Standing Sound Waves: Videos & Practice Problems

Longitudinal standing waves, such as sound waves in pipes, differ from transverse standing waves primarily at their endpoints. In an open pipe, both ends are antinodes, while in a closed pipe, one end is a node and the other an antinode. The fundamental frequency for an open pipe is given by , and for a closed pipe, , with allowed values of n being odd integers for closed pipes. Understanding these principles is crucial for solving problems related to acoustics and wave behavior.

Standing Sound Waves

Understanding the behavior of longitudinal standing waves is essential, especially when analyzing sound waves in pipes. Sound waves are classified as longitudinal waves, and their behavior varies depending on whether the pipe is open or closed. In an open pipe, both ends are open, resulting in antinodes at each endpoint. This configuration allows for the formation of standing waves, where the wave pattern resembles that of transverse standing waves, but with key differences in endpoint behavior.

For an open pipe, the fundamental frequency can be calculated using the formula:

\( f_n = \frac{n v}{2L} \)

where \( n \) is the harmonic number (1, 2, 3, ...), \( v \) is the speed of sound (approximately 343 m/s), and \( L \) is the length of the pipe. The wavelength for an open pipe is given by:

\( \lambda_n = \frac{2L}{n} \)

In contrast, a closed pipe has one end closed and the other open. In this case, the open end is an antinode, while the closed end is a node. The fundamental frequency for a closed pipe is determined by the formula:

\( f_n = \frac{n v}{4L} \)

Here, the allowed values for \( n \) are restricted to odd integers (1, 3, 5, ...). The wavelength for a closed pipe is expressed as:

\( \lambda_n = \frac{4L}{n} \)

To illustrate these concepts, consider a pipe that is 5 meters long. For an open pipe, the fundamental frequency (where \( n = 1 \)) can be calculated as follows:

\( f_1 = \frac{1 \times 343}{2 \times 5} = 34.3 \text{ Hz} \)

For a closed pipe, if we want to find the third overtone (which corresponds to \( n = 7 \)), the frequency is calculated using:

\( f_7 = \frac{7 \times 343}{4 \times 5} = 120 \text{ Hz} \)

These calculations highlight the differences in frequency and wavelength between open and closed pipes, emphasizing the importance of understanding the boundary conditions that dictate wave behavior in different environments.

The fundamental frequency of your closed organ pipe is 200 Hz. The second overtone of this pipe has the same frequency as the 3^rd harmonic of an open pipe. What is the length of this open pipe?

• A. 0.85 m
• B. 0.51 m
• C. 0.69 m
• D. 0.43

Answer: (B)

Example 1

In this discussion, we explore the calculation of frequencies associated with standing waves in a closed pipe. The length of the pipe is given as 2.75 meters, which is denoted as \( L \). To find the frequency \( f \) of the standing wave, we first need to determine the harmonic number \( n \) that corresponds to the standing wave pattern in the closed pipe.

For closed pipes, the fundamental frequency and its harmonics can be calculated using the formula:

\[ f_n = n \cdot f_1 \]

where \( f_n \) is the frequency of the \( n \)-th harmonic, and \( f_1 \) is the fundamental frequency. The speed of sound in air is approximately \( v = 343 \, \text{m/s} \). For a closed pipe, the wavelength is determined by the equation:

\[ f_n = \frac{n \cdot v}{4L} \]

In this case, we identify that the closed end of the pipe must have a node, while the open end has an antinode. For the standing wave pattern described, we find that \( n = 3 \), indicating that this is the third harmonic. Substituting the known values into the equation gives:

\[ f_3 = \frac{3 \cdot 343}{4 \cdot 2.75} \approx 93.5 \, \text{Hz} \]

With \( f_3 \) calculated, we can now find the fundamental frequency \( f_1 \) by rearranging the relationship:

\[ f_1 = \frac{f_3}{3} \approx \frac{93.5}{3} \approx 31.17 \, \text{Hz} \]

Thus, the frequency of the standing wave in the closed pipe is approximately 93.5 Hz, while the fundamental frequency is approximately 31.17 Hz. Understanding these concepts is crucial for analyzing wave behavior in closed systems.

Here’s what students ask on this topic:

What is the fundamental frequency of a standing sound wave in an open pipe?

The fundamental frequency of a standing sound wave in an open pipe is given by the equation:

$f=\frac{nv}{2l}$

where $n$ is the harmonic number (1 for the fundamental frequency), $v$ is the speed of sound in the medium (typically 343 m/s in air), and $l$ is the length of the pipe. For the fundamental frequency, $n$ is 1, so the equation simplifies to:

$f=\frac{v}{2l}$

How do standing sound waves differ in open and closed pipes?

Standing sound waves in open and closed pipes differ primarily at their endpoints. In an open pipe, both ends are antinodes, meaning the air particles at the ends oscillate with maximum amplitude. In a closed pipe, one end is a node (where air particles do not move) and the other end is an antinode. This difference affects the harmonic frequencies and wavelengths. For open pipes, the fundamental frequency is given by:

$f=\frac{nv}{2l}$

For closed pipes, the fundamental frequency is:

$f=\frac{nv}{4l}$

where $n$ is an odd integer.

What are the allowed harmonics in a closed pipe?

In a closed pipe, the allowed harmonics are only the odd integers. This is because one end of the pipe is a node and the other end is an antinode. The fundamental frequency (first harmonic) is given by $n=1$. The next allowed harmonics are $n=3$, $n=5$, and so on. The frequency for these harmonics is given by:

$f=\frac{nv}{4l}$

where $v$ is the speed of sound and $l$ is the length of the pipe.

How do you calculate the third overtone in a closed pipe?

To calculate the third overtone in a closed pipe, you need to understand that overtones are the harmonics above the fundamental frequency. In a closed pipe, only odd harmonics are allowed. The fundamental frequency is the first harmonic ($n=1$), the first overtone is the third harmonic ($n=3$), the second overtone is the fifth harmonic ($n=5$), and the third overtone is the seventh harmonic ($n=7$). The frequency is calculated using:

$f=\frac{nv}{4l}$

where $v$ is the speed of sound and $l$ is the length of the pipe.

What is the difference between longitudinal and transverse standing waves?

Longitudinal and transverse standing waves differ in the direction of particle displacement relative to wave propagation. In longitudinal waves, such as sound waves, particles oscillate parallel to the direction of wave travel. In transverse waves, like those on a string, particles oscillate perpendicular to the direction of wave travel. For standing waves, the key difference lies in the boundary conditions. In an open pipe (longitudinal), both ends are antinodes, while in a closed pipe, one end is a node and the other an antinode. In transverse standing waves on a string, both ends are typically nodes. The equations for frequency and wavelength also differ based on these boundary conditions.