Solving Wave Equation with D'Alembert: Step by Step

In summary, the person is seeking help with using the D'Alembert method to solve the wave equation. They have found the general form to be y(x,t) = f(x+ct) + g(x-ct) and have been given the condition that the time dependence at x=0 is sinusoidal. They are wondering how to deduce the full x,t dependence, which is given by y(x,t) = Asin(kx+wt) + Bsin(kx-wt) where k = w/c. They are unsure of how to proceed and are seeking assistance.
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Homework Statement



Ok so hope someone will be able to help...

I've used the D'Alembert method to solve the wave equation and have got that the general form should be

y(x,t) = f(x+ct) + g(x-ct)

Now I am also told that the time dependence at x=0 is sinusoidal..

that is, y(x,0) = sin(wt)...

How can i deduce that the full x,t dependence is given by:

y(x,t) = Asin(kx+wt) + Bsin(kx-wt)

where k = w/c?

Thank you

Homework Equations





The Attempt at a Solution



Not sure where to go..

It seems the conditions tell me that f(x) + g(x) = sinwt..cant see how that helps :S
 
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  • #2
anyone :S?
 

1. What is the D'Alembert solution to the wave equation?

The D'Alembert solution is a method for solving the wave equation, which is a partial differential equation that describes how waves propagate through a medium. It involves decomposing the equation into two simpler equations, which can then be solved using standard techniques.

2. How does the D'Alembert solution work?

The D'Alembert solution works by using the principle of superposition, which states that the total solution to a linear partial differential equation is the sum of the individual solutions to simpler equations. This allows us to break down the complex wave equation into two simpler equations, which can then be solved using standard techniques.

3. What are the steps for solving the wave equation using D'Alembert's method?

The steps for solving the wave equation using D'Alembert's method are as follows:

  1. Decompose the wave equation into two simpler equations using the principle of superposition.
  2. Solve each equation separately using standard techniques.
  3. Combine the solutions to get the final solution to the wave equation.

4. What are the benefits of using D'Alembert's method to solve the wave equation?

Using D'Alembert's method to solve the wave equation has several benefits:

  • It allows us to solve the wave equation using standard techniques that are well-understood and widely applicable.
  • It simplifies the solution process by breaking down the complex equation into two simpler equations.
  • It provides a general solution that can be applied to a wide range of problems involving wave propagation.

5. What are some real-world applications of solving the wave equation using D'Alembert's method?

D'Alembert's method for solving the wave equation has many practical applications, including:

  • Predicting the behavior of electromagnetic waves, such as radio waves and light, in various media.
  • Understanding the propagation of seismic waves in earthquakes.
  • Modeling the behavior of sound waves in different environments, such as in musical instruments or underwater.

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