Problem 74 - Calculus Early Transcendentals

Question

Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension (for example, a wave on a string) wave motion is governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. $$u(x, t)=5 \cos (2(x+c t))+3 \sin (x-c t)$$

Step-by-Step Solution

Answer

Answer: Yes, the function u(x, t) = 5cos(2(x + ct)) + 3sin(x - ct) is a solution of the given wave equation.

Step 1: Compute the first partial derivatives of u(x, t)

To find the first partial derivatives of u(x, t) with respect to x and t, we compute: $$ \frac{\partial u}{\partial x} = -10\sin(2(x + ct)) + 3\cos(x - ct) $$ and $$ \frac{\partial u}{\partial t} = -10c\sin(2(x + ct)) - 3c\cos(x - ct) $$

Step 2: Compute the second partial derivatives of u(x, t)

Now, we compute the second partial derivatives of u(x, t) with respect to x and t: $$ \frac{\partial^2 u}{\partial x^2} = -20\cos(2(x + ct)) - 3\sin(x - ct) $$ $$ \frac{\partial^2 u}{\partial t^2} = -20c^2\cos(2(x + ct)) - 3c^2\sin(x - ct) $$

Step 3: Substitute the second partial derivatives in the wave equation

The wave equation is: $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}$$ Substituting the second partial derivatives, we get: $$-20c^2\cos(2(x + ct)) - 3c^2\sin(x - ct) = c^2(-20\cos(2(x + ct)) - 3\sin(x - ct))$$

Step 4: Verify that the equation holds true

Now, we can observe that the both sides of the equation are equal: $$-20c^2\cos(2(x + ct)) - 3c^2\sin(x - ct) = -20c^2\cos(2(x + ct)) - 3c^2\sin(x - ct)$$ Since the equation holds true, we can conclude that the function u(x, t) = 5cos(2(x + ct)) + 3sin(x - ct) is a solution of the given wave equation.

Key Concepts

Partial Derivatives

In calculus, partial derivatives play a crucial role. They allow us to understand how a function changes with respect to one variable while keeping the other variables constant. Imagine you're holding a piece of fabric - if you pull on one side, the fabric's shape changes. Partial derivatives are like looking at each pull separately.

For the wave function \( u(x, t) \), the partial derivative with respect to \( x \) gives you the rate of change in the horizontal direction, affecting the wave's shape. On the other hand, the partial derivative with respect to \( t \) reveals how the wave shape changes over time, almost like observing each frame of a wave moving.

• First Partial Derivatives: Calculating these help in understanding basic changes in the wave function.
• Second Partial Derivatives: These are essential for applying the wave equation, revealing deeper wave dynamics.

In our wave equation, these derivatives help verify if a function is a solution. It's like confirming that a guessed pattern in a puzzle fits perfectly. By matching both sides of the wave equation using these derivatives, we confirmed our wave function solution was correct.

Wave Function

A wave function describes how waves, like sound or light, behave in space and time. If you think of throwing a stone into a pond, the ripples spreading out are like the wave function in motion.

In the example given, the wave function is \( u(x, t) = 5 \cos(2(x + ct)) + 3 \sin(x - ct) \). This function shows how the wave displaces at any point \( x \) in space and any time \( t \). It's essentially the blueprint of how a wave looks at any given moment.

• Components: It combines cosine and sine waves, each with its own amplitude and frequency.
• Cosine Term: Represents one part of the wave, defining certain peaks and troughs.
• Sine Term: Provides variation, altering the overall wave pattern dynamically.

Together, these functions explain the holistic motion of the wave - much like describing the path and speed of ripples in water.

Solutions to Differential Equations

Solutions to differential equations are fundamental in predicting natural phenomena, from weather patterns to wave behaviors. It's like finding a balance point where all factors are satisfied.

Differential equations involve functions and their derivatives, forming a framework to solve complex dynamic systems. The wave equation \( \frac{\partial^{2} u}{\partial t^{2}} = c^{2} \frac{\partial^{2} u}{\partial x^{2}} \) expresses this relationship for traveling waves.

• Verification Process: By substituting potential solutions, you ensure the equation holds true, confirming correct solutions.
• Practical Uses: Understanding how waves function aids in areas like engineering and physics, where accurate predictions are essential.
• Interdependence: The equality in the wave equation means both sides rely on each other, showing the interdependence of wave speed, shape, and movement.

Knowing that our function \( u(x, t) = 5 \cos(2(x + ct)) + 3 \sin(x - ct) \) satisfies this equation shows it's an accurate depiction of wave behavior, aligning theory with reality.