Boundary Value Problem for the 1-D Wave

In summary, the problem at hand is to find solutions to the 1-D Wave equation u_{tt} = u_{xx} with initial conditions u(x,0) = g(x) and u_t(x,0) = h(x). However, the additional condition u_t(0,t) = A*u_x(0,t) causes issues, as setting A = -1 leads to an overdetermined problem. The D'alembert formula can solve the usual problem, but with this added restriction, it becomes overdetermined unless A = +/- 1. This question may be better suited for a homework forum.
  • #1
29
0
So here's the problem:

I'm asked to find the solutions to the 1-D Wave equation

[tex]u_{tt} = u_{xx}[/tex]

subject to

[tex] u(x,0) = g(x), u_t(x,0) = h(x)[/tex]

but also

[tex] u_t(0,t) = A*u_x(0,t)[/tex]

and discuss why A = -1 does not allow valid solutions. I can't figure it out at all. The solutions to the usual problem with initial conditions is just the D'alembert formula... but adding this restriction seems to overdetermine the problem. We can turn that second condition into the full 1-D wave equation but it involves squaring A, which means that if A = +/- 1 it works... and otherwise it is overdetermined. Can anyone else help me?
 
Physics news on Phys.org
  • #2
should be moved to homework... sorry!
 

1. What is the 1-D Wave Boundary Value Problem?

The 1-D Wave Boundary Value Problem is a mathematical model used to describe the behavior of a wave in one dimension. It involves finding the solution to a differential equation with specified boundary conditions.

2. What are boundary conditions?

Boundary conditions are constraints placed on a differential equation that define the behavior of the solution at the boundaries of the domain. In the case of the 1-D Wave Boundary Value Problem, the boundary conditions specify the behavior of the wave at the endpoints of the one-dimensional space.

3. What are some common applications of the 1-D Wave Boundary Value Problem?

The 1-D Wave Boundary Value Problem has many applications in physics and engineering, including the study of sound and seismic waves, the behavior of strings and membranes, and the analysis of electrical circuits.

4. How is the 1-D Wave Boundary Value Problem solved?

The 1-D Wave Boundary Value Problem is typically solved using mathematical techniques such as separation of variables, Fourier series, or Laplace transforms. These methods allow for the determination of the solution that satisfies both the differential equation and the boundary conditions.

5. What are the limitations of the 1-D Wave Boundary Value Problem?

The 1-D Wave Boundary Value Problem assumes that the wave is propagating in a straight line in one dimension and does not take into account certain physical factors such as dispersion or nonlinearity. Therefore, it may not accurately model more complex wave phenomena and may require additional modifications or assumptions for more accurate results.

Suggested for: Boundary Value Problem for the 1-D Wave

Back
Top