I'm considering the wave equation $u_{tt}=u_{xx}$ defined on $x\in [0,1], \,\, t\geq 0$ with initial profile

$$ u(x,0)= I\left(\frac{1}{4}\leq x \leq \frac{3}{4}\right),$$

where $I$ is the indicator function.

Initial condition: $u_t(x,0)=0$

Boundary condition: $u(0,t)=u(1,t)=0$

My problem is with the initial condition. My intuition is to say if the time derivative is 0 initially, then the equation won't be time dependent ever. Is this correct?



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