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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 \(98 \mathrm{Hz}\). 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 6.125 meters deep.

Step by step solution

01

Identify Possible Harmonics

Standing waves occur at specific frequencies that are harmonics of a fundamental frequency, meaning the given frequencies might be multiples of this fundamental frequency. Since these frequencies are equally spaced, they form an arithmetic sequence with a common difference of 28 Hz, which suggests the fundamental frequency could be 14 Hz.
02

Calculate the Wavelengths

The frequencies are 42 Hz, 70 Hz, and 98 Hz. If we assume 14 Hz as the fundamental frequency, then one of these is likely to be the second harmonic (2nd overtone), third harmonic (3rd overtone), or fourth harmonic. Determine the wavelengths for each by using the speed of sound formula. For example, for 42 Hz:\[ \text{Wavelength at } 42 \text{Hz} = \frac{343}{42} \approx 8.167 \text{m} \]
03

Determine the Mode Shape

The mode shape gives us the relation between well depth and the wavelength. If the well acts like a closed-end tube (one open end), the wavelengths for each harmonic will be related to the well's depth by odd multiples of a quarter wavelength. Therefore, if 42 Hz is the 3rd harmonic:\[ \text{Depth of well} = \frac{3 \times 343}{4 \times 42} \] Assuming it's the third harmonic (i.e., two full wavelengths plus one-quarter wavelength form the wave pattern).
04

Calculate the Well's Depth

Using the relation for the third harmonic, the depth of the well can be calculated as:\[ \text{Depth} = \frac{3 \times 343}{4 \times 42} = 6.125 \text{m} \] Thus, by determining the proper mode shape and harmonic form, the well's depth is estimated.

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Key Concepts

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

Harmonics
When we talk about harmonics, we are referring to the specific frequencies at which standing waves can form. In musical terms, these are the overtones that resonate along with the fundamental frequency. Harmonics occur at integer multiples of the fundamental frequency. For example, if the fundamental frequency is 14 Hz, the harmonics could be 28 Hz (2nd harmonic), 42 Hz (3rd harmonic), and so on.

In the context of the problem, harmonics are important since they guide us to understand which frequencies will produce standing waves in the well. By identifying that the given frequencies form an arithmetic sequence with a difference of 28 Hz, we can infer that the initial assumption of 42 Hz not necessarily being fundamental is correct. Thus, this can help identify the fundamental frequency and harmonics present.
Fundamental Frequency
The fundamental frequency is the lowest frequency at which a system naturally vibrates to create a standing wave-like state. It is often referred to as the first harmonic. All other harmonics are based upon this fundamental frequency.

In the problem provided where we've got frequencies of 42 Hz, 70 Hz, and 98 Hz, identifying the fundamental frequency becomes crucial to solving the problem. Since these frequencies differ by 28 Hz, it suggests the possibility of a fundamental frequency at 14 Hz, causing the 42 Hz to be the 3rd harmonic. Understanding this is key to connecting how the overall sequence of harmonics relates back to the system's vibrational characteristics.
Speed of Sound
The speed of sound is a critical concept, especially when discussing standing waves. It is the rate at which sound waves travel through a medium. Commonly, in air at room temperature, it is approximately 343 m/s.

In this exercise, the speed of sound is utilized to calculate the wavelength of sound in the well. When given the frequency of sound, you can find the wavelength using the formula: \[ \text{Wavelength} = \frac{\text{Speed of Sound}}{\text{Frequency}} \]The knowledge of this speed helps translate between frequency and wavelength, enabling the determination of the physical characteristics of the well where the sound is traveling.
Wavelength Calculation
Wavelength calculation is a straightforward yet essential step to understanding the relationship between frequency and the spatial dimensions of the medium that supports wave motion, such as the well in this problem. Knowing the wavelength allows us to make connections between the concepts of harmonics and the dimensions of the medium in which the waves are sustained.

To find the wavelength of a particular frequency, you use the familiar equation: \[ \text{Wavelength} = \frac{\text{Speed of Sound}}{\text{Frequency}} \]For instance, with a frequency of 42 Hz and the speed of sound at 343 m/s, the wavelength would be approximately 8.167 meters. By understanding how wavelength ties into harmonics and the physical setup, such as the well, you can deduce various properties like the depth of the well through these calculations. This connection is essential in the given problem, where the concept of wavelength helps directly to estimate the well's depth by relating to the mode shapes involved.

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Most popular questions from this chapter

Two wires, each of length \(1.2 \mathrm{m},\) are stretched between two fixed supports. On wire \(A\) there is a second-harmonic standing wave whose frequency is \(660 \mathrm{Hz}\). However, the same frequency of \(660 \mathrm{Hz}\) is the third harmonic on wire B. Find the speed at which the individual waves travel on each wire.

A pipe open only at one end has a fundamental frequency of 256 Hz. A second pipe, initially identical to the first pipe, is shortened by cutting off a portion of the open end. Now, when both pipes vibrate at their fundamental frequencies, a beat frequency of \(12 \mathrm{Hz}\) is heard. How many centimeters were cut off the end of the second pipe? The speed of sound is \(343 \mathrm{m} / \mathrm{s}\)

A thin \(1.2-\mathrm{m}\) 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?

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?

To review the concepts that play roles in this problem, consult Multiple- Concept Example \(4 .\) Sometimes, when the wind blows across a long wire, a low-frequency "moaning" sound is produced. This sound arises because a standing wave is set up on the wire, like a standing wave on a guitar string. Assume that a wire (linear density \(=\) \(0.0140 \mathrm{kg} / \mathrm{m}\) ) sustains a tension of \(323 \mathrm{N}\) because the wire is stretched between two poles that are \(7.60 \mathrm{m}\) apart. The lowest frequency that an average, healthy human ear can detect is \(20.0 \mathrm{Hz}\). What is the lowest harmonic number \(n\) that could be responsible for the "moaning" sound?
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