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.

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!

 

[dslowik]

It's a little confusing to see y(x,t) = ysin(kx - wt) since the y on the left hand side (LHS) is a function of x and t, but the y on the rhs is a constant, the wave's amplitude.
Beyond that though, the lhs y is a function of x and t. It's a function of kx - wt, the sine function. kx -wt is a function of x and t, representing a wave traveling to the right (or is it left? think how kx - wt changes with space and time).
The waves amplitude, the rhs y, is constant. What are it's units/dimenstions? That speaks to what's 'waving'.

 

[FCPancakeIII]

Right, I gathered that rhs y is a constant, it should be ymax, the maximum displacement caused by the wave and also the waves amplitude. I guess what I'm having difficulty understanding is how to understand the behavior of a function with two variables (x and t). My normal method is to graph the function, which helps me imagine better what it is doing, whether it's a particle in free fall or a particle oscillating up and down. But I don't understand whether or not I can graph this function with both x and t as variables. I figured I am supposed to examine the wave at different times t=0, 1, 2... and so on, but it seems that if the wave is moving at a particular speed, than only specific values of x will associated with those times, and I'm confused as to how to find the actual positions of different points in the medium as a function of time.

 

[dslowik]

Yep, I remember going through that..
Try:
Assume k and w are positive constants.
As time increases, the value of x such that the phase, (kx-wt), remains the same must also increase so wave moves to right. (kx+wt) would be phase of wave moving to the left.
kx - wt = a , where a is an arbitrary number (phase)
x = (wt + a) / k = (w/k)t + a/k gives the location of that particular phase at any time t. So a/k is the initial (t=0) position of the phase a, w/k is the wave's velocity (=wavelength/Period)

So ANY function of kx - wt is a 'wave' moving to the right with velocity w/k; that functions shape marches to increasing x as t increases.
Think of a function of one variable which is zero everywhere except at some point, a, where it is equal to 1. Now let that same function act on kx - wt, and imagine what it looks like as time goes by.
Any function of x - (w/k)t, (k/w)x - t , i/h*(kx-wt), etc is a 'wave'.

The 2-d graph of y vs. x moves uniformly to the right with time,
Or you could picture a graphical surface in 3-d where y depends on both x and t, the wave velocity being the slope of the parallel lines of translation symmetry of this surface in the x, t plane (delta x / delta t thereof).


One property of these 'wave functions' is that the derivative wrt time is the wave speed, w/k, times the derivative wrt space (by the chain rule).

 

[FCPancakeIII]

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.

 

[dslowik]

Yea that all sounds right. So the answer to their question would be independent of x. You would want to know when sin(wt)=A and sin(wt)=B. so first get wt=arcsin(A), etc

At what time does x=5 reach a displacement of y=3?
3 = ymax * sin(kx + wt + phi)
kx + wt + phi = arcsin(ymax/3)
Solve this for t.

Note it gives only one t, but t + NT (with N integer, T period) are all the possible solutions.
hmmm, actually there's usually twice that many...