Solving Two PDEs To Derive Traveling Wave Solutions

In summary, solving two PDEs (partial differential equations) to derive traveling wave solutions is a mathematical technique used to simplify and analyze the behavior of complex physical systems. This technique involves combining two PDEs into a single equation and solving for the values that describe the wave-like behavior. It can be applied to a wide range of systems and has numerous benefits, including gaining a deeper understanding and predicting future behavior. Real-world examples of its use include studying atmospheric and seismic waves, heat and mass transfer, and chemical reactions.
  • #1
sigmund
23
0
I have a system of two PDEs:

[tex]y_t+(h_0v)_x=0 \quad (1a)[/tex]
[tex]v_t+y_x=0 \quad (1b)[/tex],

where [itex]h_0[/itex] is a constant.

Then I want to show that (1) has traveling wave solutions of the form

[tex]y(x,t)=f(x-ut) \quad (2a)[/tex]
[tex]v(x,t)=g(x-ut) \quad (2b)[/tex],

where [itex]u[/itex] is the propagation velocity.

Differentiating (1a) w.r.t. [itex]x[/itex] and (1b) w.r.t. [itex]t[/itex] I conclude that
[tex]\frac{\partial^2v}{\partial t^2}=h_0\frac{\partial^2v}{\partial x^2}[/tex],
but I am not sure that this is useful. I have tried a linear change of variables in order to arrive at a solution. Introducing the variables [itex]\alpha=ax+bt[/itex] and [itex]\beta=cx+dt[/itex] I arrive at a solution for [itex]v[/itex], namely either [itex]v=C(x+t)[/itex] or [itex]v=C(x-t)[/itex], depending on how the constants [itex]a,b,c,d[/itex] are chosen. Here [itex]C[/itex] is an arbitrary function. Indeed this looks like (2b), but I would like to put forward a more rigorous argument.

I would like some thoughts on this problem, and any hint on how to solve it is welcome. And is it possible to say anything about the relationship between [itex]f[/itex] and [itex]g[/itex] in (2)?
 
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  • #2
sigmund said:
I have a system of two PDEs:

[tex]y_t+(h_0v)_x=0 \quad (1a)[/tex]
[tex]v_t+y_x=0 \quad (1b)[/tex],

where [itex]h_0[/itex] is a constant.

Then I want to show that (1) has traveling wave solutions of the form

[tex]y(x,t)=f(x-ut) \quad (2a)[/tex]
[tex]v(x,t)=g(x-ut) \quad (2b)[/tex],

where [itex]u[/itex] is the propagation velocity.

Differentiating (1a) w.r.t. [itex]x[/itex] and (1b) w.r.t. [itex]t[/itex] I conclude that
[tex]\frac{\partial^2v}{\partial t^2}=h_0\frac{\partial^2v}{\partial x^2}[/tex],
but I am not sure that this is useful. I have tried a linear change of variables in order to arrive at a solution. Introducing the variables [itex]\alpha=ax+bt[/itex] and [itex]\beta=cx+dt[/itex] I arrive at a solution for [itex]v[/itex], namely either [itex]v=C(x+t)[/itex] or [itex]v=C(x-t)[/itex], depending on how the constants [itex]a,b,c,d[/itex] are chosen. Here [itex]C[/itex] is an arbitrary function. Indeed this looks like (2b), but I would like to put forward a more rigorous argument.

I would like some thoughts on this problem, and any hint on how to solve it is welcome. And is it possible to say anything about the relationship between [itex]f[/itex] and [itex]g[/itex] in (2)?

I'm not sure what more you want. Yes, differentiating as you did eliminates y from the equation leaving
[tex]\frac{\partial^2v}{\partial t^2}=h_0\frac{\partial^2v}{\partial x^2}[/tex]
which is the wave equation. It is well known that
[tex]f(x-\sqrt{h_0}t)[/tex]
and
[tex]f(x+\sqrt{h_0}t)[/tex]
are solutions for any twice differentiable function f, and you can show that by substituting them into your equation.
Of course, if you differentiate the first equation with respect to t and the second with respect to x you can eliminate v, getting the same equation for y.

If all you want to do is show that
(1) has traveling wave solutions of the form

[tex]y(x,t)=f(x-ut) \quad (2a)[/tex]
[tex]v(x,t)=g(x-ut) \quad (2b)[/tex]

you don't really need to do all that. Just substitute those functions: yx= f ' and yt= -u f '; vx= g' and vt= -ug'. Substitute those into the equations and you can see that they satisfy both equations as long as u2= h0.
 
  • #3
HallsofIvy, thank you for the reply.

I guess it suffices to substitute the assumed solutions into the system, and show that they satisfy the differential equations. This is of course the straightforward way to do it. As Einstein said: "Everything should be done as simple as possible, but no simpler."

What can we say about the relationship between f and g? Obviously, we can read it from the equation, but can we say something in general?
 
  • #4
Not without initial conditions. For example, if we have some given initial shape, F(x), and are told that the wire or string or whatever is held in that shape and then released (the partial derivative with respect to t is 0 when t=0, a very common situation) then we know that f(x)+ g(x)= F(x) and that cf '(x)- cg'(x)= 0. From the second equation, f '(x)= g'(x) so f(x)= g(x)+ C. From the first then g(x)+ C+ g(x)= F(x) or 2g(x)= F(x)- C so g(x)= (1/2)F(x)- (1/2)C. Then f(x)= g(x)+ C= (1/2)F(x)+ (1/2)C. The solution is (1/2)F(x+ ct)+ (1/2)F(x-ct) (The constants cancel). That is, the initial shape divides into two 1/2 size waves going in opposite directions.

For any functions, f and g, it is possible to come up with initial conditions that will yield them so nothing can be said in general.
 

1. What is the purpose of solving two PDEs to derive traveling wave solutions?

Solving two PDEs (partial differential equations) to derive traveling wave solutions is a mathematical technique used to find specific solutions to complex physical phenomena. It allows scientists to simplify and analyze the behavior of a system by reducing it to a simpler form.

2. How does solving two PDEs to derive traveling wave solutions work?

This technique involves using two PDEs, one for the wave speed and one for the wave shape, and combining them to form a single equation. This equation can then be solved to find the values of the variables that describe the behavior of the traveling wave.

3. What types of systems can be analyzed using this technique?

Solving two PDEs to derive traveling wave solutions can be applied to a wide range of physical systems, including fluid dynamics, heat transfer, and electromagnetism. It is particularly useful for systems that exhibit wave-like behavior, such as waves on the ocean or sound waves in air.

4. What are the benefits of using this technique?

By solving two PDEs to derive traveling wave solutions, scientists are able to gain a deeper understanding of the behavior of complex systems. It also allows for the prediction and analysis of future behavior, making it a valuable tool in many areas of science and engineering.

5. What are some real-world examples of this technique being used?

This technique has been used in various fields, including meteorology to study the propagation of atmospheric waves, geophysics to model seismic waves, and engineering to analyze the behavior of heat and mass transfer in materials. It has also been applied in biology to study the behavior of neurons and in chemistry to investigate chemical reactions.

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