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Sound enters the ear, travels through the auditory canal, and reaches the eardrum. The auditory canal is approximately a tube open at only one end. The other end is closed by the eardrum. A typical length for the auditory canal in an adult is about \(2.9\) \(\mathrm{cm} .\) The speed of sound is \(343\) \(\mathrm{m} / \mathrm{s} .\) What is the fundamental frequency of the canal? (Interestingly, the fundamental frequency is in the frequency range where human hearing is most sensitive.)

Short Answer

Expert verified
The fundamental frequency of the auditory canal is approximately 2957 Hz.

Step by step solution

01

Understanding the Problem

In this problem, we are asked to find the fundamental frequency of the auditory canal, which acts like a tube that is open at one end and closed at the other. This type of tube supports a standing wave with a fundamental frequency that creates a quarter wavelength (\(\lambda/4\)) inside the tube.
02

Formula for Fundamental Frequency

The formula for the fundamental frequency \(f\) of a tube open at one end and closed at the other is given by:\[ f = \frac{v}{4L} \]where \(v\) is the speed of sound, and \(L\) is the length of the tube.
03

Plug in Known Values

Insert the given values into the formula. Here, \(v = 343\, \text{m/s}\) and \(L = 2.9\, \text{cm} = 0.029\, \text{m}\). Thus,\[ f = \frac{343}{4 \times 0.029} \]
04

Calculate the Fundamental Frequency

Perform the calculation:\[ f = \frac{343}{0.116} \approx 2956.9 \]So, the fundamental frequency is approximately \(2957\, \text{Hz}\).
05

Conclusion

The fundamental frequency of the auditory canal, calculated as a tube open at one end, is approximately \(2957\) Hz. This frequency lies within the range where human hearing is most sensitive, highlighting the efficiency of the ear's natural design.

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Key Concepts

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

Auditory Canal
The auditory canal is an essential part of the human ear that serves as a pathway for sound waves to reach the eardrum. Shaped like a tube, it is open at one end where it collects sound and closed at the eardrum end where it transmits these vibrations. On average, for adults, the length of this canal is about 2.9 centimeters.
This structure is crucial because it supports the formation of standing waves. Specifically, because of its design, it behaves like a tube open at one end and closed at the other, which is ideal for creating certain sound frequencies that are naturally amplified inside this space.
By understanding the dimensions and properties of the auditory canal, we can predict and calculate how it influences the perception of sound, notably affecting the fundamental frequency—which is the lowest frequency at which the canal vibrates and significantly contributes to how we hear sounds.
Standing Wave
A standing wave is a pattern of waves that stays in a constant position. In the context of the ear, standing waves occur in the auditory canal. This is due to its structure, where sound waves reflect off the closed end (the eardrum) and interact with incoming waves.
When we talk about the fundamental frequency, we're referring to the first harmonic, which is the simplest standing wave pattern that can exist within the canal. In a tube that's open at one end and closed at the other, this pattern occurs when the tube holds one-fourth of a wave—known as a quarter wavelength ( λ/4 ).
This phenomenon is crucial because it establishes the primary frequency at which the auditory canal resonates, aiding in the amplification and transmission of sound to the eardrum and beyond, maintaining the integrity of the sound's pitch as we hear it.
Speed of Sound
The speed of sound is a key factor in determining the fundamental frequency of the auditory canal. It refers to the rate at which sound waves travel through the air and is typically valued at 343 meters per second at room temperature.
This speed is influenced by various factors such as the medium through which the sound travels—air in this case—and environmental conditions like temperature and humidity. For the auditory canal, knowing the speed of sound allows us to calculate the fundamental frequency using the formula: \[ f = \frac{v}{4L} \] Where \( v \) is the speed of sound and \( L \) is the length of the auditory canal.
Understanding how the speed of sound impacts our calculation is vital for determining the frequencies that the auditory canal naturally enhances, thus affecting our overall hearing experience.
Human Hearing Sensitivity
Human hearing sensitivity refers to the range of frequencies that the human ear can perceive most efficiently. This range chiefly falls between 20 Hz and 20,000 Hz, but our ears are especially sensitive to sounds between 2,000 Hz and 5,000 Hz.
Interestingly, the fundamental frequency of the auditory canal, at approximately 2957 Hz, lies right within this highly sensitive range. This greatly enhances how effectively we hear and process sound. The auditory canal's natural ability to resonate at these frequencies amplifies sounds in this range, making speech and music more distinct and comprehensible without additional amplification.
By leveraging the natural acoustics of the ear, we are able to more acutely detect subtleties and nuances in sound, optimizing the ear's functionality in environments rich in auditory information.

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Most popular questions from this chapter

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 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 Hz. What is the lowest harmonic number \(n\) that could be responsible for the "moaning" sound?

The A string on a string bass vibrates at a fundamental frequency of \(55.0 \mathrm{Hz}\). If the string's tension were increased by a factor of four, what would be the new fundamental frequency?

The E string on an electric bass guitar has a length of \(0.628\) \(\mathrm{m}\) and, when producing the note \(\mathrm{E},\) vibrates at a fundamental frequency of \(41.2\) \(\mathrm{Hz}\) Players sometimes add to their instruments a device called a "D-tuner." This device allows the \(\mathrm{E}\) string to be used to produce the note \(\mathrm{D},\) which has a fundamental frequency of \(36.7\) \(\mathrm{Hz} .\) The D-tuner works by extending the length of the string, keeping all other factors the same. By how much does a D-tuner extend the length of the E string?

A string that is fixed at both ends has a length of \(2.50\) \(\mathrm{m}\). When the string vibrates at a frequency of \(85.0\) \(\mathrm{Hz},\) a standing wave with five loops is formed. (a) What is the wavelength of the waves that travel on the string? (b) What is the speed of the waves? (c) What is the fundamental frequency of the string?

Two out-of-tune flutes play the same note. One produces a tone that has a frequency of \(262\) \(\mathrm{Hz}\), while the other produces \(266\) \(\mathrm{Hz}\). When a tuning fork is sounded together with the 262 - Hz tone, a beat frequency of 1 Hz is produced. When the same tuning fork is sounded together with the \(266-\mathrm{Hz}\) tone, a beat frequency of \(3\) \(\mathrm{Hz}\) is produced. What is the frequency of the tuning fork?

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