Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

The wave equation of physics is the partial differential equation $$ \frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}} $$ where \(c\) is a constant. Show that if \(f\) is any twice differentiable function then $$ y(x, t)=\frac{1}{2}[f(x-c t)+f(x+c t)] $$ satisfies this equation.

Step-by-Step Solution

Answer

The function satisfies the wave equation as both sides are equal under the given conditions.

Step 1: Differentiate with respect to Time

First, compute the derivative of the function \( y(x,t) = \frac{1}{2}[f(x-ct) + f(x+ct)] \) with respect to time \( t \). Use the chain rule for differentiation.\[\frac{\partial y}{\partial t} = \frac{1}{2} \left[ -c f'(x-ct) + c f'(x+ct) \right]\] Substituting ensures the differentiation respects \( c \)'s role as a constant.

Step 2: Differentiate with respect to Time Again

Next, compute the second derivative with respect to time \( t \). Apply the chain rule again to the expression from Step 1.\[\frac{\partial^{2} y}{\partial t^{2}} = \frac{1}{2} \left[ c^{2} f''(x-ct) + c^{2} f''(x+ct) \right] = c^{2} \left[ \frac{1}{2} (f''(x-ct) + f''(x+ct)) \right]\] This results in expressions for the second time derivative.

Step 3: Differentiate with respect to Space

Now, differentiate the function \( y(x,t) \) with respect to space \( x \). Use the chain rule again.\[\frac{\partial y}{\partial x} = \frac{1}{2} \left[ f'(x-ct) + f'(x+ct) \right]\] Differentiating the position shows the combination of effects from both arguments related to \( x \).

Step 4: Differentiate with respect to Space Again

Compute the second derivative with respect to space \( x \). Use the chain rule for the previously found \( \frac{\partial y}{\partial x} \).\[\frac{\partial^{2} y}{\partial x^{2}} = \frac{1}{2} \left[ f''(x-ct) + f''(x+ct) \right]\] You observe this mirrors the form of the second time derivative.

Step 5: Verify the Equation

Now compare \( \frac{\partial^{2} y}{\partial t^{2}} \) and \( c^{2} \frac{\partial^{2} y}{\partial x^{2}} \). We have:\[\frac{\partial^{2} y}{\partial t^{2}} = c^{2} \left[ \frac{1}{2} (f''(x-ct) + f''(x+ct)) \right]\] \[c^{2} \frac{\partial^{2} y}{\partial x^{2}} = c^{2} \left[ \frac{1}{2} (f''(x-ct) + f''(x+ct)) \right]\] Since these are equal, the function \( y(x,t) \) satisfies the wave equation.

Key Concepts

Wave Equation

The wave equation is a central concept in both mathematics and physics. It describes the behavior of waves, such as sound, light, and water waves, which can propagate through different media. The standard form of the wave equation in one dimension is:\[\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}\]This equation consists of second partial derivatives with respect to time \( t \) and space \( x \), indicating changes in wave amplitude across time and space.
The constant \( c \) represents the speed of wave propagation through the medium. A higher \( c \) means that the wave moves faster. Understanding this is key because the wave equation has applications across various fields, such as acoustics, electromagnetism, and fluid dynamics.
By solving the wave equation, we determine the positions and shapes of waves over time, which is essential for engineers and scientists when analyzing wave behavior in complex systems.

Chain Rule

The chain rule is an essential tool in calculus when dealing with composite functions, like those involving the wave equation. It allows us to differentiate functions that are nested within each other. Suppose you have a function \( f(g(x)) \), the chain rule for differentiation states:\[\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)\]In the context of the wave equation, the chain rule is vital when we encounter functions of the form \( f(x-ct) \) or \( f(x+ct) \).
For example, when differentiating \( y(x,t) = \frac{1}{2}[f(x-ct) + f(x+ct)] \) with respect to time, the chain rule helps to determine:

• The derivative of \( f(x-ct) \) results in \( -cf'(x-ct) \) since \( x-ct \) is a composite function.
• Similarly, differentiating \( f(x+ct) \) leads to \( cf'(x+ct) \).

These derivatives capture how the wave's shape shifts over time, a crucial component when proving that a given function satisfies the wave equation.

Second Derivatives

Second derivatives provide information about the curvature or concavity of a function. In the realm of the wave equation, second derivatives describe how the wave function's slope changes over time and space. This is crucial for understanding wave motion.
When differentiating the function \( y(x, t) \) in the wave equation, both second derivatives with respect to time \( \frac{\partial^{2} y}{\partial t^{2}} \) and space \( \frac{\partial^{2} y}{\partial x^{2}} \) are obtained. They are components of the fundamental wave equation:

• The second time derivative, \( \frac{\partial^{2} y}{\partial t^{2}} \), indicates how the wave's velocity changes with time.
• The second spatial derivative, \( \frac{\partial^{2} y}{\partial x^{2}} \), reflects the wave's acceleration along the spatial dimension.

Recognizing these connections allows one to see how waves bend and alter their shape. This is pivotal for accurately modeling the behavior of mechanical and electromagnetic waves in diverse settings.

Physics

Physics is a vast field concerned with understanding the natural world, and wave phenomena are a significant part of it. Waves are a way energy is transferred through different media, with sound waves traveling through air or seismic waves through the earth.
In physics, waves can be classified into different types like mechanical, electromagnetic, and matter waves. The wave equation we've been discussing primarily pertains to mechanical and electromagnetic waves, where understanding wave dynamics is crucial.
For instance, in acoustics, analyzing sound waves via the wave equation helps in enhancing sound quality in auditoriums. In electromagnetism, understanding light waves is vital for developing technologies like lasers and fiber optics.
In essence, physics uses the wave equation to predict and explain the behavior of waves under various physical circumstances, making it an indispensable study area for anyone delving into natural sciences.