Solve Polar Wave Equation with Boundary Conditions

In summary, you have been given a problem to solve involving the azimuthally symmetric wave equation. You have used the separation of variables method to find a solution, but you are unsure how to use the boundary conditions to solve for the value of λ. Some suggestions for solving for λ include using the boundary conditions to eliminate variables and using numerical methods to approximate the value. Good luck!
  • #1
DesZoras
1
0
Hi!

I've been given the following problem to solve:

Consider the azimuthally symmetric wave equation:

[itex]\frac{∂2u}{∂t2}[/itex] = [itex]\frac{c2}{r}[/itex][itex]\frac{∂}{∂r}[/itex](r[itex]\frac{∂u}{∂r}[/itex]) where u(r,0)=f(r), ut(r,0)=g(r), u(0,t)=1 and u(L,0)=0.

Use the separation of variables method to find the solution to this PDE.

Using a substitution of u(r,t)=T(t)R(r), I obtained the following answers:

T=Asin(λt)+Bcos(λt)
R=CJ0([itex]\frac{λ}{c}[/itex]r)+DY0([itex]\frac{λ}{c}[/itex]r)
However, we know as r→0, the Y0 function tends to -∞, thus
R=CJ0([itex]\frac{λ}{c}[/itex]r)

So, u becomes:

u=Dsin(λt)J0([itex]\frac{λ}{c}[/itex]r)+Ecos(λt)J0([itex]\frac{λ}{c}[/itex]r)

My problem is I'm unsure how to use my boundary conditions to solve for what λ should be. Substituting the boundary conditions in, I get:

J0([itex]\frac{λ}{c}[/itex]L)=0
and
Dsin(λt)+Ecos(λt)=1

Does anyone have any suggestions/hints on how to solve for λ here? I don't necessarily need a full answer, even just a hint will do, and then I should be able to figure the rest out myself :smile:
 
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  • #2


Hi there,

Thank you for sharing your problem with us. It seems like you are on the right track with using the separation of variables method to solve this PDE. However, you are correct in noting that you need to use the boundary conditions to solve for the value of λ. Here are a few suggestions that may help you solve for λ:

1. Use the boundary condition u(L,0)=0 to eliminate one of the variables (D or E) in your solution for u. This will give you an equation with only one unknown variable (λ).

2. You can then use the boundary condition u(0,t)=1 to further simplify the equation and solve for λ. You may need to use trigonometric identities to manipulate the equation into a form that allows you to solve for λ.

3. If you are still having trouble solving for λ, you can try using a numerical method such as the bisection method or Newton's method to approximate the value of λ.

I hope these suggestions will help you solve for λ and complete your solution to the PDE. Best of luck!
 

1. What is a polar wave equation?

A polar wave equation is a mathematical equation that describes the propagation of a wave in a circular or spherical coordinate system. It is commonly used in physics and engineering to model the behavior of waves in polar coordinates.

2. How do you solve a polar wave equation?

To solve a polar wave equation, you first need to apply the proper boundary conditions, which define the behavior of the wave at the boundaries of the coordinate system. Then, you can use techniques such as separation of variables or Fourier series to obtain a solution to the equation.

3. What are boundary conditions?

Boundary conditions are mathematical expressions that specify the behavior of a wave at the boundaries of a coordinate system. They are essential for solving a polar wave equation, as they help determine the unique solution to the equation that satisfies the given conditions.

4. Why are boundary conditions important in solving a polar wave equation?

Boundary conditions are important because they help determine the unique solution to a polar wave equation. Without them, the equation would have an infinite number of solutions, making it impossible to accurately model the behavior of a wave in a specific system.

5. What are some common examples of boundary conditions in polar wave equations?

Some common examples of boundary conditions in polar wave equations include fixed boundaries, where the wave amplitude is zero at the boundary, and periodic boundaries, where the wave repeats itself at regular intervals. Other examples include open boundaries, where the wave is allowed to propagate freely, and impedance boundaries, where the wave amplitude is proportional to the derivative of the wave at the boundary.

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