- #1

Palindrom

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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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