Proving a Solution of \[u_{tt} - c^2 u_{xx} = 0\]

  • Thread starter Palindrom
  • Start date
In summary, the conversation discusses the solution of a wave equation and the condition where the solution is constant along a specific line. The participants discuss different methods of proving this condition and whether it can be extended to all points in the solution space.
  • #1
Palindrom
263
0
Hi everyone.

I tried a bit, but got stuck.

Let [tex]\[u\left( {x,t} \right)\][/tex] be a solution of [tex]\[u_{tt} - c^2 u_{xx} = 0
\][/tex], and suppose [tex]\[u\left( {x,t} \right)\][/tex] is constant along the line [tex]\[
x = 2 + ct
\]
[/tex]. Then [tex]\[u\left( {x,t} \right)\][/tex] must keep:[tex]\[
u_t + cu_x = 0
\]
[/tex]
I can prove it for any point [tex]\[
\left( {x,t} \right)
\]
[/tex] which is right to the line [tex]\[
x = 2 - ct
\]
[/tex]. I don't see any way to prove it for the points left to that line.
Is there a simpler way, or more general one that doesn't make that last line special?
Thanks in advance.
 
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  • #2
U know that

[tex] u\left(2+ct,t\right)=C [/tex]

Take the PD wrt "t"

[tex]\frac{\partial u}{\partial (2+ct)}\frac{d(2+ct)}{dt}+\frac{\partial u}{\partial t} =0 [/tex]

Equivalently,using that [itex]2+ct=x [/itex]

[tex] \frac{\partial u}{\partial x} c+\frac{\partial u}{\partial t} = 0 [/tex]

Q.e.d.

Daniel.
 
  • #3
Well it's the first thing I did, but then it only proves it along the line [tex]\[
x = 2 + ct
\]
[/tex].
[tex]\[
\begin{array}{l}
\frac{d}{{dt}}\left( {u\left( {2 + ct,t} \right)} \right) = 0 \\
\frac{{\partial \left( {u\left( {x\left( t \right),t} \right)} \right)}}{{\partial x}}\frac{{\partial x\left( t \right)}}{{\partial t}} + u_t \left( {2 + ct,t} \right) = 0 \\
u_t \left( {2 + ct,t} \right) + cu_x \left( {2 + ct,t} \right) = 0 \\
\end{array}
\]
[/tex]
I need to prove it for all [tex]\[
\left( {x,t} \right) \in \Re ^2
\]
[/tex]

It's driving me a little crazy...
 

Related to Proving a Solution of \[u_{tt} - c^2 u_{xx} = 0\]

What is the equation for "Proving a Solution of utt - c2 uxx = 0"?

The equation is a second-order partial differential equation known as the wave equation, where u is a function of two variables (x and t), c is a constant representing the propagation speed of the wave, and the subscripts t and x denote partial derivatives with respect to time and space, respectively.

What does it mean to prove a solution of the wave equation?

Proving a solution of the wave equation means finding a function u(x,t) that satisfies the equation for all values of x and t. This function represents a physical wave that satisfies the laws of motion described by the wave equation.

What is the significance of the wave equation in science?

The wave equation is a fundamental equation in physics and is used to describe a wide range of phenomena, including sound waves, electromagnetic waves, and seismic waves. It is also a key equation in the study of differential equations and has applications in many fields such as engineering, mathematics, and geophysics.

What are some techniques used to prove a solution of the wave equation?

There are various techniques used to prove a solution of the wave equation, including separation of variables, Fourier series, and the method of characteristics. These techniques involve manipulating the equation and using boundary conditions to find a solution that satisfies the equation and any given initial or boundary conditions.

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

The wave equation has numerous real-world applications, including predicting and analyzing the behavior of waves in various systems such as musical instruments, water waves, and electromagnetic waves. It is also used in the development of technologies such as ultrasound imaging, radar, and earthquake prediction.

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