Problem 75 - 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. \(u(x, t)=A f(x+c t)+B g(x-c t),\) where \(A\) and \(B\) are constants and \(f\) and \(g\) are twice differentiable functions of one variable

Step-by-Step Solution

Answer

Question: Verify that the given solution \(u(x, t) = Af(x+ct) + Bg(x-ct)\) indeed satisfies the one-dimensional wave equation, \(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\). Answer: After finding the second order partial derivatives of \(u(x, t)\) with respect to time and position and substituting them into the wave equation, we see that the equality holds. Therefore, the given solution \(u(x, t) = Af(x+ct) + Bg(x-ct)\) satisfies the one-dimensional wave equation.

Step 1: Find \(\frac{\partial u}{\partial t}\)

To find the first partial derivative of \(u(x, t)\) with respect to time \(t\), we can use the chain rule: \(\frac{\partial u}{\partial t} = A \frac{\partial f(x+ct)}{\partial t} + B \frac{\partial g(x-ct)}{\partial t}\). Now, we further apply the chain rule on both terms: \(\frac{\partial u}{\partial t} = Ac f'(x+ct) - Bc g'(x-ct)\).

Step 2: Find \(\frac{\partial^2 u}{\partial t^2}\)

Now, we compute the second partial derivative of \(u(x, t)\) with respect to time \(t\) by differentiating the expression from Step 1 with respect to \(t\) again. Applying the chain rule, we get: \(\frac{\partial^2 u}{\partial t^2} = Ac^2 f''(x+ct) + Bc^2 g''(x-ct)\).

Step 3: Find \(\frac{\partial u}{\partial x}\)

Next, we find the first partial derivative of \(u(x, t)\) with respect to position \(x\). Using the chain rule, we have: \(\frac{\partial u}{\partial x} = A f'(x+ct) + B g'(x-ct)\).

Step 4: Find \(\frac{\partial^2 u}{\partial x^2}\)

Finally, we compute the second partial derivative of \(u(x, t)\) with respect to position \(x\) by differentiating the expression from Step 3 with respect to \(x\) again. Applying the chain rule, we get: \(\frac{\partial^2 u}{\partial x^2} = A f''(x+ct) + B g''(x-ct)\).

Step 5: Verify that the given solution satisfies the wave equation

Now we have the second order partial derivatives of \(u(x, t)\). We can verify whether the given solution satisfies the wave equation by substituting these partial derivatives into the wave equation: \(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\). Substitute the expressions for the partial derivatives found in steps 2 and 4: \(Ac^2 f''(x+ct) + Bc^2 g''(x-ct) = c^2 (A f''(x+ct) + B g''(x-ct))\). We can now observe that both sides of the equation are equal, meaning that the given solution \(u(x, t) = Af(x+ct) + Bg(x-ct)\) indeed satisfies the one-dimensional wave equation.

Key Concepts

Partial Derivatives

Partial derivatives play a vital role in understanding how functions change with respect to one variable while keeping others constant. In the case of the wave equation, we use partial derivatives to find out how the wave function changes over time, keeping the position constant, and vice versa. When working on the wave equation, we take derivatives of the function \( u(x, t) = Af(x+ct) + Bg(x-ct) \). We start by finding the first partial derivative with respect to \( t \), which involves differentiating with respect to time:

• The result is \( \frac{\partial u}{\partial t} = Ac f'(x+ct) - Bc g'(x-ct) \), which tells us how rapidly the wave changes over time.

Then, we find the second partial derivative with respect to \( t \) to understand the acceleration or the rate of change of the wave's velocity:

• The expression becomes \( \frac{\partial^2 u}{\partial t^2} = Ac^2 f''(x+ct) + Bc^2 g''(x-ct) \).

Similarly, understanding how the wave changes along the position \( x \) involves finding the partial derivatives with respect to \( x \). This gives us insight into the spatial behavior of the wave.

Chain Rule

The Chain Rule is an essential tool in calculus when dealing with the differentiation of composite functions. In the context of the wave equation, where we have functions nested inside other functions (such as \( f(x+ct) \) and \( g(x-ct) \)), the Chain Rule helps us handle these derivatives.Here’s a quick breakdown of how it works for the wave equation:

• When differentiating \( f(x+ct) \), we treat \( x+ct \) as a single variable and multiply by the derivative of its inside function, giving us \( c \cdot f'(x+ct) \).
• For \( g(x-ct) \), similarly, we get \( -c \cdot g'(x-ct) \).

Using the Chain Rule allows us to correctly compute these derivatives to thoroughly solve the wave equation problem. This rule simplifies how we approach the process of calculating higher-order derivatives of complex expressions, which is crucial for demonstrating that a function satisfies a differential equation like the wave equation.

Differentiable Functions

To prove that the given functions are solutions to the wave equation, we assume they are differentiable functions. A function being differentiable means that it is smooth enough to have derivatives.For the wave equation solution, the functions \( f \) and \( g \) need to be twice differentiable:

• This means we must be able to compute both the first and second derivatives of these functions, like \( f'(x+ct) \) and \( f''(x+ct) \).
• Being differentiable is crucial as it ensures that the partial derivative computations will not break due to discontinuities or undefined behaviors.

Differentiable functions provide the mathematical foundation needed to ensure the wave motion modeled by the differential equation is correctly described. This requirement guarantees not only that the function exists but that its behavior is smooth and predictable over time and space.

One-Dimensional Waves

One-dimensional waves, such as a wave moving along a string, are beautifully described by the wave equation. The quintessential form of this equation, \( \frac{\partial^{2} u}{\partial t^{2}} = c^{2} \frac{\partial^{2} u}{\partial x^{2}} \), encapsulates how waves evolve over time and space.In this setting:

• \( u(x, t) \) represents the displacement of the wave at any given point \( x \) over time \( t \).
• The constant \( c \) is the speed of the wave which indicates how fast the wave propagates through the medium.

The solutions to this equation, like \( u(x, t) = Af(x+ct) + Bg(x-ct) \), depict the wave moving in two possible directions at once:

• \( Af(x+ct) \) describes the wave moving in the positive \( x \)-direction.
• \( Bg(x-ct) \) describes the wave moving in the negative \( x \)-direction.

Understanding one-dimensional waves through this frame allows us to break down complex wave behaviors into simpler components, helping visualize and predict how real-world wave phenomena behave.