Problem 104 - Pearson Physics

Question

The fundamental frequency of an organ pipe that is closed at one end and open at the other end is \(261.6 \mathrm{~Hz}\) (middle \(C\) ). The second harmonic of an organ pipe that is open at both ends has the same frequency. What are the lengths of these two pipes?

Step-by-Step Solution

Answer

The closed pipe is approximately 0.328 meters, and the open pipe is approximately 1.31 meters long.

Step 1: Formula for Fundamental Frequency of Closed Pipe

The fundamental frequency of an organ pipe closed at one end is given by the formula \( f_1 = \frac{v}{4L_c} \), where \( v \) is the speed of sound (approximately 343 m/s in air), and \( L_c \) is the length of the closed pipe. Rearrange to solve for \( L_c \): \( L_c = \frac{v}{4f_1} \).

Step 2: Calculate Length of Closed Pipe

Substitute the given frequency \( f_1 = 261.6 \) Hz into the equation. \( L_c = \frac{343}{4 \times 261.6} = \frac{343}{1046.4} \approx 0.328 \text{ meters} \).

Step 3: Formula for Second Harmonic of Open Pipe

The second harmonic frequency of an organ pipe open at both ends is given by the formula \( f_2 = \frac{2v}{2L_o} = \frac{v}{L_o} \), where \( L_o \) is the length of the open pipe. Since the second harmonic frequency is the same as the fundamental frequency of the closed pipe, set \( f_2 = 261.6 \) Hz equal to \( \frac{v}{L_o} \) and solve for \( L_o \): \( L_o = \frac{v}{f_2} \).

Step 4: Calculate Length of Open Pipe

Substitute \( f_2 = 261.6 \) Hz into the equation. \( L_o = \frac{343}{261.6} \approx 1.31 \text{ meters} \).

Key Concepts

Closed Pipe Frequency

In an organ pipe that is closed at one end, the air column inside is able to resonate at specific frequencies. This is because the closed end creates a node where there is no movement of air particles, and the open end is an anti-node, where there's maximum movement. The fundamental frequency, or first harmonic, of a closed pipe can be calculated using the formula:

• Fundamental Frequency ( f ₁) = \[ \frac{v}{4L_c} \]

Here, \( v \) represents the speed of sound in air, commonly approximated to 343 m/s, and \( L_c \) stands for the length of the closed pipe. You can rearrange this formula to find the length of the pipe,

• Pipe Length ( L _c) = \[ \frac{v}{4f_1} \]

This equation highlights how the frequency relates to the length of the pipe and the speed of sound. The unique aspect of a closed pipe is that it only supports odd harmonics, meaning its second harmonic is actually three times the fundamental frequency.

Open Pipe Frequency

Unlike a closed pipe, an open pipe allows both ends to serve as anti-nodes, where air particles move freely. The fundamental frequency of an open pipe is determined differently, given both ends allow unrestricted movement. The formula for the frequency of any open pipe harmonic is:

• Frequency ( f _n) = \[ \frac{nv}{2L_o} \]

Where \( n \) is the harmonic number, \( v \) is the speed of sound, and \( L_o \) is the length of the open pipe. Significantly, the open pipe supports both even and odd harmonics. For the second harmonic, the formula simplifies and is:

• Second Harmonic ( f ₂) = \[ \frac{v}{L_o} \]

This ability to resonate at both even and odd harmonics means the pipe naturally resonates at half the wavelengths compared to a closed pipe for corresponding harmonics, making these pipes create richer sound patterns and tonal quality.

Fundamental Frequency

The fundamental frequency, also known as the first harmonic, is the lowest frequency at which a system vibrates. It serves as the building block for all other harmonics, effectively dictating the natural pitch of the sound produced by a musical instrument like an organ pipe.

• The fundamental frequency of a closed pipe is \[ f_1 = \frac{v}{4L_c} \]
• The fundamental frequency of an open pipe is \[ f_1 = \frac{v}{2L_o} \]

Both these equations describe how the geometry of the pipe influences its pitch. Given the speed of sound is constant, the key variable remains the length and whether the pipe is closed or open. The relationship shows how tightly sound waves can be packed within the confines of the pipe, affecting the resultant sound frequency.

Speed of Sound in Air

The speed of sound is a critical element that influences the frequency of harmonics in both open and closed pipes. At room temperature, the speed of sound in air is approximately 343 meters per second. This speed can change based on various factors such as air temperature, humidity, and pressure.

• Higher temperatures generally increase the speed of sound, because heated air molecules move faster.
• Increased humidity also slightly raises the speed because water vapor is less dense than dry air, allowing sound waves to travel faster.
• Pressure changes have minimal direct effect under constant temperature conditions, as the pressure tends to affect density and temperature equally.

Understanding the speed of sound is crucial when calculating the frequencies and harmonics of organ pipes. It ensures accurate and consistent results given the conditions in which the instrument operates. This explains why calculations are often slightly adjusted for rooms with different ambient conditions.