Expressing horizontal velocity as a function of time for a wave

[JoeyBob]

Homework Statement

y(x, t) = 0.2 sin(0.25x-12t). Find horizontal velocity as a function of time at x=4

Relevant Equations

none

This is more of a conceptual question. To find the horizontal velocity as a function of time for the above wave function, you take its partial t derivative and insert x=4. In other words the function would be -2.4sin(1-12t).

Im wondering why you take the partial t derivative and not to partial x derivative? I understand how partial derivatives work, but I still don't think I am entirely sure what they represent. Like, why can't you take the partial x derivative? Wouldnt the change in x be about velocity? What does a partial t derivative actually mean / represent in this context? Horizontal velocity, but why/how?

 

[haruspex]

Homework Statement:: y(x, t) = 0.2 sin(0.25x-12t). Find horizontal velocity as a function of time at x=4
Relevant Equations:: none

This is more of a conceptual question. To find the horizontal velocity as a function of time for the above wave function, you take its partial t derivative and insert x=4. In other words the function would be -2.4sin(1-12t).

Im wondering why you take the partial t derivative and not to partial x derivative? I understand how partial derivatives work, but I still don't think I am entirely sure what they represent. Like, why can't you take the partial x derivative? Wouldnt the change in x be about velocity? What does a partial t derivative actually mean / represent in this context? Horizontal velocity, but why/how?

Maybe some more explanation is needed regarding "horizontal" and x, y.
I would have taken a wave written as y(x,t) to mean x is horizontal and y is vertical. In that case, taking partial wrt t gives vertical velocity at a given (x,t).

Taking partial wrt x gives the slope at a given (x,t).

The horizontal velocity would be a constant, not dependent on x or t. If you increase t by Δt then that decreases the argument of the sin by 12Δt, so compensate and get the same value of the argument you need to increase x by 12Δt/0.25=48Δt. So velocity is 48 (of whatever units were used for the 0.25 and 12).