Problem 35 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

Yes, the function satisfies the wave equation.

Step 1: Compute the First Partial Derivatives

The given function is \( u = \cos(at) \sin(x) \). To verify if it satisfies the wave equation, we first compute the partial derivatives. Compute \( \frac{\partial u}{\partial x} = \cos(at) \cos(x) \) and \( \frac{\partial u}{\partial t} = -a\sin(at) \sin(x) \).

Step 2: Compute the Second Partial Derivatives

Now, compute the second partial derivatives needed for the wave equation. \( \frac{\partial^2 u}{\partial x^2} = -\cos(at) \sin(x) \) and \( \frac{\partial^2 u}{\partial t^2} = -a^2\cos(at) \sin(x) \).

Step 3: Substitute into the Wave Equation

Substitute the second partial derivatives into the wave equation. Substitute \( -a^2 \cos(at) \sin(x) \) for \( \frac{\partial^2 u}{\partial t^2} \) and \( -\cos(at) \sin(x) \) for \( \frac{\partial^2 u}{\partial x^2} \).

Step 4: Verify Equality

Compare the expressions obtained by substituting into the wave equation: \[ a^2 \left( -\cos(at) \sin(x) \right) = -a^2 \cos(at) \sin(x) \]Since both sides of the equation are equal, the given function satisfies the wave equation.

Key Concepts

Partial Derivatives

Imagine you have a function that depends on more than one variable. In such cases, when we want to find how the function changes as one of these variables changes, we compute a partial derivative. For example, for the function \( u = \cos(at) \sin(x) \), you might be curious about how \( u \) changes with respect to \( x \) while keeping \( t \) constant. This "slicing" method provides a snapshot of the function's curve along that direction.
To find \( \frac{\partial u}{\partial x} \), treat \( t \) as a constant and differentiate \( \sin(x) \) to get \( \cos(at) \cos(x) \). Similarly, to find \( \frac{\partial u}{\partial t} \), treat \( x \) as a constant and differentiate \( \cos(at) \) to derive \( -a \sin(at) \sin(x) \). These initial derivatives are the first step in exploring how the function behaves when influenced by small changes in \( x \) or \( t \).

Second Partial Derivatives

Second partial derivatives let us understand how the rate of change itself changes, akin to how acceleration measures changes in velocity. For the wave equation, these derivatives reveal whether a function maintains a consistent form under the equation's constraints. Let's take a closer look at our function again: \( u = \cos(at) \sin(x) \).
For \( \frac{\partial^2 u}{\partial x^2} \), differentiate \( \frac{\partial u}{\partial x} = \cos(at) \cos(x) \) with respect to \( x \) once more to get \( -\cos(at) \sin(x) \). Similarly, for \( \frac{\partial^2 u}{\partial t^2} \), differentiate \( \frac{\partial u}{\partial t} = -a \sin(at) \sin(x) \) with respect to \( t \) once more to arrive at \( -a^2 \cos(at) \sin(x) \).
These calculations showcase the dynamic nature of changes in the function, echoing how the function aligns with the mathematical model of a wave.

Mathematical Verification

Verification is like being a detective in mathematics, where we check if a function fulfills a certain equation or model. In this scenario, we aim to verify if \( u = \cos(at) \sin(x) \) satisfies the wave equation: \[ a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \] By substituting our derived second partial derivatives, \( -a^2 \cos(at) \sin(x) \) for \( \frac{\partial^2 u}{\partial t^2} \) and \( -\cos(at) \sin(x) \) for \( \frac{\partial^2 u}{\partial x^2} \), we get: \[ a^2 (-\cos(at) \sin(x)) = -a^2 \cos(at) \sin(x) \]
Both sides of the equation match perfectly, confirming that our function meets the wave equation's requirements. This step solidifies our understanding that the given function indeed behaves like a wave solution, validating our previous mathematical procedures.