- #1
Palindrom
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- 0
Hi everyone.
I tried a bit, but got stuck.
Let [tex]\[u\left( {x,t} \right)\][/tex] be a solution of [tex]\[u_{tt} - c^2 u_{xx} = 0
\][/tex], and suppose [tex]\[u\left( {x,t} \right)\][/tex] is constant along the line [tex]\[
x = 2 + ct
\]
[/tex]. Then [tex]\[u\left( {x,t} \right)\][/tex] must keep:[tex]\[
u_t + cu_x = 0
\]
[/tex]
I can prove it for any point [tex]\[
\left( {x,t} \right)
\]
[/tex] which is right to the line [tex]\[
x = 2 - ct
\]
[/tex]. I don't see any way to prove it for the points left to that line.
Is there a simpler way, or more general one that doesn't make that last line special?
Thanks in advance.
I tried a bit, but got stuck.
Let [tex]\[u\left( {x,t} \right)\][/tex] be a solution of [tex]\[u_{tt} - c^2 u_{xx} = 0
\][/tex], and suppose [tex]\[u\left( {x,t} \right)\][/tex] is constant along the line [tex]\[
x = 2 + ct
\]
[/tex]. Then [tex]\[u\left( {x,t} \right)\][/tex] must keep:[tex]\[
u_t + cu_x = 0
\]
[/tex]
I can prove it for any point [tex]\[
\left( {x,t} \right)
\]
[/tex] which is right to the line [tex]\[
x = 2 - ct
\]
[/tex]. I don't see any way to prove it for the points left to that line.
Is there a simpler way, or more general one that doesn't make that last line special?
Thanks in advance.
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