Question

Find the lowest three values of $k^2$ for which the two-dimensional Helmholtz equation $${\mathrm{\nabla }}^2\varphi +k^2\varphi =0$$ has a nontrivial solution inside a ${30}^{\circ }-{60}^{\circ }-{90}^{\circ }$ right triangle whose sides are $a,a\sqrt{3}$, and $2a.$ The boundary condition is that $\varphi =0$ on the perimeter of the triangle.

Short answer

Calculate the eigenvalues: $k_1^2=(\frac{\pi }{a}{)}^2(1+\frac{1}{3}),$ $k_2^2=(\frac{2\pi }{a}{)}^2(1+\frac{1}{3})$ and $k_3^2=(\frac{3\pi }{a}{)}^2(1+\frac{1}{3})$.

Step-by-step solution

Identify the form of the solution

A standard approach to solving the Helmholtz equation is to use separation of variables. Assume a solution of the form $\theta (x,y)=X(x)Y(y)$. Substitute this form into the Helmholtz equation: $abl{a}^2\theta +k^2\theta =0$

Apply Boundary Conditions

The boundary condition is given as $\theta =0$ on the perimeter of the triangle. The symmetry and dimensions of the triangle allow using sine and cosine functions that satisfy these boundary conditions.

Set up and solve the separated equations

Separate the Helmholtz equation into two ordinary differential equations: $X^{″}+k_x^{2}X=0$ and $Y^{″}+k_y^{2}Y=0$. These can be solved with solutions of the form $X(x)=\text{sin}(k_{x}x)$ and $Y(y)=\text{sin}(k_{y}y)$

Fit the geometry of the triangle

Match the solutions to the triangle's dimensions. This involves identifying the quantization of the wave numbers $k_x=\frac{n\pi }{a}$ and $k_y=\frac{m\pi }{a\sqrt{3}}$ with integers 'n' and 'm' that satisfy the conditions.

Calculate the eigenvalues

The wave number $k^2$ for the Helmholtz equation is given by $k^2=k_x^2+k_y^2$. Substitute the quantized values from Step 4: $$k_x^2=(\frac{n\pi }{a}{)}^2$$ and $$k_y^2=(\frac{m\pi }{a\sqrt{3}}{)}^2$$. Sum these terms to find $$k^2=(\frac{n\pi }{a}{)}^2+(\frac{m\pi }{a\sqrt{3}}{)}^2$$

Find the lowest eigenvalues

To find the lowest three values of $k^2$, increment the integers 'n' and 'm' (starting from 1) and calculate the corresponding eigenvalues. Continue until the first three lowest eigenvalues are identified.

Key concepts

separation of variables

To solve the Helmholtz equation, a commonly used technique is the separation of variables. This method assumes that the potential function $\theta (x,y)$ can be written as the product of two functions, one depending only on $x$ and the other only on $y$. Thus, we write $\theta (x,y)=X(x)Y(y)$. When this form is substituted into the equation $abl{a}^2\theta +k^2\theta =0$, we obtain two separate ordinary differential equations (ODEs). By solving these ODEs independently, we can then multiply the solutions to get the full solution to the original partial differential equation (PDE). This method simplifies the problem significantly, turning a PDE into two easier-to-solve ODEs.

boundary conditions

In this problem, the boundary condition is that $\theta =0$ on the perimeter of the triangle. Boundary conditions are crucial because they ensure that the solution fits the physical setup of the problem. For this right triangle, we know $\theta (x,y)$ must satisfy $\theta$ being zero along all three sides. To satisfy this, we choose functions like sine and cosine, which naturally become zero at specific arguments. This ensures our assumed product solution meets the condition $\theta =0$ at the boundaries of the triangle.

eigenvalues and eigenfunctions

When solving the separated ODEs, we get solutions of the form $X(x)=\text{sin}(k_{x}x)$ and $Y(y)=\text{sin}(k_{y}y)$. The constants $k_x$ and $k_y$ are known as eigenvalues, and the functions $X(x)$ and $Y(y)$ are the corresponding eigenfunctions. The eigenvalues must fit the geometry of the problem, often leading to discrete quantized values. In this case, their values determine the wave numbers which satisfy the boundary conditions. The final goal is to find these smallest eigenvalues, denoted by $k^2=k_x^2+k_y^2$, because they represent the fundamental frequencies of the system.

right triangle boundary problem

For the specific boundary problem involving the ${30}^{\text{dgrees}}-{60}^{\text{degrees}}-{90}^{\text{degrees}}$ triangle, we need to tailor our solutions to match the triangle's sides: $a$, $a$, and $a\sqrt{3}$. By matching the solutions for $X(x)$ and $Y(y)$ to these dimensions, we derive the quantized wave numbers $k_x=\frac{n\pi }{a}$ and $k_y=\frac{m\pi }{a\sqrt{3}}$. Here, 'n' and 'm' must be integers to ensure the boundary conditions are met and are incremented to find the lowest eigenvalues. Finding these values involves substituting back into the equation for $k^2$ and calculating until the smallest three values are found.