Mixing
Question of two functions which are continuous at a certain point
$begingroup$
The electric field $$E_x$$ and its time derivative are continuous at time $t = 0$. Show that $$k = k^,$$
if for $t < 0$
$$E_x = E_0 cos(kz-wt)$$
and if for $t > 0$
$$E_x = E_1 cos(k^,z-wt) + E_2 cos(k^,z-wt).$$
I tried doing this by setting the electric field at $t < 0$ equal to that at $t > 0$ when $t$ is equal to zero and doing the same for the time derivative but I get a whole number of trigonometric functions that will not disappear. Any advice would be appreciated.
derivatives trigonometry continuity vector-fields
edited Feb 1 at 16:06
jvdhooft
5,65961641
asked Feb 1 at 15:51
David AbrahamDavid Abraham
1149
$endgroup$
add a comment |
$begingroup$
The electric field $$E_x$$ and its time derivative are continuous at time $t = 0$. Show that $$k = k^,$$
if for $t < 0$
$$E_x = E_0 cos(kz-wt)$$
and if for $t > 0$
$$E_x = E_1 cos(k^,z-wt) + E_2 cos(k^,z-wt).$$
I tried doing this by setting the electric field at $t < 0$ equal to that at $t > 0$ when $t$ is equal to zero and doing the same for the time derivative but I get a whole number of trigonometric functions that will not disappear. Any advice would be appreciated.
derivatives trigonometry continuity vector-fields
edited Feb 1 at 16:06
jvdhooft
5,65961641
asked Feb 1 at 15:51
David AbrahamDavid Abraham
1149
$endgroup$
$begingroup$
What did you try specifically?
$endgroup$
– Alex
Feb 1 at 15:52
$begingroup$
Although I can't say for sure from the information you've given, it is almost certain that $z$ is a variable and the two relations then have to hold for all $z$. Ergo, you can differentiate the two equations with respect to $z$ (repeatedly) to obtain additional relations.
$endgroup$
– Paul Sinclair
Feb 1 at 23:34
add a comment |
$begingroup$
The electric field $$E_x$$ and its time derivative are continuous at time $t = 0$. Show that $$k = k^,$$
if for $t < 0$
$$E_x = E_0 cos(kz-wt)$$
and if for $t > 0$
$$E_x = E_1 cos(k^,z-wt) + E_2 cos(k^,z-wt).$$
I tried doing this by setting the electric field at $t < 0$ equal to that at $t > 0$ when $t$ is equal to zero and doing the same for the time derivative but I get a whole number of trigonometric functions that will not disappear. Any advice would be appreciated.
derivatives trigonometry continuity vector-fields
edited Feb 1 at 16:06
jvdhooft
5,65961641
asked Feb 1 at 15:51
David AbrahamDavid Abraham
1149
$endgroup$
The electric field $$E_x$$ and its time derivative are continuous at time $t = 0$. Show that $$k = k^,$$
if for $t < 0$
$$E_x = E_0 cos(kz-wt)$$
and if for $t > 0$
$$E_x = E_1 cos(k^,z-wt) + E_2 cos(k^,z-wt).$$
I tried doing this by setting the electric field at $t < 0$ equal to that at $t > 0$ when $t$ is equal to zero and doing the same for the time derivative but I get a whole number of trigonometric functions that will not disappear. Any advice would be appreciated.
derivatives trigonometry continuity vector-fields
derivatives trigonometry continuity vector-fields
edited Feb 1 at 16:06
jvdhooft
5,65961641
asked Feb 1 at 15:51
David AbrahamDavid Abraham
1149
edited Feb 1 at 16:06
jvdhooft
5,65961641
asked Feb 1 at 15:51
David AbrahamDavid Abraham
1149
edited Feb 1 at 16:06
jvdhooft
5,65961641
edited Feb 1 at 16:06
jvdhooft
5,65961641
edited Feb 1 at 16:06
jvdhooft
5,65961641
5,65961641
asked Feb 1 at 15:51
David AbrahamDavid Abraham
1149
asked Feb 1 at 15:51
David AbrahamDavid Abraham
1149
asked Feb 1 at 15:51
David AbrahamDavid Abraham
1149
1149
$begingroup$
What did you try specifically?
$endgroup$
– Alex
Feb 1 at 15:52
$begingroup$
Although I can't say for sure from the information you've given, it is almost certain that $z$ is a variable and the two relations then have to hold for all $z$. Ergo, you can differentiate the two equations with respect to $z$ (repeatedly) to obtain additional relations.
$endgroup$
– Paul Sinclair
Feb 1 at 23:34
add a comment |
$begingroup$
What did you try specifically?
$endgroup$
– Alex
Feb 1 at 15:52
$begingroup$
Although I can't say for sure from the information you've given, it is almost certain that $z$ is a variable and the two relations then have to hold for all $z$. Ergo, you can differentiate the two equations with respect to $z$ (repeatedly) to obtain additional relations.
$endgroup$
– Paul Sinclair
Feb 1 at 23:34
$begingroup$
What did you try specifically?
$endgroup$
– Alex
Feb 1 at 15:52
$begingroup$
What did you try specifically?
$endgroup$
– Alex
Feb 1 at 15:52
$begingroup$
Although I can't say for sure from the information you've given, it is almost certain that $z$ is a variable and the two relations then have to hold for all $z$. Ergo, you can differentiate the two equations with respect to $z$ (repeatedly) to obtain additional relations.
$endgroup$
– Paul Sinclair
Feb 1 at 23:34
$begingroup$
Although I can't say for sure from the information you've given, it is almost certain that $z$ is a variable and the two relations then have to hold for all $z$. Ergo, you can differentiate the two equations with respect to $z$ (repeatedly) to obtain additional relations.
$endgroup$
– Paul Sinclair
Feb 1 at 23:34
add a comment |
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$begingroup$
What did you try specifically?
$endgroup$
– Alex
Feb 1 at 15:52
$begingroup$
Although I can't say for sure from the information you've given, it is almost certain that $z$ is a variable and the two relations then have to hold for all $z$. Ergo, you can differentiate the two equations with respect to $z$ (repeatedly) to obtain additional relations.
$endgroup$
– Paul Sinclair
Feb 1 at 23:34