Traveling Waves and a Question about Functions

[FCPancakeIII]

I tried to post this in the right spot but if it isn't, feel free to move it and let me know what to do next time. Edit: (Woops! This should be in the calculus physics section! Sorry!)

This isn't really a homework question. I'm having trouble understanding the equation y(x,t)=ysin(kx - wt) that describes the displacement caused by a traveling wave, where y is the maximum displacement, k is the angular wave number, and omega (w) is the angular frequency.

What I don't really understand is the notation that says (I think) "y as a function of x and t." The concept is throwing me for a loop and I don't really understand how to manipulate the equation and solve it to help me understand what a traveling wave does to a particular medium.

I've had up to integral calculus and the physics I'm enrolled in is calculus based and dealing with heat, pressure, waves, and optics.

This is my first post and I'm very excited about this forum, I'm really hoping it will be a valuable learning tool for me. Thanks a bunch!

 

[CompuChip]

First of all, welcome to PF.

To get to your question... you are right that y(x, t) means that y is a function of both x and t. So it describes the amplitude of a wave at every time, at every point. Two common ways to visualize the wave is to fix t and plot it as a function of x (that is, supposing the wave travels on a piece of string for example, you draw what the wave would look like when you take a photo of the whole string at some instant) and to fix x and plot it as a function of t (i.e. sit on the string at a particular position and see what it does in time).

It's really just that, no different than you would treat a function f(x) or s(t), except that you have two things to vary. Can you give a specific example of an operation or concept that's giving you problems?

 

[FCPancakeIII]

The photograph idea helps considerably with understanding what the shape of the function actually means. So let's see, the kinds of problems that are confusing me at the moment are asking things like how long does it take a portion of the string (or whatever medium) to move between displacement A and displacement B for

y(x,t) = ymax * Sin (kx + wt + phi).

phi being the phase.

So that means they are asking for t when y(x,t) = A, and B, right? I know that for a displacement A, only certain points on the string will be displaced distance A at any given time, but that's as far as I've gotten.

 

[CompuChip]

So in this case you would pick a portion of the string (if I'm not mistaken, any part will do, because they all move in the same manner just at different phases; so you can take x = 0) and then consider y as a function of t only:
d(t) = ymax * sin(wt + phi)

Now you can solve for d(t) = A and d(t) = B using your mathematics tools. A little thought reveals that you can even take phi to be zero (on physical grounds: the phase of the wave does not influence how long a point will take to go from A to B) but if you don't see that you do that math and check that it works (similarly for taking x = 0: you can take x arbitrary but fixed and show that it cancels out of the equation).

 

[turin]

A little thought reveals that you can even take phi to be zero (on physical grounds: the phase of the wave does not influence how long a point will take to go from A to B) ...

You should be careful about this. The direction in which the displacement is changing makes a difference. You have probably encountered a similar concept: throwing a ball up vs. throwing it down. The ball starts at the same height, and it could even start with the same speed, but the ball thrown up will take longer to reach the ground. For waves, every displacement has two directions in which it can change (except of course the peak and trough).