PDE: Nontrivial solution to the wave equation

In summary, the conversation discusses finding a nontrivial solution for the wave equation u_{tt} - c^2u_{xx} = f(x,t) with given boundary conditions. The solution is obtained using a series solution and the use of orthogonal functions.
  • #1
RJLiberator
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Homework Statement


Consider the wave equation:

[itex]
u_{tt} - c^2u_{xx} = f(x,t),
\hspace{1cm}
for
\hspace{1cm}
0 < x < l \\
u(0,t) = 0 = u(l,t) \\
u(x,0) = g(x), u_t(x,0) = f(x) \\
[/itex]
Find a nontrivial solution.

Homework Equations

The Attempt at a Solution



Here's what I did, but I have little understanding of it other than I know that I am using boundary conditions and some previous material to get here:

We form a series solution:

[itex]
u(x,t) = \sum_{n=1}^{\infty} u_n(x,t) = \sum_{n=1}^{\infty} sin{\frac{nπx} {l}}u_n(t) \\
g(x) = \sum_{n=1}^{\infty}b_nsin{\frac{nπx}{l}} \\
f(x) = \sum_{n=1}^{\infty} \frac{cnπb_n*}{l} sin{\frac{nπx}{l}} \\
\\
b_n = \frac{2}{l} \int_{0}^{l} g(x)sin{\frac{nπx}{l}}dx \\
b_n* = \frac{2}{l} \int_{0}^{l} f(x)sin{\frac{nπx}{l}}dx \\
[/itex]

All together:

[itex]
u(x,t) = \sum_{n=1}^{\infty} sin{\frac{nπx} {l}} [b_n cos(\frac{cnπt}{l}) + b_n*sin(\frac{cnπt}{l})]
[/itex]

From my understanding, inputting b_n and b_n* would give us the nontrivial solution. Yes?
 
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  • #2
Yes, although you should consider using the notation ##b_n^*## instead, it would be less confusing.

If you would like a better understanding of what is going on, I suggest reading up on function spaces and orthogonal functions.
 
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  • #3
Thank you kindly for the confirmation and suggestion.
 

1. What is a PDE?

A PDE, or partial differential equation, is a mathematical equation that involves partial derivatives of an unknown function with respect to multiple independent variables.

2. What is the wave equation?

The wave equation is a specific type of PDE that describes the behavior of waves, such as light, sound, or water waves. It can be written in the form of d²u/dt² = c²(d²u/dx²), where u represents the wave function, t represents time, and x represents distance.

3. What is a nontrivial solution?

A nontrivial solution to the wave equation is a solution that is not simply a constant or zero. It represents a distinct, non-uniform wave pattern that satisfies the equation.

4. Why is finding a nontrivial solution important?

Nontrivial solutions to the wave equation are important because they represent real-world phenomena, such as sound waves or electromagnetic waves. These solutions can provide valuable insights and predictions about the behavior of these waves.

5. How is a nontrivial solution found?

Finding nontrivial solutions to the wave equation involves solving the equation using various techniques, such as separation of variables or applying boundary conditions. It requires a deep understanding of mathematical methods and physical principles.

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