Wave Equation, stuck on a partial calculation

In summary, the conversation is about a student struggling with basic concepts in their PDE I class. They are trying to verify a solution using direct substitution and are stuck on finding the second partial with respect to x. They receive help from another student and use Leibniz' formula to correctly apply the chain rule. They also reference a helpful website for taking derivatives of integrals.
  • #1
Somefantastik
230
0
Hey everybody,

My professor started our PDE I class in Chapter six, so I am having a hard time with the really basic stuff to get the theory down.

One of my questions to answer is to verify a solution by using direct substitution.

[tex]u(x,t) \ = \ \frac{1}{2}\left[\phi(x+t) \ + \ \phi(x-t) \right] \ + \ \frac{1}{2} \int^{x+t}_{x-t}\Psi(s)ds [/tex]

With initial conditions

[tex] u(x,t_{0}) = \phi(x) \ , \ \frac{\partial u}{\partial t} (x,t_{0}) = \Psi(x), \ and \ t_{0} = 0 [/tex]

satisfies [tex]\frac{\partial^{2}u}{\partial x^{2}} - \frac{\partial^{2}u}{\partial t^{2}} = 0 [/tex]

It was easy for me to plug and chug to show that [tex] u(x,t_{0}) = \phi(x) \ and \ \frac{\partial u}{\partial t} (x,t_{0}) = \Psi(x) [/tex]

Clearly my next step is to find [tex] \frac{\partial^{2}u}{\partial x^{2}} [/tex]

but that's the step on which I'm stuck. Can someone get me started? If someone can show me how to do the second partial w.r.t x, it would be a good exercise for me to figure out the second partial w.r.t t. I apply the chain rule and just get a bunch of garbage back, which means I'm messing it up somewhere.

Any input is appreciated.
 
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  • #2
You need "Leibniz' formula":
[tex]\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} \phi(x,t)dt= \frac{d\beta}{dx}\phi(x,\beta(x))- \frac{d\alpha}{dx}\phi(x,\alpha(x))+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial \phi}{\partial x}dt[/tex]

It's really just applying the chain rule correctly.
 
  • #3
[tex]
\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} \phi(x,t)dt= \frac{d\beta}{dx}\phi(x,\beta(x))- \frac{d\alpha}{dx}\phi(x,\alpha(x))+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial \phi}{\partial x}dt
[/tex]

I'm having trouble understanding this. It looks like the integrand you gave is a function of 2 variables, but the integrand I have is a function of one variable. I'm not sure what to do with that.

Also:
[tex] \frac{\partial}{\partial x} (\phi(x+t)) = ? [/tex]

is it [tex] = \phi '(x + t), \ or \ \phi_{x}(x+t) \ ? [/tex]

In the notation of a function, how do I write that? I guess I'm having a notational brain fart.
 
Last edited:
  • #4
I found this worked out in Walter Strauss's book, so I guess I don't need help anymore. Thanks for looking :)
 
  • #5
Somefantastik said:
[tex]
\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} \phi(x,t)dt= \frac{d\beta}{dx}\phi(x,\beta(x))- \frac{d\alpha}{dx}\phi(x,\alpha(x))+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial \phi}{\partial x}dt
[/tex]

I'm having trouble understanding this. It looks like the integrand you gave is a function of 2 variables, but the integrand I have is a function of one variable. I'm not sure what to do with that.
In that case,
[tex]\frac{\partial\phi}{\partial x}= 0[/tex]
and the formula becomes
[tex] \frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} \phi(x,t)dt= \frac{d\beta}{dx}\phi(x,\beta(x))- \frac{d\alpha}{dx}\phi(x,\alpha(x)) [/tex]

Also:
[tex] \frac{\partial}{\partial x} (\phi(x+t)) = ? [/tex]

is it [tex] = \phi '(x + t), \ or \ \phi_{x}(x+t) \ ? [/tex]

In the notation of a function, how do I write that? I guess I'm having a notational brain fart.
Use the chain rule. If [itex]\phi'(u)[/itex] is the derivative of [itex]\phi[/itex] as a function of the single variable u, then
[tex]\frac{\partial\phi(x+t)}{\partial x}= \phi'(x+t)(1)= \phi'(x+t)[/tex]
 
  • #6

1. What is the wave equation?

The wave equation is a mathematical formula that describes the behavior of waves in a medium. It is commonly used in physics and engineering to model a wide range of wave phenomena, such as sound, light, and water waves.

2. How is the wave equation derived?

The wave equation is derived from the fundamental principles of physics, including Newton's laws of motion and conservation of energy. It is also based on the assumption that the medium in which the wave is propagating is homogeneous and isotropic.

3. What is a partial calculation in the wave equation?

A partial calculation in the wave equation refers to a step in the overall calculation process where only part of the equation is solved or simplified. This is often necessary in more complex problems, as it allows for easier manipulation of the equation before arriving at a final solution.

4. How can I solve a partial calculation in the wave equation?

Solving a partial calculation in the wave equation involves using algebraic and calculus techniques to manipulate the equation and isolate the desired variable. It may also involve using boundary conditions or initial conditions to help solve for the unknown variable.

5. What are some real-world applications of the wave equation?

The wave equation has numerous applications in various fields, including acoustics, electromagnetics, seismology, and fluid dynamics. It is used to study and predict the behavior of waves in different media, which is crucial in understanding and designing technologies such as sonar, antennas, and seismographs.

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