PDE : Can not solve Helmholtz equation

In summary: Good luck.In summary, the conversation discusses the problem of solving the Helmholtz equation in the context of an anti-plane elasticity problem. The equation is formulated in the frequency domain and boundary conditions are specified. The process of separation of variables is used to obtain an infinite series solution, but the problem arises in finding the coefficients. After a detailed mathematical derivation, it is concluded that the solution is independent of the x direction and can be represented as a Fourier cosine series with the constant term being equal to half of the given force value.
  • #1
sompongt
2
0
PDE : Can not solve Helmholtz equation

(This is not a homework. I doing my research on numerical boundary integral. I need the analytical solution to compare the results with my computer program. I try to solve this equation, but it not success. I need urgent help.)

I working on anti-plane elasticity problem. The physical problem is described on a rectangular domain,
[itex] 0 \leq x \leq a [/itex] and [itex]0 \leq y \leq b.[/itex]

Let [itex]u_z = u(x,y;t)[/itex] is the displacement function in [itex]z[/itex] direction.

[tex]u_z = u(x,y;t)[/tex] is the displacement function in [itex]z[/itex] direction.
The boundary condition are specified by:

(1) [itex]u = 0[/itex] on the lower edge ([itex]y=0[/itex])

(2) [itex]\text{Traction} = 0[/itex] on the left edge ([itex]x=0[/itex])
(3) [itex]\text{Traction} = 0[/itex] on the right edge ([itex]x=a[/itex])
(4) [itex]\text{Traction} = P e^{i \omega t}[/itex] on the top edge ([itex]y=b[/itex]) in the [itex]z[/itex] direction

This problem can be formulate in the frequency domain to obtain the Helmholtz equation, as below.

The problem seem not difficult. But I can not solve it. Could anybody help me?
Any suggestion are welcome.

Mathematical derivation start here. The problem statement describe as,

\begin{equation}
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial x^2} + k^2 u = 0,
0 \leq x \leq a, 0 \leq y \leq b \text{...(1)}
\end{equation}

where
\begin{equation}
k^2 = \frac{\omega^2}{c^2}
\end{equation}

[itex]\omega[/itex] = angular frequency of the excited force
[itex]c[/itex] = speed of wave = function of material properties

subjected by the following boundary conditions,

\begin{align}
\frac{\partial u}{\partial x}(x=0, y) &= 0 \text{...(2)}\label{bc1}\\
\frac{\partial u}{\partial x}(x=a, y) &= 0 \text{...(3)}\label{bc2} \\
u(x, y = 0) &= 0 \text{...(4)}\label{bc3}\\
\frac{\partial u}{\partial y}(x, y=b) &= \text{constant}\label{bc4} = P \text{...(5)}
\end{align}

Solution:

By separation of variables, let
\begin{equation}
u(x,y) = X(x)Y(y)
\end{equation}

Then, working on the standard process,

\begin{align}
X'' Y + X Y'' + k^2 X Y = 0 \\
\frac{X''}{X} + \frac{Y''}{Y} + k^2 = 0
\end{align}

let

\begin{align}
\frac{X''}{X} &= -\alpha^2 &\rightarrow X''+ \alpha^2 X &= 0 \\
\frac{Y''}{Y} &= -\beta^2 &\rightarrow Y''+ \beta^2 Y &= 0
\end{align}

Relation between [itex]\alpha[/itex] and [itex]\beta[/itex],
\begin{equation}
\alpha ^2 + \beta^2 = k^2 \label{abrelation} \text{...(6)}
\end{equation}

Solve for [itex]X(x)[/itex] and [itex]Y(y)[/itex]

\begin{align}
X(x) &= C_1 \cos \alpha x + C_2 \sin \alpha x \\
Y(y) &= C_3 \cos \beta y + C_4 \sin \beta y
\end{align}

So that,

\begin{align}
u(x,y) &= X(x)Y(y) \\
&= (C_1 \cos \alpha x + C_2 \sin \alpha x)(C_3 \cos \beta y + C_4 \sin \beta y) \label{reduce1}
\end{align}

Determine coefficients [itex]C_1, C_2, C_3, C_4[/itex] by using boundary conditions (BC) eq(2-5),

First, use boundary condition in eq(4):
\begin{align}
u(x,y=0) &= 0 \\
(C_1 \cos \alpha x + C_2 \sin \alpha x)(C_3 \cdot 1 + C_4 \cdot 0) &= 0 \\
C_3(C_1 \cos \alpha x + C_2 \sin \alpha x) &= 0
\end{align}

