Green's function for the wave equation

In summary, the author introduces the Green's function for the 1-dimensional wave equation in chapter 3 of "Wave Physics" by S. Nettel. Using the separation of variables method, the author focuses on the spatial component and states that if the spatial solution ##u(x)## is specified at an end point, the Green's function ##G(x|x_0)## must vanish at that end point. This is necessary for the solution to satisfy the boundary conditions. The solution is written as an integral involving ##G(x|x_0)##, the inhomogeneity ##\kappa##, and a function ##f(x)## that satisfies the homogeneous wave equation and the boundary conditions. The integral will not be zero at
  • #1
dRic2
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Hi, I'm reading "Wave Physics" by S. Nettel and in chapter 3 he introduces the Green's function for the 1-dimensional wave equation. Using the separation of variables method he restricts his attention to the spatial component only. Let ##u(x)## be the spatial solution to the wave equation and ##G(x|x_0)## the Green's function. The author now states the following:

Let us now suppose that it is ##u(x)## that is specified at an end point. Then we must choose ##G(x|x_o)## to vanish at that end point.

I don't really understand this connection. Can someone help me ?

Thanks
Ric
 
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  • #2
You are looking to write down the solution on the form
$$
u(x) = \int G(x|x_0) \kappa(x_0) dx_0 + f(x)
$$
where ##\kappa## is the inhomogeneity in your wave equation and ##f(x)## solves the homogeneous wave equation and satisfies your boundary conditions. If ##G(x|x_0)## does not vanish at the boundary, then the integral will generally not be zero there and your function will not satisfy the boundary conditions.
 
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  • #3
In the book I'm reading there is a slightly different version of the solution:

244190


(forget about the ##\frac 1 {\rho}##)

BTW I also do not understand your answer. If, for example, ##k## vanishes at the end point the integral could also be zero and the boundary condition would be verified.
 
  • #4
dRic2 said:
If, for example, ##k## vanishes at the end point the integral could also be zero and the boundary condition would be verified.
No, this is incorrect. The point is to evaluate ##x## at the boundary, not ##x_0##, which is an integration variable.
 
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  • #5
Sorry, dumb mistake. Now I get your argument, thanks.
 

Related to Green's function for the wave equation

1. What is the Green's function for the wave equation?

The Green's function for the wave equation is a mathematical tool used to solve the wave equation in physics and engineering. It represents the response of a system to an impulse or point source, and can be used to find the solution for any initial conditions.

2. How is the Green's function for the wave equation derived?

The Green's function for the wave equation is derived by solving the wave equation with a delta function as the source term. This results in a solution that represents the response of the system to an impulse at a specific point in space and time.

3. What is the significance of the Green's function for the wave equation?

The Green's function for the wave equation is significant because it allows us to solve for the response of a system to any arbitrary source term, not just an impulse. This makes it a powerful tool in understanding and predicting the behavior of waves in various physical systems.

4. How is the Green's function for the wave equation used in practical applications?

The Green's function for the wave equation is used in practical applications such as acoustics, electromagnetics, and seismology. It can be used to model and predict the behavior of waves in these systems, and is also used in signal processing and image reconstruction techniques.

5. Are there any limitations to using the Green's function for the wave equation?

While the Green's function for the wave equation is a powerful tool, it does have some limitations. It is only applicable to linear systems, and it assumes that the system is homogeneous and isotropic. It also requires the system to have a well-defined boundary and initial conditions.

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