# Division by 0 when calculating wave transmission and reflection coefficients

Perhaps more like a mathematical issue here...

When we consider a wave propagating from medium 1 to a medium 2, boundary located at $$x=0$$, we have the following wave equations for the incident, reflected and transmitted waves:

$$y_i (x,t) = A \sin (k_1 x - \omega t)$$

$$y_r (x,t) = B \sin (k_1 x + \omega t)$$

$$y_t (x,t) = C \sin (k_2 x + \omega t)$$

And we have $$y(x,t) = y_i + y_r$$ for $$x<0$$ and $$y(x,t) = y_t$$ for $$x>0$$

The boundary conditions are: $$y(0^-,t) = y(0^+,t)$$ and $$\frac{\partial y(0^-,t)}{\partial x}=\frac{\partial y(0^+,t)}{\partial x}$$.

The first boundary conditions gives:

$$(A-B) \sin(\omega t) = C \sin (\omega t)$$

while the second gives:

$$(A+B) k_1 \cos (\omega t) = C k_2 \cos (\omega t)$$

In all derivations I found, they cancelled the sines and cosines, concluding that:

$$A - B = C \tag{1}$$

and

$$(A+B) k_1 = C k_2.\tag{2}$$

However, mathematically speaking, these conclusions are only valid for those values of $$t$$ in which the sines/cosines are $$\ne 0$$, correct?

Why can we conclude the last two equations in general? And, if not, what can we physically interpret about the points of time where either sine or cosine = $$0$$?

Right, but the constants don't depend on $$t$$, so in particular OP's relations (1) & (2) hold for all $$t$$ anyway.
Let's look at $$(A-B) \sin(\omega t) = C \sin (\omega t)$$. This equation needs to be valid for all times $$t$$. When $$t=n\pi/\omega$$ the equation is certainly still valid. The equation becomes $$0=0$$. Therefore at times when $$\sin(\omega t)=0$$ we don't have any issues. $$A-B=C$$ doesn't suddenly make the equation invalid when $$\sin(\omega t)=0$$
Another way to see this is to just not think about it as dividing by $$\sin(\omega t)$$. Certainly $$\sin(\omega t)=\sin(\omega t)$$, so we need $$A-B=C$$ in order for the equation to remain true. If $$A-B\neq C$$, then our equation is true only when $$\sin(\omega t)=0$$, which is not true for all times and is not what we want.