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Chapter 5: Problem 25

The frequency of a wave is the number of cycles that occur in 1 s. Adding two sinusoidal functions with similar, but unequal, frequencies results in a function that pulsates, or exhibits beats. Piano tuners often use this phenomenon to help them tune a piano. a) Graph the function \(y=\cos x+\cos 0.9 x\). b) Determine the amplitude and the period of the resulting wave.

Short Answer

Expert verified
The amplitude of the resulting wave is 2, and the period is \( 20\backslashpi \).

Step by step solution

01

- Understand the problem

We are given a function that is the sum of two cosine functions with similar frequencies. We need to graph this function and find the amplitude and period of the resulting wave.
02

- Graphing the function

Use a graphing tool or graphing calculator to plot the function \( y = \cos x + \cos 0.9 x \). Use an appropriate range for x, such as from 0 to 20, to see the behavior of the function over multiple cycles.
03

- Determine the wavelength of each component wave

The function consists of two cosines: \( \cos x \) and \( \cos 0.9 x \). The period of \( \cos x \) is \( 2\backslashpi \), and the period of \( \cos 0.9 x \) is \( \frac{2\backslashpi}{0.9} \).
04

- Find the beat frequency

The beat frequency is the difference between the frequencies of the two cosine functions. The frequency of \( \cos x \) is \( \frac{1}{2\backslashpi} \), and the frequency of \( \cos 0.9 x \) is \( \frac{0.9}{2\backslashpi} \). The beat frequency is then \( \left| \frac{1}{2\backslashpi} - \frac{0.9}{2\backlashpi}\right| = \frac{0.1}{2\backslashpi} \).
05

- Determine the amplitude of the resulting wave

The amplitude of the resultant wave is the sum of the amplitudes of the two cosines. Both cosines have an amplitude of 1, so the maximum amplitude of the resulting wave is 2.
06

- Determine the period of the resulting wave

The period of the resulting wave is determined by the beat frequency. The period is the reciprocal of the beat frequency: \( \frac{1}{\frac{0.1}{2\backslashpi}} = \frac{2\backslashpi}{0.1} = 20\backslashpi \).

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Key Concepts

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

Trigonometric Functions
Trigonometric functions like sine and cosine are fundamental in understanding waves. They describe how a point on a circle rotating at a constant speed projects onto the x-axis (for cosine) or y-axis (for sine). These functions are periodic, meaning they repeat their values in regular intervals. For example, the cosine function is given by \(\text{cos}(x)\), and it completes one full cycle every \(\text{2\pi}\) radians. This periodic nature helps in modeling various wave-related phenomena, from sound waves to light waves, perfectly.
Wave Amplitude
The amplitude of a wave is the height from the central axis to the peak of the wave. It indicates the energy and intensity of the wave. For a function like \(\text{cos}(x)\), the amplitude is 1, because the highest and lowest points of the curve reach +1 and -1, respectively. When you add two waves together, like in the exercise, the amplitudes add up, leading to a combined amplitude. So, if you have \(\text{cos}(x) + \text{cos}(0.9x)\), and each wave has an amplitude of 1, the maximum combined amplitude will be 2.
Wave Period
The period of a wave is the time it takes to complete one full cycle and return to its starting position. For a cosine function \(\text{cos}(x)\), the period is \(\text{2\pi}\). For the function \(\text{cos}(0.9x)\), the period changes because the frequency is different. The period for \(\text{cos}(0.9x)\) becomes \(\frac{2\pi}{0.9}\). When you add two waves with different periods, the resulting wave's period is influenced by the beat frequency, which is based on the difference in their individual frequencies.
Beat Frequency
Beat frequency occurs when two waves of slightly different frequencies interfere with each other. This interference causes a new wave pattern that pulsates or 'beats'. Piano tuners use this principle to tune pianos by listening to the beats created by two similar notes. The beat frequency is calculated as the absolute value of the difference between the two original frequencies. In the exercise, the beat frequency of \(\text{cos}(x)\) and \(\text{cos}(0.9x)\) is \(\frac{0.1}{2\pi}\). This means the beats occur with a frequency of \(\frac{0.1}{2\pi}\), leading to a period of \(\frac{2\pi}{0.1} = 20\pi\).
Graphing Trigonometric Functions
Graphing trigonometric functions helps visualize their behavior. For \(\text{cos}(x)\), the graph is a wave starting at 1, dipping to -1, and returning to 1, completing a cycle every \(\text{2\pi}\). When you graph \(\text{cos}(x) + \text{cos}(0.9x)\), you can see how these two waves interact and create a new waveform. Using a graphing tool allows you to plot these functions and observe the resultant pattern visually. Adjusting the x-axis range, for example from 0 to 20, helps better analyze wave characteristics including amplitude and period. This visual representation is essential for understanding complex wave interactions in physics and engineering.

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Most popular questions from this chapter

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