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Chapter 10: Problem 16

$$f ( x ) = \sin 3 x + 5 \sin 7 x - 2 \sin 13 x$$

Short Answer

Expert verified
The function \( f(x) = \sin 3x + 5\sin 7x -2\sin 13x \) is a complex combination of three sine waves with varying frequencies.

Step by step solution

01

Identify Components of the Function

The function is a combination of three sine functions: \( \sin 3x \), \( 5\sin 7x \), and \( -2\sin 13x \). Each of these functions oscillates between -1 and 1, but at different frequencies: 3, 7, and 13 respectively.
02

Combine the Functions

To combine these functions, simply add or subtract them as specified in the original function. At each value of \( x \), compute the value of each sine function and then add or subtract as appropriate to get the value of the total function.
03

Analyze the Combined Function

The function \( f(x) = \sin 3x + 5\sin 7x -2\sin 13x \) will oscillate in a more complex pattern than any of the individual sine functions. The maximum and minimum values will not necessarily be -1 and 1, and the frequencies of the oscillations will also be determined by a complex combination of the frequencies of the individual sine functions.

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Key Concepts

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

Understanding the Sine Function
The sine function is one of the most fundamental trigonometric functions. It represents a smooth, periodic wave that repeats every 2蟺.
  • The basic sine function is given by the expression \( \sin x \), where \( x \) is an angle usually in radians.
  • This wave has an amplitude of 1, reaching from -1 to 1, and it crosses the x-axis at every multiple of \( \pi \).
  • The graph of the sine function looks like a wave that cycles through its full period every 2蟺 radians.
The sine function is crucial in describing natural phenomena such as sound waves, light waves, and even the motion of pendulums. In our function, \( f(x) = \sin 3x + 5\sin 7x - 2\sin 13x \), each term like \( \sin 3x \) is a sine function with its own frequency and amplitude, adding layers of complexity to the resultant wave.
Exploring Function Combination
Function combination involves manipulating separate functions by adding, subtracting, multiplying, or dividing them. This can create a wholly new function.
  • In our function \( f(x) = \sin 3x + 5\sin 7x - 2\sin 13x \), we combine three distinct sine functions.
  • The operations here are straightforward: addition and subtraction.
  • Each sine term is evaluated separately for any given \( x \), and their values are either added or subtracted.
Combining functions like this can significantly alter the characteristics of the wave, including its amplitude and overall shape. For example, \( 5\sin 7x \) has a larger amplitude than \( \sin 3x \), which means its influence on \( f(x) \) is greater. The negative coefficient in \(-2\sin 13x\) flips its wave downward, adding further complexity.
Dive into Frequency Analysis
Frequency refers to how often the periodic functions repeat per interval. It is a crucial factor in defining the behavior of sine functions and their combinations.
  • For a sine function \( \sin kx \), the frequency is determined by \( k \). Higher values of \( k \) result in more rapid oscillations.
  • In the equation \( f(x) = \sin 3x + 5\sin 7x - 2\sin 13x \), the frequencies are 3, 7, and 13 respectively.
Analyzing the frequencies involved is particularly important when these functions are combined.
  • The resulting function inherits a complex pattern based on the superimposed frequencies.
  • Each component contributes uniquely, leading to a pattern more intricate than any single sine wave could produce alone.
Thus, frequency analysis helps us understand the periodic nature of our combined function, giving insights into its oscillation patterns and helping predict its behavior over time.

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Most popular questions from this chapter

$$\frac{\partial u}{\partial t}=3 \frac{\partial^{2} u}{\partial x^{2}}+x, 0<\mathcal{x}<\pi,t>0,$$ $$u(0, t)=u(\pi, t)=0, \quad t>0,$$ $$u(x, 0)=\sin x,0<\mathcal{x}<\pi$$

Find a solution to the following Neumann problem for an exterior domain: \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}=0, \quad 1< r, \quad-\pi \leq \theta \leq \pi,\) \(\frac{\partial u}{\partial r}(1, \theta)=f(\theta), \quad-\pi \leq \theta \leq \pi,\) \(u(r, \theta) \quad\) remains bounded as \(r \rightarrow \infty\)

\(f(\theta)=\cos ^{2} \theta, \quad-\pi \leq \theta \leq \pi\)

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Vibrating Drum. A vibrating circular membrane of unit radius whose edges are held fixed in a plane and whose displacement \(u(r, t)\) depends only on the radial distance \(r\) from the center and on the time \(t\) is governed by the initial boundary value problem. $$ \frac{\partial^{2} u}{\partial t^{2}} $$ \(\alpha^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)\), \(0\)<\(r\)<1,\( \quad t>0\), $$ u(1, t)=0, \quad t>0 $$ $$ u(r, t) \quad \text { remains finite as } r \rightarrow 0^{+} $$ Show that there is a family of solutions of the form $$ u_{n}(r, t)=\left[a_{n} \cos \left(k_{\pi} \alpha t\right)+b_{n} \sin \left(k_{n} \alpha t\right)\right] J_{0}\left(k_{\pi} r\right) $$ where \(J_{0}\) is the Bessel function of the first kind of order zero (see page 478\()\) and \(0\)
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