Problem 80 - Pearson Physics

Question

The human ear canal is much like an organ pipe that is closed at one end (at the tympanic membrane, or eardrum) and open at the other (see Figure 14.20). A typical ear canal has a length of about \(2.4 \mathrm{~cm}\). (a) What are the fundamental frequency of the ear canal and the wavelength of that standing wave? (b) Find the frequency and wavelength of the ear canal's third harmonic. (Recall that the third harmonic in this case is the standing wave with the second-lowest frequency.) (c) Suppose a person has an ear canal that is shorter than \(2.4 \mathrm{~cm}\). Is the fundamental frequency of that person's ear canal greater than, less than, or the same as the value found in part (a)? Explain. [Notice that the frequencies found in parts (a) and (b) correspond closely to the frequencies of enhanced sensitivity in Figure 14.19.]

Step-by-Step Solution

Answer

(a) 3572 Hz, 0.096 m; (b) 10718 Hz, 0.032 m; (c) Fundamental frequency increases.

Step 1: Understanding the Problem

The ear canal behaves like a closed-end organ pipe, which means that the fundamental frequency has a wavelength four times the length of the pipe. This is because a closed-end pipe supports a standing wave with a node at the closed end and an antinode at the open end.

Step 2: Convert Length to Meters

The length of the ear canal is given as 2.4 cm. To use it in calculations, we convert it to meters: 2.4 cm = 0.024 m.

Step 3: Fundamental Frequency and Wavelength Calculation

The fundamental wavelength \(\lambda_1\) for a closed-end pipe is\(4 \times \text{length} = 4 \times 0.024 m = 0.096 m.\)The speed of sound in air is approximately 343 m/s, so the fundamental frequency \(f_1\) is calculated as:\[f_1 = \frac{v}{\lambda_1} = \frac{343}{0.096} \approx 3572\, \text{Hz}.\]

Step 4: Third Harmonic Calculation

The third harmonic \(f_3\) corresponds to the standing wave pattern within the ear canal expressed as 3/4 of a wavelength fitting within the canal. Thus, the third harmonic wavelength \(\lambda_3\) is calculated as:\[\lambda_3 = \frac{4}{3} \times \text{length} = \frac{4}{3} \times 0.024 m = 0.032 m.\]Then, the frequency is:\[f_3 = \frac{v}{\lambda_3} = \frac{343}{0.032} \approx 10718\, \text{Hz}.\]

Step 5: Shortened Ear Canal Frequency Comparison

If the ear canal is shorter than 2.4 cm, the fundamental frequency will increase. This is because the fundamental frequency is inversely proportional to the length, as seen in the formula \(f_1 = \frac{v}{4L}\). Thus, a shorter length results in a higher frequency.

Key Concepts

Fundamental Frequency

The concept of fundamental frequency is crucial in understanding how sound behaves in a closed-end structure like the human ear canal. It refers to the lowest frequency at which a system can naturally vibrate. In the case of the ear canal, which acts like an organ pipe closed at one end, the fundamental frequency occurs when the length of the ear canal equals one quarter of the wavelength of the sound wave.

- **Calculation**: For an ear canal length of 2.4 cm, the fundamental wavelength (\(\lambda_1\)) is calculated to be four times this length, owing to the quarter-wavelength condition. This results in a wavelength of 0.096 meters.
- **Frequency**: The speed of sound in air (343 m/s) helps determine the fundamental frequency (\(f_1\)) by dividing the speed by the wavelength: \(f_1 = \frac{v}{\lambda_1} \approx 3572\, \text{Hz}\). This is the lowest resonant frequency perceived by the ear.

Harmonics

Harmonics are higher frequencies that accompany the fundamental frequency; they are integer multiples of this base frequency. They define the richness or timbre of the sound.

- **Definition**: In the ear canal, which behaves as a closed pipe, the harmonics are determined by the fitting patterns of standing waves within the canal.
- **Third Harmonic**: For the ear canal, the third harmonic is particularly interesting as it is the next prominent frequency. It corresponds to a standing wave with 3/4 of its wavelength fitting the canal. Calculations show that the frequency of this harmonic is approximately 10718 Hz.
These higher frequencies, like the third harmonic, add complexity to sounds we hear and help differentiate between different sound sources.

Sound Waves

Sound waves are oscillations of pressure transmitted through air (or any medium), and they are the primary carriers of sound.

- **Nature**: They are longitudinal waves, meaning that the oscillation of particles is parallel to the direction the wave travels. This creates alternating areas of compression and rarefaction in the medium.
- **Influence on Ear Canal**: Within the ear canal, these sound waves set up standing wave patterns due to reflections off the eardrum (closed end) and the open ear aperture. The ear canal selectively amplifies sound at its resonant frequencies, like the fundamental and third harmonics, enhancing our sensitivity to these frequencies.
Sound waves are integral to how we perceive auditory information through changes in pitch and volume.

Standing Wave

A standing wave is formed when two identical waves traveling in opposite directions interfere with each other, creating nodes and antinodes.
- **In the Ear Canal**: The ear canal supports standing waves because it is closed at the eardrum and open at the ear opening. This configuration allows waves reflecting off the eardrum to meet incoming waves, establishing nodes (points of no movement) and antinodes (points of maximum movement).
- **Characteristics**: For the fundamental frequency, there is one node at the eardrum and an antinode at the open end. Higher harmonics establish additional nodes and antinodes within the canal.
The phenomenon of standing waves is vital in the process of sound amplification and resonance, facilitating the detection of specific frequencies by the ear.

Ear Canal

The ear canal, part of the outer ear, is a tubular passage that leads from the outer ear to the eardrum.
- **Structure**: Approximately 2.4 cm in length, it functions like a closed-end pipe, crucially influencing the acoustics of hearing by naturally amplifying specific frequencies.
- **Acoustic Function**: The shape and length of the ear canal determine its resonant frequencies, like the fundamental frequency and its harmonics, making it selectively sensitive to certain pitches.

- **Impact of Canal Length Change**: If the ear canal is shorter, the fundamental frequency is higher because the wavelength of the resonant sound is shorter, given the formula \(f_1 = \frac{v}{4L}\).
Understanding the acoustics of the ear canal is key to grasping how it contributes to the clarity and quality of sounds we perceive.