Page 23 below is from Garrity's book "Electricity and Magnetism for Mathematicians". He is discussing how changing to a coordinate frame moving with some speed relative to the original coordinate frame results in our solution to the one dimensional wave equation having a corresponding change in speed.
However, in the discussion, he uses a fact I don't understand. If y(x,t) is our solution to the one dimensional wave equation, then why should 0 = dy/dt? Please see picture below. Thanks for the help!
It seems like dy/dt = 0 because we are restricting attention to just values of x and t that give us y0 = y(x,t). If this is the case, then our result is only valid for those special x and t values?
It seems he's trying to derive the wave equation. He's doing that by "following" a point on a wavefront, and showing how the derivatives of are then related.
Here's what's happening: he has some function $y(x,t)$ which tells us the amplitute at a specific place $x$ and time $t$. Then he's asking us to follow, say, the node of a wave over time. This corresponds to setting $y(x,t)=y_0$. Then we no longer have $x$ and $t$ as independent variables; rather $x$ secretly depends on $t$, through the function $y$. Then we can take derivatives of both sides with respect to $t$. Clearly the derivative of the right side is $0$, which is what that first line says. Then the derivative of the second side is found using the chain rule. From this, and the fact that given these constraints $\partial x/\partial t$ is the velocity of the wave, you can derive the equation at the bottom.
All of this math is meant to constrain the form that $y$ can take, if we want it to represent a wave.