Problem 73 - 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. $$u(x, t)=\cos (2(x+c t))$$

Step-by-Step Solution

Answer

Answer: Yes, the function $u(x, t) = \cos(2(x + ct))$ satisfies the one-dimensional wave equation.

Step 1: Compute partial derivatives

First, let's find the first and second partial derivatives of the given function with respect to time (t) and position (x). The given function is: $$u(x, t) = \cos (2(x + ct))$$ First partial derivatives: $$\frac{\partial u}{\partial x} = -2\sin(2(x+ct))$$ $$\frac{\partial u}{\partial t} = -2c\sin(2(x+ct))$$ Second partial derivatives: $$\frac{\partial^2 u}{\partial x^2} = -4\cos(2(x+ct))$$ $$\frac{\partial^2 u}{\partial t^2} = -4c^2\cos(2(x+ct))$$ Now that we have the second partial derivatives, we can insert them into the wave equation and check if it holds true.

Step 2: Verify if the wave equation holds true

Now, let's substitute the second partial derivatives into the wave equation and check if it is true: $$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$ Substituting the values: $$-4c^2\cos(2(x+ct)) = c^2(-4\cos(2(x+ct)))$$ Simplifying the equation, we see that both sides are equal, meaning the given function indeed satisfies the wave equation: $$-4c^2\cos(2(x+ct)) = -4c^2\cos(2(x+ct))$$ Therefore, it's confirmed that the given function, $$u(x, t) = \cos(2(x + ct))$$ is a solution of the one-dimensional wave equation.

Key Concepts

Partial Derivatives

In calculus, partial derivatives are used to analyze functions with multiple variables. These derivatives help us understand how a function changes as one variable changes, while keeping others constant. When examining the wave equation, we often deal with a function like \( u(x, t) \), which represents the displacement of a wave at position \( x \) and time \( t \). We then consider partial derivatives with respect to each variable.

• For the function \( u(x, t) = \cos(2(x+ct)) \), the first partial derivative with respect to \( x \) is \( -2\sin(2(x+ct)) \).
• The first partial derivative with respect to \( t \) is \( -2c\sin(2(x+ct)) \).

Taking the process one step further, the second partial derivatives provide insights into the curvature or acceleration of the wave. They are:

• \( \frac{\partial^2 u}{\partial x^2} = -4\cos(2(x+ct)) \)
• \( \frac{\partial^2 u}{\partial t^2} = -4c^2\cos(2(x+ct)) \)

These derivatives are essential to verify solutions for differential equations, particularly in wave motion scenarios.

Periodic Motion

Periodic motion refers to motion that repeats itself at regular time intervals. In the context of waves, both water and electromagnetic waves exhibit periodic motion. For a function like \( u(x, t) = \cos(2(x+ct)) \), the periodic nature is defined by the cosine function, which oscillates between -1 and 1. This means:

• The displacement of the wave follows a predictable and repeating pattern.
• The term \( 2(x+ct) \) introduces a phase shift, affecting how the wave appears over time and space.
• The speed of the wave, \( c \), influences the rate of oscillation in time.

Overall, periodic motion is a critical concept in understanding how waves behave and propagate in different mediums. It allows us to predict the position and movement of waves over time.

Solutions of Differential Equations

Differential equations are fundamental in describing natural phenomena, especially when it comes to wave motion. A common type of differential equation seen in physics is the wave equation, which involves second-order partial derivatives. Ensuring that a function like \( u(x, t) = \cos(2(x+ct)) \) is a solution involves plugging the second derivatives into the equation and verifying equality.
The wave equation, \( \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \), must hold true for any valid wave function. Once the second partial derivatives are computed, substitution confirms that:\[ -4c^2\cos(2(x+ct)) = c^2(-4\cos(2(x+ct))) \] This simplification shows both sides of the equation are equivalent, verifying that the function is indeed a solution.
Finding solutions to such equations helps in predicting the movement and behavior of waves. By understanding how displacement changes with time and location, we gain insights into wave dynamics as well as practical applications, such as signal transmission and acoustic design.