Ch 17: Superposition

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 17: Superposition

Problem 21

Chapter 17, Problem 21

A 170-cm-long open-closed tube has a standing sound wave at 250 Hz on a day when the speed of sound is 340m/s . How many pressure antinodes are there, and how far is each from the open end of the tube?

Verified step by step guidance

Step 1: Understand the problem. An open-closed tube supports standing waves where the open end is a displacement antinode (pressure node) and the closed end is a displacement node (pressure antinode). The wavelength of the standing wave is determined by the harmonic condition for an open-closed tube.

Step 2: Recall the harmonic condition for an open-closed tube. The length of the tube (L) is related to the wavelength (λ) by the formula: \( L = \frac{(2n-1)\lambda}{4} \), where \( n \) is the harmonic number (1 for the fundamental, 2 for the first overtone, etc.).

Step 3: Calculate the wavelength of the wave using the wave equation \( v = f \lambda \), where \( v \) is the speed of sound (340 m/s) and \( f \) is the frequency (250 Hz). Solve for \( \lambda \): \( \lambda = \frac{v}{f} \).

Step 4: Determine the number of pressure antinodes. For each harmonic, there is one pressure antinode at the closed end and additional antinodes spaced at intervals of \( \frac{\lambda}{2} \) along the tube. Use the tube length (170 cm or 1.7 m) and the calculated wavelength to find how many antinodes fit within the tube.

Step 5: Calculate the positions of the pressure antinodes. Starting from the closed end (0 m), the positions are spaced at intervals of \( \frac{\lambda}{2} \). List the positions up to the length of the tube (1.7 m).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. In an open-closed tube, the open end allows for maximum displacement (antinodes), while the closed end has no displacement (nodes). Understanding the formation of standing waves is crucial for analyzing sound waves in tubes.

Harmonics in Open-Closed Tubes

In an open-closed tube, only odd harmonics are present due to the boundary conditions imposed by the open and closed ends. The fundamental frequency (first harmonic) has one antinode at the open end and one node at the closed end. Higher harmonics can be calculated as odd multiples of the fundamental frequency, which helps determine the number of pressure antinodes in the tube.

Wave Speed and Frequency Relationship

The speed of a wave in a medium is related to its frequency and wavelength by the equation v = fλ, where v is the wave speed, f is the frequency, and λ is the wavelength. In this scenario, knowing the speed of sound and the frequency allows us to calculate the wavelength, which is essential for determining the positions of the pressure antinodes in the tube.

Recommended video:

Guided course

07:19

Intro to Waves and Wave Speed

Related Practice

Textbook Question

A bass clarinet can be modeled as a 120-cm-long open-closed tube. A bass clarinet player starts playing in a 20° C room, but soon the air inside the clarinet warms to where the speed of sound is 352m/s . Does the fundamental frequency increase or decrease? By how much?

views

Textbook Question

Two loudspeakers emit sound waves along the x-axis. The sound has maximum intensity when the speakers are 20 cm apart. The sound intensity decreases as the distance between the speakers is increased, reaching zero at a separation of 60 cm. What is the wavelength of the sound?

views

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?

views

Textbook Question

What is the thinnest film of MgF₂ (n=1.39) on glass that produces a strong reflection for orange light with a wavelength of 600 nm?

views

Textbook Question

Two loudspeakers in a 20°C room emit 686 Hz sound waves along the x-axis. If the speakers are out of phase, what is the smallest distance between the speakers for which the interference of the sound waves is maximum constructive?

views

Textbook Question

The lowest note on a grand piano has a frequency of 27.5 Hz. The entire string is 2.00 m long and has a mass of 400 g. The vibrating section of the string is 1.90 m long. What tension is needed to tune this string properly?

views