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Chapter 12: Problem 64

A periodic wave is composed of the superposition of three sine waves whose frequencies are \(36,60,\) and 84 Hz. The speed of the wave is 180 m/s. What is the wavelength of the wave? [Hint: The \(36 \mathrm{Hz}\) is not necessarily the fundamental frequency.]

Short Answer

Expert verified
Answer: The wavelength of the periodic wave is 15 meters.

Step by step solution

01

Find the Fundamental Frequency

Note that the three given frequencies are 36, 60, and 84 Hz. We want to find the fundamental frequency, which is the lowest frequency that divides all of these frequencies. Let's find the greatest common divisor (GCD) of these three numbers, as it corresponds to the fundamental frequency: GCD(36, 60, 84) = 12 Hz So, the fundamental frequency is 12 Hz.
02

Calculate the Wavelength

We can now use the wave speed formula to calculate the wavelength of the wave. The wave speed formula is given by: speed = frequency 脳 wavelength We know the wave speed is 180 m/s, and we just found the fundamental frequency to be 12 Hz. We can rearrange the formula to find the wavelength: wavelength = speed / frequency Now, we can plug in the values: wavelength = 180 m/s / 12 Hz Wavelength = 15 meters The wavelength of the wave is 15 meters.

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Most popular questions from this chapter

A musician plays a string on a guitar that has a fundamental frequency of \(330.0 \mathrm{Hz}\). The string is \(65.5 \mathrm{cm}\) long and has a mass of \(0.300 \mathrm{g} .\) (a) What is the tension in the string? (b) At what speed do the waves travel on the string? (c) While the guitar string is still being plucked, another musician plays a slide whistle that is closed at one end and open at the other. He starts at a very high frequency and slowly lowers the frequency until beats, with a frequency of \(5 \mathrm{Hz}\), are heard with the guitar. What is the fundamental frequency of the slide whistle with the slide in this position? (d) How long is the open tube in the slide whistle for this frequency?
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