Solving the Helmholtz Equation Using the Finite Analytic Method: A Step-by-Step Walk Through

What if we could solve complicated wave patterns by thinking locally inside little squares?

That's exactly what the Finite Analytic Method (FAM) does. Even though this comes from engineering and advanced math, we’re going to break it down to something you can follow with just algebra and logical steps.

In this blog, I’ll show you step by step how to build a solution to the Helmholtz equation — this is an equation that describes wave behaviors, like sound waves, water waves, or even electromagnetic waves. And best of all: you don’t need calculus to understand the flow of this method.

Let’s dive in.

Step 1: What Are We Trying to Solve?

The Helmholtz equation is like the "go-to" rulebook for how steady waves behave. Written out, it looks like this:

∂²u/∂x² + ∂²u/∂y² + k²u = 0

But since we aren’t working with calculus here, let’s think of it this way:

  • The letter u is the quantity we care about (like wave height at a point).

  • The x and y are positions in space.

  • The k is like a "wave frequency number" — it tells us how wiggly the wave is.

Our goal is to figure out what u is at different points in space.

Step 2: Assume a Pattern for the Solution

Instead of guessing random numbers, let’s assume that u(x, y) behaves like a combination of patterns made by hyperbolic functions (you can think of these like fancy versions of exponential growth or decay).

We assume this pattern for every small square (cell) in our grid:

u(x, y) = A * cosh(λx) * cosh(μy)
        + B * cosh(λx) * sinh(μy)
        + C * sinh(λx) * cosh(μy)
        + D * sinh(λx) * sinh(μy)

Here:

  • A, B, C, D are numbers we will figure out.

  • cosh() and sinh() are just special kinds of functions that show wave-like behavior.

Step 3: Use Points on the Grid

To find A, B, C, D, we look at points around the cell we care about.
Imagine a small square:

  • East point: u_E = u(x + Δx/2, y)

  • West point: u_W = u(x - Δx/2, y)

  • North point: u_N = u(x, y + Δy/2)

  • South point: u_S = u(x, y - Δy/2)

We’ll use these to write equations that let us solve for A, B, C, and D.

Step 4: Derive Our Equation for Each Cell

By plugging the points into our assumed pattern, and doing some algebra, we eventually get this neat-looking rule for the value of u at the center of the cell (we'll call it u_P):

C1 * (u_E + u_W) + C2 * (u_N + u_S) + C3 * u_P = 0

Here’s what the C's mean:

C1 = (1 / Δx²) * (tanh(μ * Δy / 2) / (μ / 2))
C2 = (1 / Δy²) * (tanh(λ * Δx / 2) / (λ / 2))
C3 = -2 * (C1 + C2) - k²

These are just numbers based on our grid size (Δx, Δy), the wave number k, and a few hyperbolic functions.

You don’t have to know what tanh() means — just know it’s like a fancy slope calculator for waves.

Step 5: Add the Boundaries (Edges of Our Grid)

Next, we handle what happens at the edges of our area.

  • If we know the value of u at the boundary, we just use it.

  • If we know the slope or how the wave behaves at the boundary, we adjust our formulas.

This is just like how you might know the height of a wave at the shore or how fast it's changing.

Step 6: Let’s Try with Numbers

Let's pick some numbers for fun:

Δx = Δy = 0.1
k = 10
λ = μ = i * k / √2

(The "i" here is the imaginary number, but you can treat it like a constant for now.)

Now compute:

λ * Δx / 2 = i * (10 / √2) * 0.1 / 2

Find C1, C2, and C3 using the formulas above.

Once we have C1, C2, and C3, we build our final equation for each cell:

C1 * (u_E + u_W) + C2 * (u_N + u_S) + C3 * u_P = 0

We repeat this for every cell in the grid

Step 7: Solving the Grid

Finally, we collect all these equations together into a big system, like this:

[C] * {u} = {b}

  • [C] is a big matrix of numbers (from C1, C2, C3).

  • {u} is our unknown list of wave values at each point.

  • {b} contains information from our boundary points.

We use a computer to solve this — it spits out the values of u at every point.

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