19.5. Solution by Separation of Variables

Assume: u(x, y,t)=X(x)Y(y)T(t)

In this step, we assume that the solution to the wave equation can be separated into three functions: X(x), Y(y), and T(t). Each function depends on a single variable: X depends only on the position along the x-axis, Y depends only on the position along the y-axis, and T depends only on time t. This assumption simplifies the problem by allowing us to focus on one variable at a time.

Think of a fruit salad where each type of fruit represents a separate function. Just as you can enjoy a strawberry, a banana, or an apple independently while still creating a fruit salad, we can analyze the behavior of each function (X, Y, T) separately to understand the entire solution.

Substituting into the wave equation:

d2T ( d2X d2Y ) XY =c2 YT +XT dt2 dx2 dy2

Here, we take the assumption of the solution and substitute it into the two-dimensional wave equation. This gives us an equation that relates the second derivatives of the functions with respect to their respective variables. This substitution is crucial because it transforms the problem into one where we can separate variables.

Imagine a cooking recipe where you need to mix different ingredients. When you substitute the ingredients into your recipe, you start to see how they combine to create a dish. Similarly, substituting the functions into the wave equation allows us to understand how the different parts of our solution interact.

Dividing both sides by XY T:

1 d2T ( 1 d2X 1 d2Y ) =c2 + T dt2 X dx2 Y dy2

After substituting into the wave equation, we divide both sides of the equation by the product XYT. This leads to a separation of the equation into distinct parts that can be analyzed separately based on time, position in the x-direction, and position in the y-direction. This is a pivotal step in using the separation of variables technique.

Consider peeling an onion layer by layer. Each layer represents a different variable in our equation. By dividing the equation this way, we effectively peel away the complexity and focus on each individual layer, making it easier to solve.

Let:

1 d2T =−λ(temporal part) T dt2

1 d2X 1 d2Y λ

• =− (spatial part) X dx2 Y dy2 c2

Split again:

1 d2X 1 d2Y ( λ ) =−μ, =− −μ X dx2 Y d y2 c2

In this step, we introduce the terms λ and μ. These constants represent spatial and temporal separation in our equation. By setting these equations for T, X, and Y, we create three ordinary differential equations (ODEs) that can each be solved independently. This splitting is fundamental to the separation of variables method.

Imagine you are planning a road trip and need to manage your time and fuel separately. By defining time (λ) for your journey and fuel consumption (μ) as two different concerns, you can optimize your trip more efficiently, just as we optimize our equations by handling each part separately.

1. X″ +μX=0, X(0)=X(a)=0 ⇒ X (x)=sin(nπx/a), μ = (nπ/a)2

2. Y″ +νY=0, Y(0)=Y(b)=0 ⇒ Y (y)=sin(mπy/b), ν = (mπ/b)2

3. T″ +λT=0, where λ=c2(μ n+ν m) ⇒ T nm(t)=A nmcos(ω nmt)+B nmsin(ω nmt), with ω =√λ=c √(nπ/a)2 + (mπ/b)2

Here we solve the separate ordinary differential equations obtained in the previous step. The solutions involve trigonometric functions (sine), which are suitable for problems with fixed boundaries (like a membrane). Each solution leads to specific terms in the final expression of the displacement function. The constants A and B relate to the initial conditions of motion.

Think of a musician tuning their instrument. Each string must hit the right pitch (like our solved equations). The correct notes produced from these ‘songs’ reflect how well each segment of the equation meets the boundary conditions, ultimately creating a harmonious overall sound from the individual notes.

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