Separation of variables wave equation

• coverband
In summary: But a much easier way is to integrate both sides ofu_t(x,0)= \sum_{n=0}^\infty 2nF_n sin(nx)= 1from x= 0 to x= \pi and use the fact that\int_0^\pi sin(mx) sin(nx)dx= \frac{\pi}{2}\delta_{mn}.In summary, the conversation discusses solving the wave equation u_(tt) = 4u_(xx) on the interval [0, π] subject to the conditions u(x, 0) = cos x, u_t(x, 0) = 1, u(0, t) = 0 = u(π
coverband
1. Solve the wave equation u_(tt) = 4u_(xx) on the interval [0, π] subject to the
conditions
u(x, 0) = cos x, u_t(x, 0) = 1, u(0, t) = 0 = u(π, t).

Homework Equations

3. Hello. This appears to be a common separation of variables question. Only problem is after using the boundary conditions and initial conditions, I am left with an unknown constant.

So, after using 2 b.c. and first i.c. I'm left with U=∑_(n=1)^(n=∞)▒〖E_n Sin(nx)Cos2nt〗+F_n/2n Sin(2nt)Sin(nx)

The problem is when i use my last i.c. this tells me the value of constant E but I'm left with F!

coverband said:
1. Solve the wave equation u_(tt) = 4u_(xx) on the interval [0, π] subject to the
conditions
u(x, 0) = cos x, u_t(x, 0) = 1, u(0, t) = 0 = u(π, t).

Homework Equations

3. Hello. This appears to be a common separation of variables question. Only problem is after using the boundary conditions and initial conditions, I am left with an unknown constant.

So, after using 2 b.c. and first i.c. I'm left with U=∑_(n=1)^(n=∞)▒〖E_n Sin(nx)Cos2nt〗+F_n/2n Sin(2nt)Sin(nx).

The problem is when i use my last i.c. this tells me the value of constant E but I'm left with F!

One problem you have is that the initial condition u(x,0)= cos(x) tells us that u(0,0)= 1 while the boundary condition u(0,t)= 0 tells us that u(0,0)= 0. That's not a crucial problem- it means that the solution cannot be continuous and so, strictly speaking, cannot satisfy the differential equation where it is not continuous.

It's impossible to tell what you might have done wrong because you haven't told us what you did! But it is not clear what you mean by "the first i.c." and "the second i.c." The two boundary conditions tell us that there is no "cos(nx}" in the solution and those alone give us
$$u(x,t)= \sum_{n=0}^\infty (E_n cos(2nt)+ F_n sin(2nt))sin(nx)$$

I would have thought of "$u_t(x,0)= 0$" as the "second" i.c. but that's the easy one:
$$u_t(x,t)= \sum_{n=0}^\infty (-2nE_n sin(2nt)+ 2nF_n cos(2nt))sin(nx)$$
so
$$u_t(x,0)= \sum_{n=0}^\infty 2nF_n sin(nx)= 0$$
so that Fn= 0 for all n.

The other i.c. becomes
$$u_t(x,0)= \sum_{n=0}^\infty E_n sin(nx)= cos(x)$$

To find E_n, for all n, do what you would do if you had
$$\sum_{n=0}^\infty E sin(nx)= F(x)$$
for any function F- write F(x) as a Fourier sine series. Here, of course, F(x)= cos(x) so you want to write cos(x) as a sine series! You are going to have discontinuities at x= 0 and $\pi$ but since you find the Fourier coefficients by integrating, that is not a problem.

Hello

There is a mistake in your proposed solution. You wrote

HallsofIvy said:
$$u_t(x,0)= \sum_{n=0}^\infty 2nF_n sin(nx)= 0$$
so that Fn= 0 for all n.

when in fact if you reread the question:

$$u_t(x,0)= \sum_{n=0}^\infty 2nF_n sin(nx)= 1$$

Now. How would one go about solving for $$F_n$$ here!? Thanks

You are right. I completely missed that. One way to find the coefficients is, of course, to expand "1" in a Fourier sine series.

1. What is the wave equation and why is it important in physics?

The wave equation is a mathematical formula used to describe the behavior of waves, such as light and sound, in a given medium. It is important in physics because it allows us to understand and predict the propagation of waves, which is crucial in various fields such as optics, acoustics, and electromagnetism.

2. What is separation of variables and how is it applied to the wave equation?

Separation of variables is a method used to solve differential equations by breaking them down into simpler equations. In the case of the wave equation, it involves separating the variables of time and space, allowing us to solve for each variable separately and then combine the solutions to obtain the overall solution.

3. What are the boundary conditions for the wave equation and why are they important?

The boundary conditions for the wave equation refer to the constraints that must be satisfied at the boundaries of the medium where the wave is propagating. These conditions are important because they help us determine the behavior of the wave and its properties, such as amplitude and wavelength, at different points in space.

4. Can the separation of variables method be applied to all types of wave equations?

No, the separation of variables method can only be applied to linear wave equations, where the wave function is directly proportional to the external forces acting on it. Nonlinear wave equations, where the wave function is affected by the square or higher powers of the external forces, cannot be solved using this method.

5. How does the solution obtained through separation of variables for the wave equation compare to the actual wave behavior?

The solution obtained through separation of variables is an approximation of the actual wave behavior. It assumes that the medium is homogeneous and that there are no external forces acting on the wave. In reality, there may be other factors that affect the wave's behavior, making the actual solution more complex.

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