We obtain,
\begin{equation}
C_3 = 0
\end{equation}

So that, [itex]u(x,y)[/itex] reduce to
\begin{align}
u(x,y) &= C_4 \sin \beta y (C_1 \cos \alpha x + C_2 \sin \alpha x) \\
&= \sin \beta y (C_4C_1 \cos \alpha x + C_4C_2 \sin \alpha x) \\
&= \sin \beta y (C_5 \cos \alpha x + C_6 \sin \alpha x) \label{reduce2} \\
\frac{\partial u}{\partial x} (x,y) &= \alpha \sin \beta y (-C_5 \sin \alpha x + C_6 \cos \alpha x)
\end{align}

Second, use boundary condition in eq(2):
\begin{align}
\frac{\partial u}{\partial x}(x=0,y) &= 0 \\
\alpha \sin \beta y (-C_5 \cdot 0 + C_6 \cdot 1) &= 0 \\
C_6 \cdot \alpha \sin \beta y &= 0
\end{align}

We obtain,

\begin{equation}
C_6 = 0
\end{equation}

So that, [itex]u(x,y)[/itex] reduce to
\begin{align}
u(x,y) &= C_5 \cos \alpha x \sin \beta y \\
\frac{\partial u}{\partial x} (x,y) &= - \alpha C_5 \sin \alpha x \sin \beta y
\end{align}

Third step, use boundary condition in eq(3):

\begin{align}
\frac{\partial u}{\partial x}(x=a,y) &= 0 \\
-\alpha C_5 \sin \alpha a \sin \beta y = 0
\end{align}

Which can be conclude that,

\begin{align}
\sin \alpha a &= 0 \\
\alpha a &= n \pi \\
\alpha_n &= \frac{n \pi}{a}
\end{align}

After we find [itex]\alpha[/itex], we can determine [itex]\beta[/itex] from eq(6)

\begin{align}
\beta_n^2 = k^2 - \alpha_n^2
\end{align}

At this point, we can represent [itex]u(x,y)[/itex] as infinite series by using principle of superposition

\begin{align}
u(x,y) &= \sum_{n = 0}^{\infty} C_n \cos \alpha_n x \sin \beta_n y \\
\frac{\partial u}{\partial y} (x,y) &= \sum_{n = 0}^{\infty} \beta_n C_n \cos \alpha_n x \cos \beta_n y
\end{align}

We can determine the unknowns [itex]C_n[/itex] by using the last boundary condition in eq(5)

\begin{align}
P &= \frac{\partial u}{\partial y} (x,y = b) \\
&= \sum_{n = 0}^{\infty} \beta_n C_n \cos \alpha_n x \cos \beta_n b \\
&= \sum_{n = 0}^{\infty} \underbrace{\left(C_n\beta_n \cos \beta_n b \right)}_{\text{constant} = \bar{C_n}} \cos \alpha_n x \\
&= \sum_{n = 0}^{\infty} \bar{C_n}\cos \alpha_n x \text{...(7)}
\end{align}

Consider [itex]\bar{C_n}[/itex] as coefficient of Fourier cosine series,

\begin{align}
P &= \frac{c_0}{2} + \sum_{ n = 1 }^{\infty} c_n \cos \frac{n\pi x}{a} \\
c_n &= \frac{2}{a} \int_{0}^{a} P \cdot \cos \frac{n\pi x}{a} dx
\end{align}

It is not so hard to find that,
\begin{align}
c_0 &= 2P \\
c_n &= 0
\end{align}

For me, the problem arise here. How can I find [itex]\bar{C_n}[/itex] in eq(7)? Could anybody help me? What point that I am wrong? I feel very headache. Please, please, please help me.
 
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  • #2
Take a step back, you separated variables correctly but you must say that:
[tex]
\frac{X''}{X}+\frac{Y''}{Y}+k^{2}=0\rightarrow \frac{X''}{X}=\mu\rightarrow \frac{Y''}{Y}+k^{2}=-\mu
[/tex]
Start from this point.
 
  • #3
You basically got it. Your equation (7) is setting a constant [itex]P[/itex] equal to a Fourier cosine series. The lowest order term in such a series is a constant, and you even correctly identified the value [itex]c_0=2P[/itex]; if you take half that, this IS your [itex]\bar C_0[/itex] value. Working that back to your original [itex]C_0[/itex], you get [itex]C_0 = \frac{\bar C_0}{\beta_0 \cos(\beta_0 b)}=\frac{P}{k\cos(k b)}[/itex], and all the other terms in the series are zero. This gives your final solution, [itex]u(x,y)=\frac{P}{k\cos(k b)}\sin(k y)[/itex].

