z e^j(Ét - kz)$ is a possible solution to:
$fracd^2 Edz^2 - E_o E_r μ_o fracd^2 Edt^2 = μ_o à frac∂E∂t $
...
$ E = E_o e^jÉt - z(jk + α')$
$fracd Edz = -(jk + α') E_o e^jÉt - z(jk + α')$
$fracd^2Edz^2 = (jk + α')^2 E_o e^jÉt - z(jk + α') = (α'^2 + j2α'k - k^2) E_o e^jÉt - z(jk + α')$
$frac∂E∂t = jÉ E_o e^jÉt - z(jk + α') $
How does jw replace the $(α'^2 + j2α'k - k^2)$ term?
Here's screenshot.
calculus partial-derivative
edited Jul 23 at 22:29
Taroccoesbrocco
3,48941431
asked Jul 15 at 0:11
Jack
311614
add a comment |Â
up vote
0
down vote
favorite
Show that $ E = E_o e^-α' z e^j(Ét - kz)$ is a possible solution to:
$fracd^2 Edz^2 - E_o E_r μ_o fracd^2 Edt^2 = μ_o à frac∂E∂t $
...
$ E = E_o e^jÉt - z(jk + α')$
$fracd Edz = -(jk + α') E_o e^jÉt - z(jk + α')$
$fracd^2Edz^2 = (jk + α')^2 E_o e^jÉt - z(jk + α') = (α'^2 + j2α'k - k^2) E_o e^jÉt - z(jk + α')$
$frac∂E∂t = jÉ E_o e^jÉt - z(jk + α') $
How does jw replace the $(α'^2 + j2α'k - k^2)$ term?
Here's screenshot.
calculus partial-derivative
edited Jul 23 at 22:29
Taroccoesbrocco
3,48941431
asked Jul 15 at 0:11
Jack
311614
FYI, you can typeset greek letters in mathjax/TeX by typing a backslash before the name of the letter (for example, surrounding alpha in dollar signs yields $alpha$, beta yields $beta$; capitalizing the first letter gives capital greek letters like $Pi$).
– Crosby
Jul 15 at 0:26
1
Are you supposed to find a value of $alpha'$ for which the given solution holds?
– AlexanderJ93
Jul 15 at 3:39
@AlexanderJ93 I'm not sure, given the solution and how steps go, how does the math check out at the asked part?
– Jack
Jul 17 at 23:49
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Show that $ E = E_o e^-α' z e^j(Ét - kz)$ is a possible solution to:
$fracd^2 Edz^2 - E_o E_r μ_o fracd^2 Edt^2 = μ_o à frac∂E∂t $
...
$ E = E_o e^jÉt - z(jk + α')$
$fracd Edz = -(jk + α') E_o e^jÉt - z(jk + α')$
$fracd^2Edz^2 = (jk + α')^2 E_o e^jÉt - z(jk + α') = (α'^2 + j2α'k - k^2) E_o e^jÉt - z(jk + α')$
$frac∂E∂t = jÉ E_o e^jÉt - z(jk + α') $
How does jw replace the $(α'^2 + j2α'k - k^2)$ term?
Here's screenshot.
calculus partial-derivative
edited Jul 23 at 22:29
Taroccoesbrocco
3,48941431
asked Jul 15 at 0:11
Jack
311614
Show that $ E = E_o e^-α' z e^j(Ét - kz)$ is a possible solution to:
$fracd^2 Edz^2 - E_o E_r μ_o fracd^2 Edt^2 = μ_o à frac∂E∂t $
...
$ E = E_o e^jÉt - z(jk + α')$
$fracd Edz = -(jk + α') E_o e^jÉt - z(jk + α')$
$fracd^2Edz^2 = (jk + α')^2 E_o e^jÉt - z(jk + α') = (α'^2 + j2α'k - k^2) E_o e^jÉt - z(jk + α')$
$frac∂E∂t = jÉ E_o e^jÉt - z(jk + α') $
How does jw replace the $(α'^2 + j2α'k - k^2)$ term?
Here's screenshot.
calculus partial-derivative
edited Jul 23 at 22:29
Taroccoesbrocco
3,48941431
asked Jul 15 at 0:11
Jack
311614
edited Jul 23 at 22:29
Taroccoesbrocco
3,48941431
edited Jul 23 at 22:29
Taroccoesbrocco
3,48941431
edited Jul 23 at 22:29
Taroccoesbrocco
3,48941431
3,48941431
asked Jul 15 at 0:11
Jack
311614
asked Jul 15 at 0:11
Jack
311614
asked Jul 15 at 0:11
Jack
311614
311614
FYI, you can typeset greek letters in mathjax/TeX by typing a backslash before the name of the letter (for example, surrounding alpha in dollar signs yields $alpha$, beta yields $beta$; capitalizing the first letter gives capital greek letters like $Pi$).
– Crosby
Jul 15 at 0:26
1
Are you supposed to find a value of $alpha'$ for which the given solution holds?
– AlexanderJ93
Jul 15 at 3:39
@AlexanderJ93 I'm not sure, given the solution and how steps go, how does the math check out at the asked part?
– Jack
Jul 17 at 23:49
add a comment |Â
FYI, you can typeset greek letters in mathjax/TeX by typing a backslash before the name of the letter (for example, surrounding alpha in dollar signs yields $alpha$, beta yields $beta$; capitalizing the first letter gives capital greek letters like $Pi$).
– Crosby
Jul 15 at 0:26
1
Are you supposed to find a value of $alpha'$ for which the given solution holds?
– AlexanderJ93
Jul 15 at 3:39
@AlexanderJ93 I'm not sure, given the solution and how steps go, how does the math check out at the asked part?
– Jack
Jul 17 at 23:49
FYI, you can typeset greek letters in mathjax/TeX by typing a backslash before the name of the letter (for example, surrounding alpha in dollar signs yields $alpha$, beta yields $beta$; capitalizing the first letter gives capital greek letters like $Pi$).
– Crosby
Jul 15 at 0:26
FYI, you can typeset greek letters in mathjax/TeX by typing a backslash before the name of the letter (for example, surrounding alpha in dollar signs yields $alpha$, beta yields $beta$; capitalizing the first letter gives capital greek letters like $Pi$).
– Crosby
Jul 15 at 0:26
1
1
Are you supposed to find a value of $alpha'$ for which the given solution holds?
– AlexanderJ93
Jul 15 at 3:39
Are you supposed to find a value of $alpha'$ for which the given solution holds?
– AlexanderJ93
Jul 15 at 3:39
@AlexanderJ93 I'm not sure, given the solution and how steps go, how does the math check out at the asked part?
– Jack
Jul 17 at 23:49
@AlexanderJ93 I'm not sure, given the solution and how steps go, how does the math check out at the asked part?
– Jack
Jul 17 at 23:49
add a comment |Â
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FYI, you can typeset greek letters in mathjax/TeX by typing a backslash before the name of the letter (for example, surrounding alpha in dollar signs yields $alpha$, beta yields $beta$; capitalizing the first letter gives capital greek letters like $Pi$).
– Crosby
Jul 15 at 0:26
1
Are you supposed to find a value of $alpha'$ for which the given solution holds?
– AlexanderJ93
Jul 15 at 3:39
@AlexanderJ93 I'm not sure, given the solution and how steps go, how does the math check out at the asked part?
– Jack
Jul 17 at 23:49