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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 × 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

An ambulance traveling at \(44 \mathrm{m} / \mathrm{s}\) approaches a car heading in the same direction at a speed of \(28 \mathrm{m} / \mathrm{s}\). The ambulance driver has a siren sounding at \(550 \mathrm{Hz}\). At what frequency does the driver of the car hear the siren?
An aluminum rod, \(1.0 \mathrm{m}\) long, is held lightly in the middle. One end is struck head-on with a rubber mallet so that a longitudinal pulse-a sound wave- -travels down the rod. The fundamental frequency of the longitudinal vibration is \(2.55 \mathrm{kHz}\). (a) Describe the location of the node(s) and antinode(s) for the fundamental mode of vibration. Use either displacement or pressure nodes and antinodes. (b) Calculate the speed of sound in aluminum from the information given in the problem. (c) The vibration of the rod produces a sound wave in air that can be heard. What is the wavelength of the sound wave in the air? Take the speed of sound in air to be \(334 \mathrm{m} / \mathrm{s}\). (d) Do the two ends of the rod vibrate longitudinally in phase or out of phase with each other? That is, at any given instant, do they move in the same direction or in opposite directions?
(a) What is the pressure amplitude of a sound wave with an intensity level of \(120.0 \mathrm{dB}\) in air? (b) What force does this exert on an eardrum of area \(0.550 \times 10^{-4} \mathrm{m}^{2} ?\)
Your friend needs advice on her newest "acoustic sculpture." She attaches one end of a steel wire, of diameter \(4.00 \mathrm{mm}\) and density $7860 \mathrm{kg} / \mathrm{m}^{3},$ to a wall. After passing over a pulley, located \(1.00 \mathrm{m}\) from the wall, the other end of the wire is attached to a hanging weight. Below the horizontal length of wire she places a 1.50 -m-long hollow tube, open at one end and closed at the other. Once the sculpture is in place, air will blow through the tube, creating a sound. Your friend wants this sound to cause the steel wire to oscillate at the same resonant frequency as the tube. What weight (in newtons) should she hang from the wire if the temperature is \(18.0^{\circ} \mathrm{C} ?\)
Find the speed of sound in mercury, which has a bulk modulus of $2.8 \times 10^{10} \mathrm{Pa}\( and a density of \)1.36 \times\( \)10^{4} \mathrm{kg} / \mathrm{m}^{3}.$
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