Problem 49 - Calculus Early Transcendentals

Question

\(49-54\) Assume that all the given functions have continuous second-order partial derivatives. Show that any function of the form $$z=f(x+a t)+g(x-a t)$$ is a solution of the wave equation $$\frac{\partial^{2} z}{\partial t^{2}}=a^{2} \frac{\partial^{2} z}{\partial x^{2}}$$ \([\)Hint: Let \(u=x+a t, v=x-a t]\)

Step-by-Step Solution

Answer

The function \( z = f(x+at) + g(x-at) \) satisfies the wave equation as both sides are equal when substituted.

Step 1: Understand the Function Form

The function is given as \( z = f(x+at) + g(x-at) \). Here, \( u = x + at \) and \( v = x - at \) are introduced, which are new variables that simplify the problem.

Step 2: Find First-Order Derivatives

Calculate the first-order derivatives of \( z \) with respect to \( t \) and \( x \).For \( t \):\[ \frac{\partial z}{\partial t} = f'(u)\cdot a + g'(v)\cdot (-a) \]For \( x \):\[ \frac{\partial z}{\partial x} = f'(u) + g'(v) \]

Step 3: Find Second-Order Derivatives

Calculate the second-order derivatives.For \( t \):\[ \frac{\partial^{2} z}{\partial t^{2}} = f''(u)\cdot a^2 + g''(v)\cdot a^2 \]For \( x \):\[ \frac{\partial^{2} z}{\partial x^{2}} = f''(u) + g''(v) \]

Step 4: Substitute into the Wave Equation

Substitute the second derivatives into the wave equation:\[ \frac{\partial^{2} z}{\partial t^{2}} = a^2 \frac{\partial^{2} z}{\partial x^{2}} \]LHS: \( f''(u)\cdot a^2 + g''(v)\cdot a^2 \)RHS: \( a^2(f''(u) + g''(v)) \)

Step 5: Verify the Equality

Observe that both sides simplify to the same expression:\[ f''(u)\cdot a^2 + g''(v)\cdot a^2 = a^2f''(u) + a^2g''(v) \]This verifies that the assumed form satisfies the wave equation.

Key Concepts

Partial Derivatives

Partial derivatives can be thought of as a way to see how a function changes when only one variable is modified, keeping others constant. Consider the function you've been working with: - Given as \( z = f(x+at) + g(x-at) \).The partial derivative of \( z \) with respect to \( t \) (denoted \( \frac{\partial z}{\partial t} \)) is determined by seeing how \( z \) changes as \( t \) changes, while keeping \( x \) constant.Similarly, the partial derivative with respect to \( x \) (\( \frac{\partial z}{\partial x} \)) represents the change in \( z \) when \( x \) changes, while keeping \( t \) constant. These calculations are foundational in understanding the behavior of functions in multiple dimensions, crucial for solving the wave equation.

Second-Order Derivatives

Second-order derivatives help describe how the rate of change itself changes. Here's how they relate to your function:- They involve taking partial derivatives twice.In your work, you calculated \( \frac{\partial^2 z}{\partial t^2} \) and \( \frac{\partial^2 z}{\partial x^2} \). This means you're looking at the acceleration or the curvature of the function in terms of both \( t \) and \( x \).These second-order derivatives are vital in the wave equation because they help assess how the wave propagates over time and space, which is what ultimately verifies if a solution is valid.

Wave Solutions

Wave solutions, like the one you're examining, model how waves travel through various media. In this context, you have:- A solution \( z = f(x+at) + g(x-at) \), where \( f \) and \( g \) represent wave fronts moving in opposite directions.The terms \( x + at \) and \( x - at \) imply a medium that's moving at speed \( a \). When plugged into the wave equation, their behavior showcases fundamental wave properties, such as speed and direction.Understanding these solutions can provide insight into real-world phenomena, such as sound or water waves. They allow us to predict and explain how waves behave under different conditions.

Mathematical Proof

Mathematical proof is the method by which you confirm that a statement or solution is true. In the exercise mentioned, the goal was to verify whether the function of a specified form satisfies the wave equation:- By substituting the second-order derivatives into the equation, the proof shows that both sides are equal.With both sides simplifying to \( a^2(f''(u) + g''(v)) \), it confirms the validity of the solution for the wave equation. Such proofs are a critical part of mathematical work as they ensure that equations and solutions are consistent with established rules.Learning to construct and understand mathematical proofs not only aids in solving equations but also deepens comprehension of mathematical principles.