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The eardrum, which transmits vibrations to the sensory organs of your ear, lies at the end of the ear canal, which can be modeled as a 2.5 cm long air column that is open on one end and closed at the other. What frequencies of standing waves can occur within the ear canal? Which are within the range of human hearing?

The eardrum, which transmits vibrations to the sensory organs of your ear, lies at the end of the ear canal, which can be modeled as a 2.5 cm long air column that is open on one end and closed at the other. What frequencies of standing waves can occur within the ear canal? Which are within the range of human hearing?

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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The eardrum, which transmits vibrations to the sensory organs of your ear, lies at the end of the ear canal, which can be modeled as a 2.5 cm long air column that is open on one end and closed at the other. What frequencies of standing waves can occur within the ear canal?
Which are within the range of human hearing?


### Understanding Resonant Frequencies in the Ear Canal **Problem Statement:** The eardrum, which transmits vibrations to the sensory organs of your ear, is situated at the end of the ear canal. This canal can be modeled as a 2.5 cm long air column that is open at one end and closed at the other. We need to determine the frequencies of standing waves that can occur within the ear canal and identify which are within the range of human hearing. **Tasks:** 1. **Sketch the Situation:** - Define all variables. - Include sketches showing relevant resonant waveforms, highlighting their nodes and antinodes. 2. **Relevant Physics Equations:** - Identify and justify the equations applicable to solving this problem. 3. **Solution Application:** - Derive an algebraic expression and calculate numerical answers. - Clearly label the units of your answers and explain your methodology. ### Visual Explanation: - **Diagrams:** - **Air Column Setup:** Depict the 2.5 cm long column, with an open end and a closed end. - **Waveforms:** Illustrate the standing waves, marking nodes (points of no displacement) and antinodes (points of maximum displacement). ### Conceptual Approach: **Physics Concepts:** - Resonant frequencies in an air column open at one end and closed at the other: Such columns support odd harmonics. - **Equation:** \( f_n = \frac{(2n - 1)v}{4L} \) - \( f_n \): Frequency of the nth harmonic - \( v \): Speed of sound in air (approximately 343 m/s at room temperature) - \( L \): Length of the air column (2.5 cm in this case) ### Steps to Solve: 1. **Identify Harmonics:** - Only odd harmonics will exist in the given setup (n = 1, 3, 5,...). 2. **Calculate Frequencies:** - Use the equation to compute the frequencies for each relevant harmonic. - Convert the ear canal length from centimeters to meters for consistency in units. 3. **Frequency Range Analysis:** - Determine which frequencies fall within the typical range of human hearing (approximately 20 Hz to 20,000 Hz). ### Example Calculation: - **First Harmonic (n=1):** \[
Transcribed Image Text:### Understanding Resonant Frequencies in the Ear Canal **Problem Statement:** The eardrum, which transmits vibrations to the sensory organs of your ear, is situated at the end of the ear canal. This canal can be modeled as a 2.5 cm long air column that is open at one end and closed at the other. We need to determine the frequencies of standing waves that can occur within the ear canal and identify which are within the range of human hearing. **Tasks:** 1. **Sketch the Situation:** - Define all variables. - Include sketches showing relevant resonant waveforms, highlighting their nodes and antinodes. 2. **Relevant Physics Equations:** - Identify and justify the equations applicable to solving this problem. 3. **Solution Application:** - Derive an algebraic expression and calculate numerical answers. - Clearly label the units of your answers and explain your methodology. ### Visual Explanation: - **Diagrams:** - **Air Column Setup:** Depict the 2.5 cm long column, with an open end and a closed end. - **Waveforms:** Illustrate the standing waves, marking nodes (points of no displacement) and antinodes (points of maximum displacement). ### Conceptual Approach: **Physics Concepts:** - Resonant frequencies in an air column open at one end and closed at the other: Such columns support odd harmonics. - **Equation:** \( f_n = \frac{(2n - 1)v}{4L} \) - \( f_n \): Frequency of the nth harmonic - \( v \): Speed of sound in air (approximately 343 m/s at room temperature) - \( L \): Length of the air column (2.5 cm in this case) ### Steps to Solve: 1. **Identify Harmonics:** - Only odd harmonics will exist in the given setup (n = 1, 3, 5,...). 2. **Calculate Frequencies:** - Use the equation to compute the frequencies for each relevant harmonic. - Convert the ear canal length from centimeters to meters for consistency in units. 3. **Frequency Range Analysis:** - Determine which frequencies fall within the typical range of human hearing (approximately 20 Hz to 20,000 Hz). ### Example Calculation: - **First Harmonic (n=1):** \[
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