Question : solve the wave equation utt = c2 uxx for −∞ < x < +∞ with the initial conditions u(x, 0) = φ(x)= sin x , ut(x, 0) = 0
Solution :
To solve wave equation \[u_{tt}\ =\ c^2\ u_{xx} for -\infty\ <\ x\ <\ +\infty \] with the initial conditions \[u\left(x,\ 0\right)=\sin{x=\phi\left(x\right)},\ u_t\left(x,\ 0\right)=\ 0=\psi(x)\ \] The general solution of this problem is \[u\left(x,\ t\right)=\frac{1}{2}\ \left[\varphi\left(x\ +\ ct\right)+\ \varphi\left(x\ -\ ct\right)\right]\ +\frac{1}{2c}\ \int_{x-ct}^{x+ct}{\psi(s)\ }\ ds\] Substituting \[\varphi\left(x\ +\ ct\right)=\sin{\left(x\ +\ ct\right)},\ \ \varphi\left(x\ -\ ct\right)=\sin{\left(x-\ ct\right)},\ \psi\left(s\right)=0\] \[u\left(x,\ t\right)=\frac{1}{2}\ \left[\sin{\left(x\ +\ ct\right)}+\sin{\left(x-\ ct\right)}\right]\] Using formula sin(A+B)=sin(A)cos(B)+cos(A)sin(B) \[u\left(x,\ t\right)=\frac{1}{2}\left[2sin\ \left(x\right)\ cos\ \left(ct\right).\right]\ \] We obtained solution of the given wave equation \[u(x,\ t)\ =sin\ \left(x\right)\ cos\ \left(ct\right)\]