Understanding the Relationship between Variables in a Wave Equation

[bluestar]

I am unable to determine the relationship between x and t in the following equation.

$$y\left(x,t\right)=A\sin\left( kx-\omega t \right)\\$$


If $$\nu=\frac{x}{{t}}$$ then the numbers within the bracket goes to zero; because $$kx=\omega t$$
for all points on y(x,t).

Can anyone enlighten me please?

 

[kanato]

x and t are independent variables; there is no relationship between them. That equation describes a wave. Pick any time t_0, then you can look at the whole wave in space (along x). Pick a point x_0, and you can see how that point oscillates in time. Both can be looked at independently.

 

[bluestar]

I set-up a spreadsheet and generated a sinusoidal wave starting at x0 which progresses parallel along the positive x-axis.
If I leave t=0, then any value I plugged in for x falls on the curve.
Likewise, if I left x=0, then any value I plugged in for t falls on the curve.

Does this mean when one variable has a value then the other must be 0?

 

[blkqi]

No. It's a wave function of two free variables, x and t--longitudinal position and time. Pick any constant t and you have a standing wave at t. Let t be a variable to see the evolution of the wave over time.

 

[bluestar]

I had trouble grasping the concept of how to graph a function depended on two variables. I found a site that presented a simple Gaussian wave and then progressed to a Gaussian Wave dependent on two variables. The page wraps up with a general equation of a cosine function dependent on two variable including axes offsets.

I found it to be a great site for explaining the implication of a function with two independent variable. Now I understand a little better what is happening in the wave equation.


http://resonanceswavesandfields.blogspot.com/2007/08/true-waves.html

 

[Bob_for_short]

I had trouble grasping the concept of how to graph a function depended on two variables.

In 3D space a function of two variables can be drawn as a surface, wavy in both directions in your case.