What is u in the Wave Equation u(x,t)?

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SUMMARY

In the wave equation u(x,t), the variable u represents the displacement of a vibrating string at a specific location x and time t. The equation can be expressed as u''(t) = c²u''(x), where c denotes the wave velocity. The discussion highlights the visualization of this equation as a three-dimensional surface, with x representing spatial dimensions, t representing time, and u representing the wave's amplitude. Additionally, it is clarified that u can also denote the wave field distribution in space, emphasizing its role in both physical and light wave contexts.

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coverband
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In the wave equation u(x,t), what does u represent?
 
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Can you write out the equation?
 
If you are representing a vibrating string, then u(x,t) represents the displacement of the string at location x at time t.
 
yungman: u''(t)=u''(x)c^2

lckurtz: so visually its like a plane/surface in 3D (x,y,z) where x = x, y = t and z = u ...?
 
coverband said:
yungman: u''(t)=u''(x)c^2

lckurtz: so visually its like a plane/surface in 3D (x,y,z) where x = x, y = t and z = u ...?

I would visualize x to the right, u vertically, and motion in time. You can see a gif on my home page: http://math.asu.edu/~kurtz" .
 
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Anyone else so visually its like a plane/surface in 3D (x,y,z) where x = x, y = t and z = u ...? !
 
coverband said:
yungman: u''(t)=u''(x)c^2

lckurtz: so visually its like a plane/surface in 3D (x,y,z) where x = x, y = t and z = u ...?

I don't think so, From your equation, it gives the wave is traveling in x direction only. I assume t is time, x is the distance. I think you switch x and t also. It should be u"(x)-(c^2)u"(t)=0.

If you want direction in x y and z, you have to use Laplace equation to have PD of all three direction:
\nabla ^{2}u - C^{2}\frac{d^{2}u}{dt^{2}} = 0

I can't get the equation look right, I hope you get what I mean!


So I look at your equation as the function u(x,t) ( of x direction and time t ) travel in x direction with the magnitude of the function u(x,t). "c" is the velocity.

This is a one dimension spatial problem of x.
 
Last edited:
coverband said:
Anyone else so visually its like a plane/surface in 3D (x,y,z) where x = x, y = t and z = u ...? !
Your original equation did not have a "y" variable so you can think of z as either the height of a string in one-dimension or as the height of a "cylindrical" wave, extending infinitely in the y direction. In any case, your original "u" is the height of the wave if you are thinking of a physical wave, the intensity if you are thinking of a light wave, etc.
 
coverband said:
In the wave equation u(x,t), what does u represent?

Actually, u is complex amplitude of the wave, which denotes the amplitude of given spatial point and time. So u represents the wave field distribution in space.
 

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