Semi-Infinite Wave, D'Alembert Solution

The 1-D wave equation

for infinite boundaries has the D'Alembert solution through a coordinate transformation

where

. The initial conditions are

and

. For the semi-infinite domain,

would be undefined for

. Therefore a boundary condition at

is used to extend to negative values

. Therefore a boundary condition at

is used to extend to negative values. Then

. Also if

this boundary condition results in the modification for

. This applet allows you to input a function for

and

. Note use x for t in this term. You can also modify the wave speed,

and then advance the time,

The boundary condition

can also be accommodated with an extension for

to negative

values. This applet shows the result for

What was the extension on

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