Problem 36 - Advanced Engineering Mathematics

Question

Verify that the given function satisfies the wave equation: $$a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}$$ $$ u=\cos (x+a t)+\sin (x-a t) $$

Step-by-Step Solution

Answer

The function satisfies the wave equation on both sides.

Step 1: Write Down the Given Function

The given function is \( u(x,t) = \cos(x + at) + \sin(x - at) \). We will use this to find the necessary derivatives.

Step 2: Compute the First Partial Derivatives

Calculate the first partial derivative of \( u \) with respect to \( x \) and \( t \):1. \( \frac{\partial u}{\partial x} = -\sin(x + at) + \cos(x - at) \)2. \( \frac{\partial u}{\partial t} = -a\sin(x + at) - a\cos(x - at) \)

Step 3: Compute the Second Partial Derivatives

Find the second partial derivatives with respect to both \( x \) and \( t \):1. \( \frac{\partial^2 u}{\partial x^2} = -\cos(x + at) - \sin(x - at) \)2. \( \frac{\partial^2 u}{\partial t^2} = -a^2\cos(x + at) - a^2\sin(x - at) \)

Step 4: Verify the Wave Equation

Check if the wave equation \( a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \) holds by substituting the second partial derivatives computed:- Left side: \( a^2 \frac{\partial^2 u}{\partial x^2} = a^2(-\cos(x + at) - \sin(x - at)) \)- Right side: \( \frac{\partial^2 u}{\partial t^2} = -a^2\cos(x + at) - a^2\sin(x - at) \)Both sides are equal, confirming that the given function satisfies the wave equation.

Key Concepts

Understanding Partial Derivatives

Partial derivatives are used to measure how a multi-variable function changes as one of its variables changes, while keeping the other variables constant.When dealing with functions of several variables, like in our wave equation, partial derivatives become crucial for understanding how the function behaves in relation to each variable independently.
For the function provided, \( u(x,t) = \cos(x + at) + \sin(x - at) \), we need to calculate the partial derivatives with respect to both \( x \) and \( t \).

• First partial derivative with respect to \( x \), \( \frac{\partial u}{\partial x} \), is found by differentiating the function while treating \( t \) as a constant.
• First partial derivative with respect to \( t \), \( \frac{\partial u}{\partial t} \), is found by treating \( x \) as a constant.

Understanding and finding these partial derivatives is fundamental, as they lay the foundation for more advanced concepts like second derivatives.

Exploring Second Derivatives

Second derivatives provide insights into the curvature and concavity of a function, giving us a deeper understanding of its behavior.
In the context of the wave equation, second derivatives play a key role, as the equation involves second derivatives with respect to both space \( x \) and time \( t \).
When we move to the second derivatives of \( u(x,t) \), we are essentially finding the rate of change of the already determined first derivatives:

• The second partial derivative with respect to \( x \) is \( \frac{\partial^2 u}{\partial x^2} = -\cos(x + at) - \sin(x - at) \).
• The second partial derivative with respect to \( t \) is \( \frac{\partial^2 u}{\partial t^2} = -a^2\cos(x + at) - a^2\sin(x - at) \).

These second derivatives are crucial for verifying whether the function satisfies the wave equation, as they directly form the components on each side of the equation.

Mathematical Verification of the Wave Equation

Mathematical verification is about checking that the solution or function satisfies a given condition—in this case, the wave equation.
The wave equation is a second-order linear partial differential equation, expressed as \( a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \).
To confirm our function fulfills this equation, we'd compute the second derivatives of the function with respect to \( x \) and \( t \) and substitute them into the wave equation:

• On the left-hand side, substitute \( a^2 \frac{\partial^2 u}{\partial x^2} = a^2(-\cos(x + at) - \sin(x - at)) \).
• On the right-hand side, substitute \( \frac{\partial^2 u}{\partial t^2} = -a^2\cos(x + at) - a^2\sin(x - at) \).

By comparing both sides, we see they are equal, hence verifying mathematically that the function satisfies the wave equation. Ensuring accuracy in each step of finding derivatives and comparing their results is key for successful verification.