5.3 - Standing Sound Waves in a Pipe
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Introduction to Standing Sound Waves
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Today we're going to explore standing sound waves in pipes. Can anyone tell me what a standing wave is?
Isn't it a wave that doesn't appear to travel in space?
Exactly! Standing waves are formed by the interference of incident and reflected waves. In the case of pipes, these waves interact at the boundaries to create nodes and antinodes.
What are nodes and antinodes?
Good question! Nodes are points where there is no displacement, while antinodes are points of maximum displacement, where the wave's energy is concentrated.
How is this related to the sound we hear?
The frequencies of these standing waves determine the pitch of the sound we hear! Let's look at the equations for different pipe types.
Frequency Equations for Open-Open Pipes
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For an open-open pipe, the frequencies of the standing waves can be expressed by the equation: $$ f_n = \frac{n v}{2L} $$ What does this tell us?
That the frequency increases with the number of harmonics, or n?
Exactly! And it also tells us that the speed of sound and the length of the pipe play critical roles in determining the frequencies of the harmonics.
So if we have a longer pipe, does that mean a lower frequency?
That's correct! A longer pipe correlates with lower frequencies for each harmonic mode.
And what happens if I change the medium?
Changing the medium affects the speed of sound, thus altering the frequencies too. Letβs move on to open-closed pipes.
Frequency Equations for Open-Closed Pipes
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For an open-closed pipe, the frequency equations change to: $$ f_n = \frac{(2n - 1)v}{4L} $$ Can anyone tell me the significance of the '(2n - 1)'?
It means only odd harmonics are present, right?
Exactly! This is because a closed end must always be a node, which results in the requirement for the odd harmonics only.
So can we get even harmonics at all?
No, in open-closed pipes we only see odd harmonic frequencies due to the constraints at the closed end.
Does that affect the sound we hear?
Yes, these only being odd harmonics can give a different tonality to the sound produced. Let's summarize!
Application and Importance of Standing Sound Waves
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Standing sound waves have practical applications. Can anyone think of some?
Like musical instruments? They use pipes to create sound!
What about organ pipes?
Great examples! Indeed, many musical instruments, as well as some technological applications, rely on these principles of standing waves.
So understanding this can help us in design, right?
Absolutely! Whether it's designing wind instruments or acoustical engineering, knowledge of these concepts is crucial.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
Standing sound waves are formed in pipes by the interference of incident and reflected waves. The section outlines the frequency equations related to different pipe types, highlighting how these waves exhibit distinct modes of vibration based on the length of the pipe.
Detailed
Standing Sound Waves in a Pipe
Standing sound waves occur in different types of pipes due to the interference between incident and reflected sound waves. When sound waves travel through a pipe, they reflect off the boundaries, creating regions of constructive and destructive interference. These regions define points of maximum (antinodes) and minimum displacement (nodes). This section delineates the different frequency equations applicable to various types of pipes:
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Open-Open Pipe: The frequency of the harmonic modes can be described by the equation:
$$ f_n = \frac{n v}{2L}, \, n = 1, 2, ... $$ where $f_n$ is the frequency of the nth harmonic, $v$ is the speed of sound in the medium, and $L$ is the length of the pipe. -
Open-Closed Pipe: The equation for the harmonics in an open-closed pipe differs, given by:
$$ f_n = \frac{(2n - 1)v}{4L} $$ where $n$ is a positive integer (1, 2, 3,...), corresponding to the odd harmonics.
This differentiation arises due to the boundary conditions at the closed end of the pipe, where a node is formed, leading to a distinct harmonic structure compared to an open-open pipe.
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Open-Open Pipe Frequency Equation
Chapter 1 of 2
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Chapter Content
For an open-open pipe, the frequency is given by:
f_n = \frac{n v}{2L}, \quad n=1,2,...
Detailed Explanation
This equation describes the frequencies of standing sound waves in a pipe that is open at both ends. In this case, the frequency (f_n) is directly related to the wave speed (v) and the length of the pipe (L). The variable n represents the harmonic number, which can take any positive integer value (1, 2, 3, ...). Each value of n corresponds to a specific frequency of a standing wave that can fit in the pipe. The pipe allows for the formation of these waves without any ends being fixed, hence 'open-open'.
Examples & Analogies
You can think of an open-open pipe like a swing that can move freely in both directions. Just as the swing can have different ways (or frequencies) to move back and forth depending on how hard you push (in this case, the speed of the wave), the sound waves can take on multiple frequencies depending on how they resonate within the length of the pipe.
Open-Closed Pipe Frequency Equation
Chapter 2 of 2
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Chapter Content
For an open-closed pipe, the frequency is given by:
f_n = \frac{(2n-1)v}{4L}
Detailed Explanation
This equation applies to a pipe that is open at one end and closed at the other. Unlike the open-open pipe, the frequencies (f_n) for the open-closed pipe are given by the formula that includes (2n - 1). The variable n again represents the harmonic number, but in this case, it can only be positive integers 1, 2, 3, etc. This results in specific frequencies for standing sound waves in this configuration, with the closed end creating a node (point of no displacement) and the open end creating an antinode (point of maximum displacement).
Examples & Analogies
Imagine a tube where one end is plugged. When you blow into the open end, the sound waves that form have a different pattern than those in a tube that is open on both ends. This is like playing a closed-end flute compared to a trumpet; each instrument produces its unique sound based on how the air moves inside them.
Key Concepts
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Standing Waves: A result of interference that creates points of no displacement, known as nodes, and points of maximum displacement, known as antinodes.
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Frequency Equation for Open-Open Pipe: Determines the frequencies of standing waves for a pipe open at both ends: f_n = n v / 2L.
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Frequency Equation for Open-Closed Pipe: Determines the frequencies of standing waves for a pipe open at one end and closed at the other: f_n = (2n - 1)v / 4L.
Examples & Applications
An open-open pipe that is 2 meters long will produce harmonics at frequencies f_1 = 87.5 Hz, f_2 = 175 Hz, and f_3 = 262.5 Hz if the speed of sound is 350 m/s.
An open-closed pipe of length 1 meter will produce harmonics at f_1 = 87.5 Hz and f_2 = 262.5 Hz using the same speed of sound.
Memory Aids
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Rhymes
In pipes that are open on each end, many harmonics we can send.
Stories
Imagine playing a flute (open-open) where every note resonates. Now picture a clarinet (open-closed) β only select notes emerge from its ranging sonorous heart.
Memory Tools
Remember: O-O stands for Many (Open-Open = all harmonics), while O-C means Odd (Open-Closed = only odd harmonics).
Acronyms
ON for Open-Closed, all odd, No even allowed.
Flash Cards
Glossary
- Standing Wave
A wave that remains in a constant position, formed by the interference of two traveling waves moving in opposite directions.
- Node
A point along a standing wave where the wave has minimum amplitude where no movement occurs.
- Antinode
A point along a standing wave where the wave has maximum amplitude.
- OpenOpen Pipe
A pipe that is open at both ends, allowing for both ends to vibrate freely.
- OpenClosed Pipe
A pipe that is open at one end and closed at the other, resulting in different harmonic modes.
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