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18. Waves & Sound
Standing Sound Waves
Problem 55
Textbook Question
A 280 Hz sound wave is directed into one end of the trombone slide seen in FIGURE P17.55. A microphone is placed at the other end to record the intensity of sound waves that are transmitted through the tube. The straight sides of the slide are 80 cm in length and 10 cm apart with a semicircular bend at the end. For what slide extensions s will the microphone detect a maximum of sound intensity?


1
Step 1: Understand the problem. The sound wave will resonate in the trombone slide when the length of the slide corresponds to a multiple of the wavelength of the sound wave. The goal is to find the slide extensions 's' that result in maximum sound intensity, which occurs at resonance.
Step 2: Calculate the wavelength of the sound wave. The frequency of the sound wave is given as 280 Hz. Use the formula for the speed of sound in air, v = f * λ, where v is the speed of sound (approximately 343 m/s at room temperature), f is the frequency, and λ is the wavelength. Rearrange the formula to solve for λ: λ = v / f.
Step 3: Determine the total length of the trombone slide. The trombone slide consists of two straight sections, each 80 cm long, and a semicircular bend with a diameter of 15 cm. The length of the semicircular bend is half the circumference of a circle, calculated as π * diameter. Add these lengths together to find the total length of the slide without extension.
Step 4: Account for the variable slide extension 's'. The total length of the trombone slide becomes L_total = L_straight + L_semi + s, where L_straight is the combined length of the straight sections, L_semi is the length of the semicircular bend, and 's' is the variable extension.
Step 5: Set up the resonance condition. Resonance occurs when the total length of the slide corresponds to a multiple of half the wavelength, i.e., L_total = n * (λ / 2), where n is an integer (1, 2, 3, ...). Solve for 's' by substituting the values of L_straight, L_semi, and λ into the equation and finding the values of 's' that satisfy the resonance condition.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sound Waves
Sound waves are longitudinal waves that propagate through a medium, such as air, by compressing and rarefying the particles in the medium. The frequency of a sound wave, measured in hertz (Hz), determines its pitch. In this scenario, a 280 Hz sound wave is introduced into the trombone slide, which will influence how the wave interacts with the tube's dimensions.
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Resonance
Resonance occurs when a system is driven at its natural frequency, leading to an increase in amplitude. In the context of the trombone slide, certain lengths of the slide will correspond to specific wavelengths of the sound wave, resulting in constructive interference and maximum sound intensity at the microphone. Understanding the relationship between the length of the slide and the wavelength is crucial for determining the slide extensions that produce resonance.
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Wavelength and Frequency Relationship
The relationship between wavelength (λ), frequency (f), and the speed of sound (v) is given by the equation v = fλ. For sound waves in air, the speed of sound is approximately 343 m/s at room temperature. By knowing the frequency of the sound wave (280 Hz), one can calculate the wavelength, which is essential for determining the conditions under which maximum sound intensity will be detected at the microphone.
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Related Practice
Textbook Question
The fundamental frequency of an open-open tube is 1500 Hz when the tube is filled with 0°C helium. What is its frequency when filled with 0°C air?
995
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