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Chapter 17: Problem 49

A person hums into the top of a well and finds that standing waves are established at frequencies of \(42,70.0,\) and 98 Hz. The frequency of \(42\) \(\mathrm{Hz}\) is not necessarily the fundamental frequency. The speed of sound is \(343\) \(\mathrm{m} / \mathrm{s} .\) How deep is the well?

Short Answer

Expert verified
The well is approximately 3.06 meters deep.

Step by step solution

01

Understand the Frequency Series

The standing waves form a series of harmonic frequencies. Given frequencies are 42 Hz, 70 Hz, and 98 Hz. Here, 42 Hz is the first in this sequence but not necessarily the fundamental. Let's denote the difference between consecutive frequencies as the fundamental, or the first harmonic frequency.
02

Determine Frequency Differences

Calculate the differences between consecutive frequencies: - From 42 Hz to 70 Hz: 70 - 42 = 28 Hz- From 70 Hz to 98 Hz: 98 - 70 = 28 HzSince both differences are 28 Hz, this frequency represents the fundamental frequency, or the first harmonic (\(f_1 = 28 \text{ Hz}\)).
03

Calculate the Length of the Well

The fundamental frequency of a closed-end air column (such as a well) is given by the formula: \[ f = \frac{v}{4L} \]where \(f = 28 \text{ Hz} \) is the fundamental frequency, \(v = 343 \text{ m/s} \) is the speed of sound, and \(L\) is the length of the well. Rearrange to solve for the length of the well:\[ L = \frac{v}{4f} \]\[ L = \frac{343}{4 \times 28} \approx 3.06 \text{ m} \]

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Key Concepts

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

Standing Waves
When a person hums into a well, sound waves reflect off the bottom and interfere with incoming waves. This interference creates standing waves. Unlike typical waves that travel in one direction, standing waves appear to stay in one place.

Standing waves are characterized by nodes and antinodes. Nodes are points where there is no motion, while antinodes have maximum motion. In the context of a well, the closed bottom creates a node, and the open top allows an antinode.
  • A node occurs where destructive interference happens, cancelling out motion.
  • An antinode occurs where constructive interference happens, enhancing motion.
Understanding these properties helps us see why certain frequencies resonate within the well.
Harmonic Frequencies
Harmonic frequencies in a standing wave system are integral multiples of the fundamental frequency. In the exercise, 28 Hz is identified as the fundamental frequency, with higher frequencies being its harmonics.

Each harmonic frequency represents a different standing wave mode. Harmonics are essentially the different tones or pitches we hear.
  • First harmonic (fundamental frequency): Produces the simplest wave pattern.
  • Second harmonic: Has a frequency twice that of the first.
  • Third harmonic: Frequency is three times the fundamental, and so on.
This pattern is useful in analyzing wave phenomena, including musical instruments and acoustical spaces. Recognizing the differences between harmonics helps determine the fundamental frequency, as different harmonics produce different musical notes.
Fundamental Frequency
The fundamental frequency is the lowest frequency at which a system resonates. It is crucial for understanding the system's response to vibration. In this case, the fundamental frequency of 28 Hz helps define the acoustical properties of the well.

This frequency correlates with the simplest standing wave pattern. In general terms, it can be considered the system's base tone.
  • Fundamental frequency forms the foundation for all other harmonics.
  • In musical contexts, it's often referred to as the "first harmonic."
  • It's directly related to the physical dimensions of the resonating system.
Knowing the fundamental frequency allows us to solve for system attributes, such as length, by implying that multiples of this frequency reproduce similar patterns.
Speed of Sound
The speed of sound is how fast sound waves travel through a medium, such as air. For the exercise, the speed of sound is given as 343 m/s. This constant value is crucial when calculating the resonating length of the well.

Sound speed is affected by several factors, such as temperature, air pressure, and humidity. For consistent results, it's important to assume standard conditions unless changes are specified.
  • At 20掳C, the speed of sound in air is about 343 m/s.
  • Higher temperatures generally increase the speed of sound.
  • Knowledge of sound speed allows for the calculation of distances based on time taken for sound to reflect.
In many acoustics problems, the speed of sound allows for the transition from time-based to distance-based understanding, thereby enabling measurement of physical parameters, like the depth of a well.

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Most popular questions from this chapter

Two loudspeakers on a concert stage are vibrating in phase. A listener is \(50.5 \mathrm{m}\) from the left speaker and \(26.0 \mathrm{m}\) from the right one. The listener can respond to all frequencies from 20 to \(20000 \mathrm{Hz},\) and the speed of sound is \(343 \mathrm{m} / \mathrm{s} .\) What are the two lowest frequencies that can be heard loudly due to constructive interference?

A thin 1.2 -m aluminum rod sustains a longitudinal standing wave with vibration antinodes at each end of the rod. There are no other antinodes. The density and Young's modulus of aluminum are, respectively, \(2700\) \(\mathrm{kg} / \mathrm{m}^{3}\) and \(6.9 \times 10^{10} \mathrm{N} / \mathrm{m}^{2} .\) What is the frequency of the rod's vibration?

The range of human hearing is roughly from twenty hertz to twenty kilohertz. Based on these limits and a value of \(343\) \(\mathrm{m} / \mathrm{s}\) for the speed of sound, what are the lengths of the longest and shortest pipes (open at both ends and producing sound at their fundamental frequencies) that you expect to find in a pipe organ?

One method for measuring the speed of sound uses standing waves. A cylindrical tube is open at both ends, and one end admits sound from a tuning fork. A movable plunger is inserted into the other end at a distance \(L\) from the end of the tube where the tuning fork is. For a fixed frequency, the plunger is moved until the smallest value of \(L\) is measured that allows a standing wave to be formed. Suppose that the tuning fork produces a \(485-\mathrm{Hz}\) tone, and that the smallest value observed for \(L\) is \(0.264 \mathrm{m} .\) What is the speed of sound in the gas in the tube?

A vertical tube is closed at one end and open to air at the other end. The air pressure is \(1.01 \times 10^{5}\) Pa. The tube has a length of \(0.75\) \(\mathrm{m} .\) Mercury (mass density \(=13600\) \(\mathrm{kg} / \mathrm{m}^{3}\) ) is poured into it to shorten the effective length for standing waves. What is the absolute pressure at the bottom of the mercury column, when the fundamental frequency of the shortened, air-filled tube is equal to the third harmonic of the original tube?
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