# D'alembert's solution to the wave equation, on Chain Rule

• kougou
In summary, the conversation is about the proof of D'alembert's solution to the wave equation. The person is having trouble understanding how to take the second partial derivative of u with respect to x and whether it is a function of z and n. The expert advises using the chain rule to convert the derivatives to be in terms of ζ and η.
kougou

## Homework Statement

Please have a look at the picture attach, which shows the proof of the D'alembert's solution to the wave equation. If you can't open the open,
https://www.physicsforums.com/attachment.php?attachmentid=54937&stc=1&d=1358917223

The part I have problem is the taking the second paritial derivative of u with respect to x.
Ux=du/dz+du/dn, does this mean that Ux is a function of z and n, that is, Ux(z,n), and z(x,t) and n(x,t)?

so later when I taken the second derivative, then I apply chain rule?

Thank you

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hi kougou!

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kougou said:
The part I have problem is the taking the second paritial derivative of u with respect to x.
Ux=du/dz+du/dn, does this mean that Ux is a function of z and n, that is, Ux(z,n), and z(x,t) and n(x,t)?

so later when I taken the second derivative, then I apply chain rule?

i'm not really following your question

∂u/∂x can be considered either a function of x and y, or a function of ζ and η

since you want a final result in terms of ζ and η, you'll have to convert the derivatives wrt x and y into derivatives wrt ζ and η (and yes, you use the chain rule)

## 1. What is D'alembert's solution to the wave equation?

D'alembert's solution is a mathematical formula that provides a general solution to the wave equation, which describes the propagation of waves through a medium.

## 2. How does D'alembert's solution work?

The solution uses the chain rule, which is a mathematical rule for differentiating composite functions. It breaks down the wave equation into two simpler equations, one for position and one for time, and then combines them to form the final solution.

## 3. What is the significance of D'alembert's solution?

D'alembert's solution is significant because it provides a general solution to the wave equation, meaning it can be applied to a wide range of physical systems that exhibit wave-like behavior. It also allows for the prediction and analysis of wave behavior in these systems.

## 4. Can D'alembert's solution be applied to all types of waves?

Yes, D'alembert's solution can be applied to various types of waves, including electromagnetic waves, sound waves, and water waves. As long as the wave equation can accurately represent the behavior of the wave, D'alembert's solution can be used.

## 5. Are there any limitations to D'alembert's solution?

While D'alembert's solution is a powerful tool, it does have some limitations. It assumes that the medium through which the wave is propagating is uniform and does not take into account any external forces acting on the wave. Additionally, it may not provide an exact solution for more complex systems where the wave equation is not known or is difficult to solve.

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