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

[coverband]

In the wave equation u(x,t), what does u represent?

 

[yungman]

Can you write out the equation?

 

[LCKurtz]

If you are representing a vibrating string, then u(x,t) represents the displacement of the string at location x at time t.

 

[coverband]

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 ...?

 

[LCKurtz]

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" .

 

[coverband]

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

 

[yungman]

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.

 

[HallsofIvy]

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.

 

[jvc]

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.