Solving the Wave Equation in semi-infinite domain with easy ICs

In summary, the conversation is about trying to solve the wave equation with certain initial conditions, but the speaker is having difficulty due to the methods they are using. After solving numerically, they realize that the waves must propagate at a speed of 1 and the solution can be simplified to H(t-x)sin(w(t-x)).
  • #1
Gengar
13
0
Hi, so the problem is this:

I am trying to solve (analytically) the wave equation with c=1:

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

on x,t>0 given the initial conditions

[tex]u(x,0)=u_{t}(x,0)=0, u(0,t)=sin(wt)[/tex]

I know how to solve on semi-infinite domains for quite a few cases using Green's Functions, Fourier Transforms, D'Alembert's solution and separation of variables. But I keep getting u=0 with these familiar methods due to the initial conditions of u being 0 and unmoving at t=0.

I feel like this is easier than I'm making it! Anyway, any help would be appreciated!
 
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  • #2
Yeh... After solving numerically I realized that I just hadn't thought about the fact that the waves must propagate at speed 1 and u=0 for all x>t. So a quick bit of algebra gives:

[tex]u(x,t)=H(t-x)sin(w(t-x))[/tex]

simples
 

1. What is the wave equation?

The wave equation is a mathematical model used to describe the behavior of waves in a given medium. It is a second-order partial differential equation that relates the second derivative of a wave function to its spatiotemporal variations.

2. What is a semi-infinite domain?

A semi-infinite domain is a mathematical concept used to describe a region that extends infinitely in one direction but is bounded in the other direction. In the context of solving the wave equation, it refers to a medium that is infinite in one direction but has a boundary at the other end.

3. What are easy ICs in the context of solving the wave equation?

Easy ICs (initial conditions) refer to a specific set of initial conditions that are relatively simple to solve for in the wave equation. This can include initial displacement, velocity, or any other parameter that is required to fully describe the behavior of the wave in the given medium.

4. How is the wave equation solved in a semi-infinite domain with easy ICs?

To solve the wave equation in a semi-infinite domain with easy ICs, one can use various techniques such as separation of variables, Fourier transforms, or Green's functions. The specific method used will depend on the specific boundary and initial conditions given in the problem.

5. What are some real-world applications of solving the wave equation in a semi-infinite domain with easy ICs?

The wave equation is widely used in fields such as physics, engineering, and geophysics to model various wave phenomena, including sound, light, and seismic waves. Solving the wave equation in a semi-infinite domain with easy ICs can be applied to real-world scenarios such as seismic imaging, ultrasonic testing, and underwater acoustics.

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