Sanity check this to make sure it solves your problem:
1. Solves the PDE? Yes.
2. Satisfies the [itex]\frac{\partial}{\partial x}[/itex] boundary conditions? Yes, it doesn't depend on [itex]x[/itex], so the [itex]x[/itex] partials are zero everywhere.
3. Zero on the bottom boundary? Yes, the sine makes sure that happens.
4. Has the right derivative value at the top boundary? Yes, the constants are such that the function outward normal derivative is equal to [itex]P[/itex] at the top.

Its unsurprising that the solution is independent of [itex]x[/itex]. There is no forcing in that direction, since the top boundary is moving up and down in unison at the frequency [itex]\omega[/itex]. This launches plane waves down the sheet. Any variation in that top boundary condition will launch more interesting waves that move side to side and interact with the side boundaries.
 
  • #4
rajb245 said:
You basically got it. Your equation (7) is setting a constant [itex]P[/itex] equal to a Fourier cosine series. The lowest order term in such a series is a constant, and you even correctly identified the value [itex]c_0=2P[/itex]; if you take half that, this IS your [itex]\bar C_0[/itex] value. Working that back to your original [itex]C_0[/itex], you get [itex]C_0 = \frac{\bar C_0}{\beta_0 \cos(\beta_0 b)}=\frac{P}{k\cos(k b)}[/itex], and all the other terms in the series are zero. This gives your final solution, [itex]u(x,y)=\frac{P}{k\cos(k b)}\sin(k y)[/itex].

Sanity check this to make sure it solves your problem:
1. Solves the PDE? Yes.
2. Satisfies the [itex]\frac{\partial}{\partial x}[/itex] boundary conditions? Yes, it doesn't depend on [itex]x[/itex], so the [itex]x[/itex] partials are zero everywhere.
3. Zero on the bottom boundary? Yes, the sine makes sure that happens.
4. Has the right derivative value at the top boundary? Yes, the constants are such that the function outward normal derivative is equal to [itex]P[/itex] at the top.

Its unsurprising that the solution is independent of [itex]x[/itex]. There is no forcing in that direction, since the top boundary is moving up and down in unison at the frequency [itex]\omega[/itex]. This launches plane waves down the sheet. Any variation in that top boundary condition will launch more interesting waves that move side to side and interact with the side boundaries.

rajb245, thank you very much.
 
  • #5


I understand your frustration and urgency in finding the solution to this problem. However, it is important to note that the Helmholtz equation is a well-known and extensively studied PDE in mathematics and physics. It is used to describe wave propagation in various physical systems, and its analytical solutions are only known for a few simple cases.

Based on your boundary conditions and the specific problem you are trying to solve, it is unlikely that an analytical solution to the Helmholtz equation exists. In such cases, numerical methods are often used to approximate the solution. These methods involve discretizing the domain and solving the resulting system of algebraic equations.

I suggest that you consult with a mathematician or a numerical analyst who has experience in solving PDEs. They can help you choose an appropriate numerical method and assist you in implementing it to obtain numerical solutions. Additionally, you can also try searching for literature or research papers on similar problems to get an idea of the solutions that have been obtained in the past. Good luck with your research.
 

1. Why is it difficult to solve the Helmholtz equation using PDE methods?

The Helmholtz equation is a second-order partial differential equation (PDE) that involves both a spatial variable and a temporal variable. This makes it a more complex equation to solve compared to other PDEs that only involve one variable. Additionally, the Helmholtz equation has many possible solutions, which can make it challenging to determine the correct solution.

2. What techniques are commonly used to solve the Helmholtz equation?

Some common techniques used to solve the Helmholtz equation include separation of variables, finite difference methods, finite element methods, and spectral methods. Each technique has its own advantages and disadvantages, and the choice of method often depends on the specific problem and the desired accuracy of the solution.

3. Can the Helmholtz equation be solved analytically?

In some cases, the Helmholtz equation can be solved analytically using separation of variables. However, this is only possible for simple boundary conditions and specific geometries. In most cases, numerical methods are needed to solve the equation.

4. How does the Helmholtz equation relate to wave propagation?

The Helmholtz equation is commonly used to model wave propagation in various physical systems, such as sound waves, electromagnetic waves, and seismic waves. It describes how a wave evolves in space and time, taking into account the properties of the medium through which the wave is propagating.

5. What are some real-world applications of the Helmholtz equation?

The Helmholtz equation has many practical applications, including the design of antennas and other electromagnetic devices, the study of acoustic waves in different environments, and the analysis of seismological data. It is also used in medical imaging, such as ultrasound and MRI, to model the propagation of waves through the human body.

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