A transverse wave `y=A sin omega (x/V_(1) -t)` is travelling in a medium with speed `V_(1)`. Plane `x = 0` is the boundary of the medium. For `x gt 0` there is a different medium in which the wave travels at a different speed `V_(2)`. Part of the wave is reflected and part is transmitted. For `x lt 0` the wave function is described as `y_(-)=A_(1) sin omega (x/V_(1)-t)+A_(2) sin omega (x/V_(2)+t)` while for `x gt 0, y_(+)=A_(3) sin omega(x/V_(2)-t)`
(a) Using the fact that the wave function must be continuous at `x = 0`, show that `A_(1) – A_(2) = A_(3)`
(b) Using the fact that `(del y)/(del x)` must be continuous at `x = 0`, prove that `V_(2)/V_(1) A_(1)=A_(3)-A_(2)`
(c) Show that `A_(3)=(2V_(2)A_(1))/(V_(1)+V_(2))` and `A_(2)=(V_(1)-V_(2))/(V_(1)+V_(2))A_(1)`