Can D'Alembert's Solution Solve the 1-Dimensional Wave Equation?

  • Thread starter DanielO_o
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In summary, the D'Alembert solution for the 1-dimensional wave equation is 1/2[f(x-ct) + f(x+ct)] + 1/2C*integral(g(s)ds), where f(x) and g(x) are initial conditions. To solve for these initial conditions, you must solve for F(x) and G(x) separately using the equations phi(x,0) = f(x) and phi_t(x,0) = g(x). Once you have solved for F(x) and G(x), you can plug them back into the D'Alembert solution to get the final solution for #(x,t).
  • #1
DanielO_o
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I have to solve the 1-dimensional wave equation using D'Alembert's solution.

I know the solution has the forum 1/2[f(x-ct) + f(x+ct)] + 1/2C*integral(g(s)ds)

and i have two intial conditions f(x) and g(x)... do i just plug in (x-ct) where i see x etc...?

also what is g(s) in the integral?

Thanks

Dan
 
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  • #2
DanielO_o said:
I have to solve the 1-dimensional wave equation using D'Alembert's solution.

I know the solution has the forum 1/2[f(x-ct) + f(x+ct)] + 1/2C*integral(g(s)ds)

and i have two intial conditions f(x) and g(x)... do i just plug in (x-ct) where i see x etc...?

also what is g(s) in the integral?

Thanks

Dan

You've written this very confusingly! You have f used in at least two different ways and g used in two different ways.

What you are giving as the "D'Alembert solution" is too specialized.

The general D'Alembert solution to the wave equation is [itex]\phi (x,t)= F(x+ ct)+ G(x- ct)[/itex] where F and G can be any two twice differentiable functions of a single variable. Then [itex]\phi_t(x,t)= cF'(x+ct)- cG'(x- ct)[/itex]. If one initial condition is [itex]\phi_t(x,0)= 0[/itex], then it follows that [itex]F'(x)= G'(x)[/itex] so F and G differ by a constant- in that case, [itex]\phi (x,t)= (1/2)(F(x+ct)+ F(x-ct))+ constant[/itex].

But that is not the case here. (It is also confusing to say just "initial values f(x) and g(x)" since functions are not "conditions". I assume you mean [itex]\phi (x,0)= f(x)[/itex] and [itex]\phi_t(x,0)= g(x)[/itex].)

Then you have [itex]\phi (x,0)= F(x)+ G(x)= f(x)[/itex] and [itex]\phi_t(x,0)= cF'(x)- cG'(x)= g(x)[/itex]. Solve those two equations for F and G.
 
  • #3
Yes, sorry to be confusing but your assumptions were right...

The initial conditions are #(x,0) = e^(-x^2) and #'(x,0) = 2cxe^(-x^2)...

I wasn't sure how to use these to get to the solution but it seems like i need to solve those two equations.

Thanks!
 
  • #4
I have got the answer #(x,t) = e^-((x-ct)^2)

Am i right?
 
  • #5
Well, it's easy to check, isn't it? Calculate #xx and #tt and see if it satisfies the differential equation. Is #(x,0)= e^(-x^2)?
Is #_t(x,0)= -2cxe^(-x^2)?
 

1. What is D'Alembert's Solution?

D'Alembert's Solution is a mathematical formula used to solve the partial differential equation known as the wave equation. It was developed by French mathematician Jean le Rond d'Alembert in the 18th century.

2. How does D'Alembert's Solution work?

D'Alembert's Solution involves separating the wave equation into two parts, known as the homogeneous and particular solutions. The homogeneous solution represents the propagation of the wave in the absence of any external forces, while the particular solution takes into account any external forces acting on the wave.

3. What are the applications of D'Alembert's Solution?

D'Alembert's Solution has various applications in physics and engineering, particularly in the study of wave phenomena such as sound, light, and electromagnetic waves. It is also used in the analysis of vibrating strings and membranes, as well as in fluid mechanics and acoustics.

4. Are there any limitations to D'Alembert's Solution?

While D'Alembert's Solution is a useful tool for solving the wave equation, it does have its limitations. It assumes that the medium through which the wave is propagating is linear, homogeneous, and isotropic. It also does not take into account any nonlinear or dissipative effects.

5. How does D'Alembert's Solution compare to other methods of solving the wave equation?

D'Alembert's Solution is one of the most widely used methods for solving the wave equation due to its simplicity and versatility. However, there are other methods such as the method of separation of variables and the Fourier transform method, which may be more suitable for certain types of problems.

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