Consider the following wave equation: u t t − u x x = 0 , 0 &lt; t &lt; t x , k &gt; 1 u | t = 0 = ϕ 0 ( x ) , x ≥ 0 u t | t = 0 = ϕ 1 ( x ) , x ≥ 0 u | t = k x = ψ ( x ) In which ϕ 0 ( 0 ) = ψ ( 0 ).The problem is that on part of the sector where t &gt; x d'Alembert's formula isn't applicable. It seems we will have to do some sort of extension or reflection, but how to start?

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Consider the following wave equation:                              u                      t            t                          −                  u                      x            x                                              =        0        ,                0        &amp;lt;        t        &amp;lt;        t        x        ,        k        &amp;gt;        1                            u                              |                                t            =            0                                              =                  ϕ          0                (        x        )        ,                x        ≥        0                                      u          t                                      |                                t            =            0                                              =                  ϕ          1                (        x        )        ,                x        ≥        0                            u                              |                                t            =            k            x                                              =        ψ        (        x        )            In which       ϕ    0    (  0  )  =  ψ  (  0  ).The problem is that on part of the sector where   t  &amp;gt;  x d&#039;Alembert&#039;s formula isn&#039;t applicable. It seems we will have to do some sort of extension or reflection, but how to start?

Eliezer Olson · Accepted Answer

Given a point   (  x  ,  t  ) in the region   x  &amp;lt;  t  &amp;lt;  k    x, construct the parallelogram formed by the characteristic curves   x  ±  t  =  1 with vertices vertices in   (  x  ,  t  ), the line   t  =  k    x and the line   t  =  0 (the fourth vertex will fall in the region   0  &amp;lt;  x  &amp;lt;  t). You know the value of   u at the last three of the vertices, and can find   u  (  x  ,  t  ) using the fact that the sums of the values of   u at opposing vertices are equal (this is a property of solutions of the wave equation.)

Consider the following wave equation: <mtable columnalign="right left right left right left righ

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