14.3.1. Step 1: Change of Variables

Introduce new variables:

𝜉 = 𝑥 + 𝑐𝑡, 𝜂 = 𝑥 − 𝑐𝑡

In this step, we are redefining our variables to make the mathematical treatment of the wave equation simpler. We introduce two new variables, ξ and η.

• ξ (xi) is defined as the sum of position x and the product of wave speed c and time t (c*t). This represents the position of a wave moving to the right.
• η (eta) represents the position of a wave moving to the left, defined as x minus the same product (x - c*t). By changing variables, we isolate the effects of waves traveling in both directions, which is crucial for solving the wave equation more easily.

Think of this change of variables as changing perspectives when analyzing a busy street. Instead of looking at each car (position) individually, you could imagine two separate lanes (one for cars going a certain direction and another for the opposite direction). This helps you effectively analyze the traffic flow as two distinct groups, making the overall situation easier to understand and manage.

Then, by chain rule:

∂𝑢/∂𝑥 = ∂𝑢/∂𝜉 ∂𝜉/∂𝑥 + ∂𝑢/∂𝜂 ∂𝜂/∂𝑥

∂2𝑢/∂𝑥2 = ∂2𝑢/∂𝜉2 (∂𝜉/∂𝑥)² +2 ∂2𝑢/∂𝜉∂𝜂 (∂𝜉/∂𝑥)(∂𝜂/∂𝑥) + ∂2𝑢/∂𝜂2 (∂𝜂/∂𝑥)²

The chain rule of calculus is applied here to express the partial derivatives of u with respect to x in terms of the new variables ξ and η. This step is essential because we want to reframe our wave equation with these new variables:

• The first equation calculates the first derivative of u with respect to x (du/dx) and uses the derivative of our new variables to express this derivative in terms of ξ and η.
• The second equation calculates the second derivative of u with respect to x (d²u/dx²), again translating this into the new variables ×, using derivatives that involve ξ and η and their relationships to x. These formulations are crucial for simplifying the wave equation going forward.

Imagine you're describing the experience of riding a wave at the beach, where the height of the wave at a position changes over time. Instead of referring to specific locations on the beach, you start talking about how far you've traveled left or right of a marker while simultaneously considering how deep or shallow the water feels. This transforms a potentially complicated series of observations (height at specific points) into clearer, more manageable ones focused on your movement relative to the wave.

Similarly,

∂2𝑢/∂𝑡2 = 𝑐2 (∂2𝑢/∂𝜉2 − 2 ∂2𝑢/∂𝜉∂𝜂 + ∂2𝑢/∂𝜂2)

In this chunk, we are expressing the second derivative of u with respect to time (d²u/dt²) through the transformed variables ξ and η. The process follows a similar chain rule approach:

• Here we calculate how the wave's characteristics change over time by also considering the interactions between the waves moving left (η) and right (ξ). The terms in this equation indicate how the changes in one direction influence the other, thus capturing the wave dynamics thoroughly. This step contributes significantly to the wave equation's simplification.

Consider a situation where you're tracking a ball being tossed upward and then coming down. By analyzing the ball's motion as a function of both its height (up) and position (sideways), you get a fuller picture of its journey. The equation here reflects that complexity as it captures the essence of time affecting the ball's position, similar to how the left-moving and right-moving waves influence one another in the wave equation.

Substitute into the wave equation:

𝑐2(𝑢𝜉𝜉 + 2𝑢𝜉𝜂 + 𝑢𝜂𝜂) = 𝑐2(𝑢𝜉𝜉 − 2𝑢𝜉𝜂 + 𝑢𝜂𝜂)

After rewriting the derivatives in terms of our new variables, we substitute them back into the original wave equation. The wave equation can now be represented entirely in terms of our new variables ξ and η. This substitution is a critical step because it transforms our complex problem into a simpler form:

• The left side represents the changes to the wave's profile coming from one direction, while the right side accounts for the other direction. The wave equation's balance reflects how waveforms interact with each other based on the defined dynamics.

Think of it like a dance where two partners influence each other's movements. When one dancer moves to the left, the other dancer responds by adjusting their position to the right. The wave equation here captures that interaction—one side informs how the other side moves, creating a beautiful balance of motion.

Cancel terms to get:

𝑢𝜉𝜂 = 0

The next step is to simplify our equation by canceling the terms. This leads to the condition that the product of the second derivatives with respect to ξ and η must equal zero. This implies that u can be expressed as a function in a separable manner, leading to the conclusion that:

• u(ξ, η) = F(ξ) + G(η), where F and G are arbitrary functions representing each wave's profile traveling in opposite directions. This transforms our problem into a workable format where we can further derive the complete solution to the wave equation.

Imagine a two-lane highway where two vehicles drive in opposite directions. Each vehicle moves independently, governed by its own rules (speed and direction). When we realize that the movements can be described separately, we simplify our analysis, focusing solely on how each influences traffic flow, instead of trying to understand the chaos of both lanes at once.

Return to original variables:

𝑢(𝑥,𝑡) = 𝑓(𝑥 + 𝑐𝑡) + 𝑔(𝑥 − 𝑐𝑡)

Finally, we revert back to our original variables by translating ξ and η back to their terms of x and t. This brings us back to the form where u(x, t) is now expressed in terms of functions f and g, which encapsulate the behavior of the wave as it travels in either direction. This result showcases the general solution to the wave equation and sets up the stage for applying initial conditions in subsequent steps.

This final step is like flipping back to the original map legend after using an extended reference. You initially traveled through new paths (changes in variables), but now you can express your journey in terms of familiar places (original variables). This way, you have a clearer overview of the entire journey while still understanding how you moved in new directions.

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