Problem 158 - Calculus 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 \(f(x, t) = \sin(x + t)\) and \(f(x, t) = \sin(x - t)\) are solutions to the wave equation.

Step 1: Find Partial Derivatives for First Function

First, verify the solution for the function \(f(x, t) = \sin(x + t)\). Compute its partial derivatives with respect to \(t\) and \(x\). The first partial derivative with respect to \(t\):\[\frac{\partial}{\partial t} f(x, t) = \cos(x + t)\]The second partial derivative with respect to \(t\):\[\frac{\partial^2}{\partial t^2} f(x, t) = -\sin(x + t)\]The first partial derivative with respect to \(x\):\[\frac{\partial}{\partial x} f(x, t) = \cos(x + t)\]The second partial derivative with respect to \(x\):\[\frac{\partial^2}{\partial x^2} f(x, t) = -\sin(x + t)\]

Step 2: Verify First Equation

For \(f(x, t) = \sin(x + t)\), substitute the second partial derivatives into the wave equation \(f_{tt} = f_{xx}\):\[-\sin(x+t) = -\sin(x+t)\]Both sides are equal, so \(f(x, t) = \sin(x + t)\) is a solution to the wave equation.

Step 3: Find Partial Derivatives for Second Function

Next, verify the solution for the function \(f(x, t) = \sin(x - t)\). Compute its partial derivatives with respect to \(t\) and \(x\). The first partial derivative with respect to \(t\):\[\frac{\partial}{\partial t} f(x, t) = -\cos(x - t)\]The second partial derivative with respect to \(t\):\[\frac{\partial^2}{\partial t^2} f(x, t) = -\sin(x - t)\]The first partial derivative with respect to \(x\):\[\frac{\partial}{\partial x} f(x, t) = \cos(x - t)\]The second partial derivative with respect to \(x\):\[\frac{\partial^2}{\partial x^2} f(x, t) = -\sin(x - t)\]

Step 4: Verify Second Equation

For \(f(x, t) = \sin(x - t)\), substitute the second partial derivatives into the wave equation \(f_{tt} = f_{xx}\):\[-\sin(x-t) = -\sin(x-t)\]Both sides are equal, so \(f(x, t) = \sin(x - t)\) is also a solution to the wave equation.

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus, useful for analyzing functions of several variables. In the context of partial derivatives, we focus on how a function changes as one of the variables it's dependent on changes, while the others are held constant.
In analyzing the functions given in the wave equation problem, \(f(x, t) = \sin(x + t)\) and \(f(x, t) = \sin(x - t)\), partial derivatives with respect to both \(x\) and \(t\) are calculated.
Here, for each function, we compute:

• The first partial derivative with respect to \(t\), which measures rate of change concerning time \(t\).
• The second partial derivative with respect to \(t\), which gives us the way the rate of change itself changes as \(t\) varies.
• The first partial derivative with respect to \(x\), which evaluates how the function changes across space \(x\).
• The second partial derivative with respect to \(x\), indicating how that spatial rate of change evolves.

Once these derivatives are found, they become the key ingredients in the solution of the differential equation. The analysis shows that if the second derivatives with respect to \(t\) and \(x\) are equal, the function is a solution to the wave equation.

Solutions to Differential Equations

Solving differential equations involves finding a function that satisfies these equations, which describe relationships involving rates of change of quantities.
The wave equation \(f_{tt} = f_{xx}\) is a classic example of a partial differential equation with broad applications, including in physics to describe waves on a string or sound waves in air.
To solve this, we test candidate solutions—like \(f(x, t) = \sin(x + t)\) and \(f(x, t) = \sin(x - t)\).
By calculating the second partial derivatives, we validate these functions against the wave equation. If both sides match, the function is indeed a solution.

• For \(f(x, t) = \sin(x + t)\), the second derivatives are both \(-\sin(x + t)\).
• Similarly, for \(f(x, t) = \sin(x - t)\), they are \(-\sin(x - t)\).

Both functions satisfy the equality \(f_{tt} = f_{xx}\), confirming them as valid solutions.

Mathematical Verification

Mathematical verification ensures that the conclusions or solutions derived in equations match the stated problems or conditions.
Here, the verification process confirms that certain functions are indeed solutions to the wave equation. We do this by substituting the calculated second partial derivatives back into the original equation \(f_{tt} = f_{xx}\).
When we substitute our derived second partial derivatives for the functions \(f(x, t) = \sin(x + t)\) and \(f(x, t) = \sin(x - t)\), both sides of the equation equate, showing correctness.
Hence:

• For our function \(f(x, t) = \sin(x + t)\), the equation holds as: \(-\sin(x + t) = -\sin(x + t)\).
• For \(f(x, t) = \sin(x - t)\), it similarly holds as: \(-\sin(x - t) = -\sin(x - t)\).

This step of verification is crucial in mathematics to ensure that solutions are not only derived but also correct and logically consistent with the given conditions of the problem.