Problem 158 - Calculus Volume 3

Question

The basic wave equation is \(f_{t t}=f_{x x}\). Verify that \(f(x, t)=\sin (x+t) \quad\) and \(\quad f(x, t)=\sin (x-t) \quad\) are solutions.

Step-by-Step Solution

Answer

Both functions verify the wave equation: \( f(x,t) = \sin(x + t) \) and \( f(x,t) = \sin(x - t) \) satisfy \( f_{tt} = f_{xx} \).

Step 1: Understand the Wave Equation

The wave equation is given by:\[ f_{tt} = f_{xx} \]This means that the second time derivative of the function equals the second spatial derivative. We need to show that both \( f(x, t) = \sin(x + t) \) and \( f(x, t) = \sin(x - t) \) satisfy this equation.

Step 2: Differentiate the First Function with Respect to Time

Take the function \( f(x, t) = \sin(x + t) \). First, find the first derivative with respect to \( t \):\[ f_t(x, t) = \cos(x + t) \]Now, find the second derivative with respect to \( t \):\[ f_{tt}(x, t) = -\sin(x + t) \]

Step 3: Differentiate the First Function with Respect to Space

Now, differentiate \( f(x, t) = \sin(x + t) \) with respect to \( x \):\[ f_x(x, t) = \cos(x + t) \]Find the second derivative:\[ f_{xx}(x, t) = -\sin(x + t) \]

Step 4: Verify the First Function is a Solution

Check if the second derivatives are equal:\[ f_{tt}(x, t) = -\sin(x + t) = f_{xx}(x, t) \]Both sides of the equation are equal, verifying that \( f(x, t) = \sin(x + t) \) is a solution.

Step 5: Differentiate the Second Function with Respect to Time

Consider the function \( f(x, t) = \sin(x - t) \). First, find the first derivative with respect to \( t \):\[ f_t(x, t) = -\cos(x - t) \]Now, find the second derivative with respect to \( t \):\[ f_{tt}(x, t) = -(-\sin(x - t)) = \sin(x - t) \]

Step 6: Differentiate the Second Function with Respect to Space

Differentiate \( f(x, t) = \sin(x - t) \) with respect to \( x \):\[ f_x(x, t) = \cos(x - t) \]Find the second derivative:\[ f_{xx}(x, t) = -\sin(x - t) \]

Step 7: Verify the Second Function is a Solution

Check if the second derivatives are equal:\[ f_{tt}(x, t) = \sin(x - t) = f_{xx}(x, t) \]Both sides of the equation are equal, verifying that \( f(x, t) = \sin(x - t) \) is a solution.

Key Concepts

Partial Derivatives

When dealing with the wave equation, partial derivatives come into play. Partial derivatives help us understand how a function changes with respect to one variable while keeping the other constant. Let's break it down using the function \(f(x, t) = \sin(x+t)\).

• First Partial Derivative (Time): Find \(f_t(x, t)\) while treating \(x\) as a constant. Calculate the derivative of \(\sin(x+t)\) with respect to \(t\), which gives us \(\cos(x+t)\).
• Second Partial Derivative (Time): Take another derivative with respect to \(t\). The result is \(f_{tt}(x, t) = -\sin(x+t)\).

We perform similar steps when finding the partial derivatives with respect to \(x\). Understanding these partial derivatives is essential for verifying solutions to the wave equation.

Solutions of Differential Equations

In the realm of calculus, especially with wave equations, confirming solutions involves understanding differential equations. Differential equations like the basic wave equation \( f_{tt} = f_{xx} \) describe relationships involving rates of change. Here's what makes them interesting:

• Equation Context: The wave equation specifies that the acceleration (second derivative with respect to time) equals the spatial curvature (second derivative with respect to space).
• Function Verification: To verify a function as a solution, its second derivatives over time and space must match exactly. For instance, \(f(x, t) = \sin(x+t)\) satisfies this as its \(f_{tt}\) and \(f_{xx}\) give the same result.

Solutions to these equations help us model phenomena like sound waves and vibrations.

Trigonometric Functions in Calculus

Trigonometric functions like sine and cosine frequently appear in calculus when dealing with periodic nature of phenomena. In our wave equation example:

• Periodic Behavior: The function \(f(x, t) = \sin(x+t)\) represents a wave, which is inherently periodic due to its trigonometrically defined nature.
• Derivatives and Trigonometry: Taking derivatives of trigonometric functions is straightforward yet crucial. The derivative of \(\sin\) is \(\cos\), and vice versa with a negative sign, leading to patterns that help solve differential equations.

Understanding these trigonometric principles is key to handling wave behaviors in various scientific fields.