Changing variable wave equation

Question

We consider the wave equation $u_{tt} - u_{xx} = 0$ in the domain $0 \leq x \leq s(t)$ where $s(t)$ be a positive function depending on $t$. We change the variable $y = \frac{x}{s(t)}$ and consider the function $w(y,t) = u(y s(t),t) = u(x,t)$. What is the equation for $w$ respected to $y$ and $t$ ? Can anyone help me to calculate?

Answer 1

Note $x=ys(t)$ and hence by the Chain Rule \begin{eqnarray} w_y&=&u_x\frac{\partial x}{\partial y}=u_xs(t),\\ w_{yy}&=&u_{xx}(\frac{\partial x}{\partial y})^2+u_x\frac{\partial^2 x}{\partial y^2}=u_{xx}s^2(t), \end{eqnarray} and hence $$u_x=\frac{1}{s(t)}w_y,u_{xx}=\frac1{s^2(t)}w_{yy}.\tag{1}$$ Note $$ w_{yt}=u_{xx}s^2(t)+u_{xt}s(t)+u_xs'(t)=w_{yy}+u_{xt}s(t)+w_y\frac{s'(t)}{s(t)}$$ and hence $$ u_{xt}=\frac{w_{yt}}{s(t)}-\frac{w_{yy}}{s(t)}-w_y\frac{s'(t)}{s^2(t)}. \tag{2}$$ Similarly \begin{eqnarray} w_t&=&u_x\frac{\partial x}{\partial t}+u_t=u_xs'(t)y+u_t,\\ w_{tt}&=&u_{xx}(s'(t))^2y^2+u_{xt}s'(t)y+u_xs''(t)y+u_{tt}.\tag{3} \end{eqnarray} Thus from (1), (2) and (3), you can get an equation for $w(y,t)$ and I omit the detail.