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Chapter 12: Problem 78

Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are 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. $$u(x, t)=5 \cos (2(x+c t))+3 \sin (x-c t)$$

Short Answer

Expert verified
Answer: Yes, the function satisfies the given wave equation.

Step by step solution

01

Find the second partial derivatives

Let's find the second partial derivatives of the function \(u(x,t)\) with respect to time \(t\) and position \(x\). First, the second partial derivative of \(u\) with respect to \(x\): $$\frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x}(\frac{\partial u}{\partial x})$$ $$\frac{\partial u}{\partial x} = -10 \sin(2(x+ct))+3 \cos(x-ct)$$ $$\frac{\partial^2 u}{\partial x^2} = -20 \cos(2(x+ct))-3 \sin(x-ct)$$ Next, the second partial derivative of \(u\) with respect to \(t\): $$\frac{\partial^2 u}{\partial t^2} = \frac{\partial}{\partial t}(\frac{\partial u}{\partial t})$$ $$\frac{\partial u}{\partial t} = -10c \sin(2(x+ct))-3c \cos(x-ct)$$ $$\frac{\partial^2 u}{\partial t^2} = -20c^2 \cos(2(x+ct))+3c^2 \sin(x-ct)$$
02

Substitute derivatives into the wave equation

Now, let's substitute the second partial derivatives we found in the previous step into the given wave equation: $$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$ Plug in the functions: $$-20c^2 \cos(2(x+ct))+3c^2 \sin(x-ct) = c^2(-20 \cos(2(x+ct))-3 \sin(x-ct))$$
03

Compare the left and right sides of the wave equation

We can see that both sides of the equation, after substitution, are the same: $$-20c^2 \cos(2(x+ct))+3c^2 \sin(x-ct) = -20c^2 \cos(2(x+ct))+3c^2 \sin(x-ct)$$ Since the left side equals the right side of the equation, it shows that the function $$u(x,t)=5 \cos(2(x+ct))+3 \sin(x-ct)$$ is a solution to the given wave equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
Partial derivatives give us a way to look at the change of multivariable functions. Here, our function depends on two variables: position \(x\) and time \(t\). When we take the partial derivative of our wave function \(u(x, t)\) with respect to \(x\), we see how the wave changes as it moves through space, but holding time constant. Similarly, taking the partial derivative with respect to \(t\) shows how the wave evolves over time, without changing position.To check if a function solves the wave equation, we need to find the second partial derivatives and substitute them into the equation. This is crucial because it tells us whether the change in time matches the change in space, scaled by the wave speed squared \(c^2\). Understanding this relationship allows us to determine if a function truly represents wave motion.
Wave Motion
Wave motion describes how waves travel through medium or space. In the case of the wave equation provided in the exercise, it expresses the relationship between the change in time and space for a wave phenomenon.In mathematical terms, wave motion can be described using functions that rely on trigonometric components, like \(\cos\) and \(\sin\). These components capture the oscillatory nature of waves, meaning they repeat themselves in a predictable pattern. The wave equation characterizes this by comparing changes over time and space, elegantly captured with second partial derivatives. When both parts of the equation match鈥攁s demonstrated in the solution鈥攖he function successfully represents wave motion. The constant \(c\) represents the speed of this movement, controlling how fast the wave oscillates through medium or space.
Periodic Functions
Periodic functions are functions that repeat their values in regular intervals or periods. Waves, whether on the ocean surface or in an electromagnetic spectrum, exhibit periodic behavior.The function \(u(x, t) = 5 \cos(2(x+ct)) + 3 \sin(x-ct)\) from the exercise is a classic example. The \(\cos\) and \(\sin\) terms highlight the periodic nature, with the wave's behavior repeating as the angles inside these functions reach multiples of \( 2\pi \).These periodic functions are at the core of describing wave phenomena. They help model scenarios where wave heights or displacements cyclically repeat over time and space, ensuring we adequately capture the essence of wave dynamics. Understanding the period and amplitude of these functions is key to analyzing waves in real-world applications.

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Most popular questions from this chapter

Flow in a cylinder Poiseuille's Law is a fundamental law of fluid dynamics that describes the flow velocity of a viscous incompressible fluid in a cylinder (it is used to model blood flow through veins and arteries). It says that in a cylinder of radius \(R\) and length \(L,\) the velocity of the fluid \(r \leq R\) units from the center-line of the cylinder is \(V=\frac{P}{4 L \nu}\left(R^{2}-r^{2}\right),\) where \(P\) is the difference in the pressure between the ends of the cylinder and \(\nu\) is the viscosity of the fluid (see figure). Assuming that \(P\) and \(\nu\) are constant, the velocity \(V\) along the center line of the cylinder \((r=0)\) is \(V=k R^{2} / L,\) where \(k\) is a constant that we will take to be \(k=1.\) a. Estimate the change in the centerline velocity \((r=0)\) if the radius of the flow cylinder increases from \(R=3 \mathrm{cm}\) to \(R=3.05 \mathrm{cm}\) and the length increases from \(L=50 \mathrm{cm}\) to \(L=50.5 \mathrm{cm}.\) b. Estimate the percent change in the centerline velocity if the radius of the flow cylinder \(R\) decreases by \(1 \%\) and the length \(L\) increases by \(2 \%.\) c. Complete the following sentence: If the radius of the cylinder increases by \(p \%,\) then the length of the cylinder must increase by approximately __________ \(\%\) in order for the velocity to remain constant.

Consider the curve \(\mathbf{r}(t)=\langle\cos t, \sin t, c \sin t\rangle\) for \(0 \leq t \leq 2 \pi,\) where \(c\) is a real number. a. What is the equation of the plane \(P\) in which the curve lies? b. What is the angle between \(P\) and the \(x y\) -plane? c. Prove that the curve is an ellipse in \(P\).

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the x y-, x z^{-}, \text {and } y z-\text {traces, when they exist. c. Sketch a graph of the surface. $$-\frac{x^{2}}{3}+3 y^{2}-\frac{z^{2}}{12}=1$$

Identify and briefly describe the surfaces defined by the following equations. $$y^{2}-z^{2}=2$$

In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) The data may be plotted as a scatterplot in the \(x y\) -plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Let the equation of the best-fit line be \(y=m x+b,\) where the slope \(m\) and the \(y\) -intercept \(b\) must be determined using the least squares condition. First assume that there are three data points \((1,2),(3,5),\) and \((4,6) .\) Show that the function of \(m\) and \(b\) that gives the sum of the squares of the vertical distances between the line and the three data points is $$ \begin{aligned} E(m, b)=&((m+b)-2)^{2}+((3 m+b)-5)^{2} \\ &+((4 m+b)-6)^{2} \end{aligned}. $$ Find the critical points of \(E\) and find the values of \(m\) and \(b\) that minimize \(E\). Graph the three data points and the best-fit line.
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