Understanding PDEs: Evaluating a Solution to the 1-Dimensional Wave Equation

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In summary, the conversation discusses the one-dimensional wave equation and a proposed solution of f(x, t) = exp(x-ivt). The summary also includes a correction of a typo and a question about the possibility of the proposed solution, which is determined to be incorrect due to the difference between exp(x-ivt) and -exp(x-ivt).
  • #1
ZedCar
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Homework Statement


I'm just trying to get an understanding of answering PDEs, so wanted to ask what you thought of my answer to this question.

The one-dimensional wave equation is given by the first equation shown in this link;

http://mathworld.wolfram.com/WaveEquation1-Dimensional.html

where Ψ = f(x, t)

Is f(x, t) = exp(x-ivt) a possible solution?

Homework Equations


The Attempt at a Solution



∂^2 f/∂x^2 = exp(x-ivt)

and

∂f/∂t = -iv exp(x-ivt)

Possible if v = -i
 
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  • #2
ZedCar said:

Homework Statement


I'm just trying to get an understanding of answering PDEs, so wanted to ask what you thought of my answer to this question.

The one-dimensional wave equation is given by the first equation shown in this link;

http://mathworld.wolfram.com/WaveEquation1-Dimensional.html

where Ψ = f(x, t)

Is f(x, t) = exp(x-ivt) a possible solution?



Homework Equations





The Attempt at a Solution



∂^2 f/∂x^2 = exp(x-ivt)

and

∂f/∂t = -ic exp(x-ivt)

Possible if v = -i

You need to compute [itex] \partial^2 f/\partial t^2, \text{ not just } \partial f/\partial t.[/itex] Anyway: what is "c"? The PDE does not have "c" in it, nor does your f.

RGV
 
  • #3
Ray Vickson said:
Anyway: what is "c"? The PDE does not have "c" in it, nor does your f.

Sorry, c should have been v. I've corrected it now.
 
  • #4
Ray Vickson said:
You need to compute [itex] \partial^2 f/\partial t^2, \text{ not just } \partial f/\partial t.[/itex]

RGV

So when I obtain the 2nd partial differentiation for 't' I obtain;

-v^2 exp(x-ivt)

So I assume this is not a possible solution since

exp(x-ivt) ≠ -exp(x-ivt)
 

Related to Understanding PDEs: Evaluating a Solution to the 1-Dimensional Wave Equation

1. What is the 1-Dimensional Wave Equation?

The 1-Dimensional Wave Equation is a mathematical equation that describes the behavior of waves in one dimension. It is commonly used in physics and engineering to model various physical phenomena, such as sound waves and electromagnetic waves.

2. How do you solve the 1-Dimensional Wave Equation?

The 1-Dimensional Wave Equation can be solved using various methods, including separation of variables and Fourier series. The specific method used depends on the initial conditions and boundary conditions of the problem.

3. What is the importance of evaluating a solution to the 1-Dimensional Wave Equation?

Evaluating a solution to the 1-Dimensional Wave Equation allows us to understand the behavior of waves in a given system. This can help in predicting and analyzing real-world phenomena, such as the propagation of seismic waves in earthquakes or the transmission of signals in communication systems.

4. Are there any limitations to the 1-Dimensional Wave Equation?

Yes, the 1-Dimensional Wave Equation has some limitations. It assumes that the medium in which the wave is traveling is homogeneous and isotropic, and that there are no external forces acting on the system. In reality, these assumptions may not hold true for all situations.

5. How is the 1-Dimensional Wave Equation used in real-world applications?

The 1-Dimensional Wave Equation is used in various fields, including physics, engineering, and mathematics, to model and analyze wave phenomena. It has practical applications in fields such as acoustics, electromagnetics, and signal processing.

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