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18. Waves & Sound
Standing Waves
5:49 minutes
Problem 17a
Textbook Question
Textbook QuestionBIO Tendons are, essentially, elastic cords stretched between two fixed ends. As such, they can support standing waves. A woman has a 20-cm-long Achilles tendon—connecting the heel to a muscle in the calf—with a cross-section area of 90 mm^2 . The density of tendon tissue is 1100 kg/m^3 . For a reasonable tension of 500 N, what will be the fundamental frequency of her Achilles tendon?
Verified step by step guidance
1
Step 1: Convert the cross-sectional area of the Achilles tendon from mm^2 to m^2. Since 1 mm^2 = 1e-6 m^2, multiply the given area by 1e-6.
Step 2: Calculate the mass per unit length (\( \mu \)) of the tendon using the formula \( \mu = \text{density} \times \text{cross-sectional area} \). Use the density of tendon tissue and the converted area from step 1.
Step 3: Use the wave speed formula for a string, \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the tendon and \( \mu \) is the mass per unit length calculated in step 2.
Step 4: Calculate the fundamental frequency (first harmonic) of the tendon using the formula for the fundamental frequency of a string fixed at both ends, \( f_1 = \frac{v}{2L} \), where \( L \) is the length of the tendon and \( v \) is the wave speed calculated in step 3.
Step 5: Substitute the values of \( v \) and \( L \) into the formula from step 4 to find the fundamental frequency of the Achilles tendon.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standing Waves
Standing waves occur when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. In the context of a tendon, these waves can form due to the tension and elasticity of the material, creating nodes and antinodes along its length. Understanding standing waves is crucial for analyzing how tendons can vibrate and transmit frequencies.
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Fundamental Frequency
The fundamental frequency is the lowest frequency at which a system can vibrate, determined by its physical properties such as length, tension, and mass per unit length. For a tendon, this frequency can be calculated using the formula f = (1/2L) * √(T/μ), where L is the length, T is the tension, and μ is the mass per unit length. This concept is essential for determining how the tendon will respond to forces.
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Mass per Unit Length
Mass per unit length (μ) is a measure of how much mass is distributed along a given length of an object, calculated as μ = density × cross-sectional area. In the case of the Achilles tendon, knowing the density of the tendon tissue and its cross-sectional area allows us to find μ, which is necessary for calculating the fundamental frequency. This concept links the physical characteristics of the tendon to its vibrational properties.
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