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

• coverband
In summary: 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.
coverband
In the wave equation u(x,t), what does u represent?

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" .

Last edited by a moderator:
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.

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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.

## 1. What is the wave equation?

The wave equation is a mathematical formula that describes how waves propagate through a medium. It is used to model a variety of physical phenomena, such as sound waves, electromagnetic waves, and water waves.

## 2. What does the variable u represent in the wave equation?

The variable u in the wave equation represents the displacement of the wave at a specific point in space and time. It is a function of both the position, x, and time, t.

## 3. How is the wave equation derived?

The wave equation is derived from the laws of motion and the conservation of energy. It can also be derived from Maxwell's equations, which describe the behavior of electromagnetic waves.

## 4. What are the units of u in the wave equation?

The units of u in the wave equation depend on the type of wave being modeled. For example, in the case of a sound wave, u would have units of pressure or displacement. In the case of an electromagnetic wave, u would have units of electric or magnetic field strength.

## 5. What role does u play in the wave equation?

The variable u in the wave equation is the solution to the equation and represents the amplitude of the wave at a given point in space and time. It is essential in understanding the behavior and propagation of waves in a medium.

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