Equation of standing wave clarification

Question

In looking for the standing wave equation y(x,t) I seem to be finding two variations.

The first variation is $y(x,t)=2A\sin(kx)\cos(\omega t)$ which I found on Wikipedia and other sites, which has a sine and a cosine.

The second variation is $y(x,t)=2A\sin(kx)\sin(\omega t)$ which I found here (scroll to 1.5.6) and on other sites. It has two sines.

It seems like all of the websites with the first variation derive the equation from the superposition of two sine waves whereas the websites with the second variation derive the equation from the superposition of two cosine waves but I don't see how that would result in a different end result.

Which is the correct variation? Or is there some factor I'm missing that differentiates the two and makes them both correct?

Answer 1

They are equivalent.

Let $y_s(x,t) = 2A \sin(k x) \sin(\omega t)$ and $y_c(x,t) = 2 A \sin(k x) \cos (\omega t)$.

Then, since $\sin(x + \pi/2) = \cos(x)$, we have that

\begin{equation} y_s(s,t+t_0) = y_c(x,t) \end{equation} where $t_0 = \frac{\pi}{2\omega}$. In other words, these two formulations differ only by a shift in time -- just wait one quarter period and $y_s$ will turn into $y_c$.