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The benefits of scaling
- It is difficult to figure out all the physical parameters of a case
- And it is not necessary because of a powerful: scaling
Introduce new x , t , and u without dimension:
\bar x = \frac{x}{L},\quad \bar t = \frac{c}{L}t,\quad
\bar u = \frac{u}{a}
Insert this in the PDE (with f=0 ) and dropping bars
u_{tt} = u_{xx}
Initial condition: set a=1 , L=1 , and
x_0\in [0,1] in (20).
In the code: set a=c=L=1, x0=0.8, and there is no need to calculate with
wavelengths and frequencies to estimate c !
Just one challenge: determine the period of the waves and an
appropriate end time (see the text for details).
