- #1
member 428835
Hi PF! Suppose we have a water wave with mean depth ##H## with disturbance ##\zeta## above/below ##H## propagating through a channel of thickness ##b##. The book parenthetically remarks that the continuity equation becomes $$\partial_t(b(H+\zeta))+\partial_x(bHu)=0.$$ However, when I try deriving this I write $$\partial_t(b(H+\zeta) \Delta x )=ub(H+\zeta)|_x-b(H+\zeta)u|_{x+\Delta x}\implies\\
\partial_t(b(H+\zeta))+\partial_x(b(H+\zeta)u)=0$$ which is not quite what they have. Any idea what I'm doing wrong?
\partial_t(b(H+\zeta))+\partial_x(b(H+\zeta)u)=0$$ which is not quite what they have. Any idea what I'm doing wrong?