Question

A closed organ pipe has fundamental frequency \(100 \mathrm{~Hz}\). What frequencies will be produced if its other end is also opened? (A) \(200,400,600,800 \ldots \ldots\) (B) \(200,300,400,500 \ldots \ldots\) (C) \(100,300,500,700 \ldots \ldots\) (D) \(100,200,300,400 \ldots \ldots\)

Short answer

When the other end of the organ pipe is opened, the frequencies produced are 200 Hz, 400 Hz, 600 Hz, 800 Hz, and so on. The correct answer is (A).

Step-by-step solution

Understanding the formula for fundamental frequency

The fundamental frequency of a closed organ pipe is given by \(f_{closed}=\frac{v}{4L}\), where \(v\) is the speed of sound and \(L\) is the length of the pipe. For an open organ pipe, it is given by \(f_{open}=\frac{v}{2L}\).

Using the given information

The problem states that the fundamental frequency of theclosed organ pipe is 100 Hz. We can use this information to find the length of the pipe. From the formula, \(f_{closed}=\frac{v}{4L}\), we can write the equation: \(100 = \frac{v}{4L}\)

Find the fundamental frequency of open organ pipe

Now we can find the fundamental frequency of the open organ pipe using the formula: \(f_{open}=\frac{v}{2L}\). Since we have the equation \(100 = \frac{v}{4L}\), we can multiply both sides by 2 to get: \(200 = \frac{v}{2L}\). Thus, when opened, the organ pipe has a fundamental frequency of 200 Hz.

Find the higher order frequencies of open organ pipe

For an open organ pipe, the higher-order frequencies are given by the formula \(f_n = n\cdot f_{open}\), where \(n\) is an integer. Using the fundamental frequency of 200 Hz for the open organ pipe, we can find the next frequencies by multiplying by integers: \(f_2 = 2 \cdot 200 = 400~\text{Hz}\) \(f_3 = 3 \cdot 200 = 600~\text{Hz}\) \(f_4 = 4 \cdot 200 = 800~\text{Hz}\) Thus, the frequencies produced when the other end of the organ pipe is also opened are 200, 400, 600, 800... The correct answer is (A).

Key concepts

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system, such as an organ pipe, can vibrate. It is often referred to as the first harmonic. For an organ pipe, this frequency depends on whether the pipe is closed at one end or open at both ends. When a pipe is closed at one end, like many traditional organ pipes, air vibrations within the pipe have a wavelength four times the length of the pipe. This results in the fundamental frequency given by the formula:

• For closed pipe: \( f_{closed} = \frac{v}{4L} \)

where \( v \) is the speed of sound and \( L \) is the length of the pipe.
Conversely, when the pipe is open at both ends, the length of the pipe supports a wave with a wavelength twice the length of the pipe, thus the formula becomes:

• For open pipe: \( f_{open} = \frac{v}{2L} \)

Understanding these formulas is crucial for predicting changes in frequency when the structure of an organ pipe is altered.

Harmonics

Harmonics are the multiple frequencies at which an object vibrates when it is already vibrating at the fundamental frequency. These frequencies are whole number multiples of the fundamental frequency and are crucial for creating complex sound waves that make musical sounds unique.

In the case of an open organ pipe, the presence of harmonics is more pronounced compared to a closed pipe because it supports a full spectrum of harmonics. The frequency of the nth harmonic for an open pipe is given by:

• \( f_n = n \cdot f_{open} \)

where \( n \) is a positive integer representing the order of the harmonic.
For instance, if the fundamental frequency \( f_{open} \) is 200 Hz for an open pipe, its harmonics would be 400 Hz (second harmonic), 600 Hz (third harmonic), 800 Hz (fourth harmonic), and so on.
Harmonics define the timbre or color of the sound, thus making different musical instruments sound distinct from one another even though they might play the same note.

Speed of Sound

The speed of sound in air is an important factor in determining the frequency of sound that an organ pipe produces. It is generally around 343 meters per second at room temperature (20°C or 68°F). This speed changes with temperature and pressure, as higher temperatures or pressures enable sound waves to travel faster. This is because the particles in the medium (air, in this case) move more rapidly, facilitating quicker transmission of sound waves.

When you are calculating the fundamental frequency and the harmonics of an organ pipe, the speed of sound \( v \) is an essential value in the formulas:

• For a closed pipe: \( f_{closed} = \frac{v}{4L} \)
• For an open pipe: \( f_{open} = \frac{v}{2L} \)

Any change in the speed of sound due to external conditions will directly affect these calculations, altering the fundamental frequency and thus impacting all related harmonics.

Acoustics

Acoustics is the branch of physics concerned with the study of sound. It involves understanding how sound is produced, transmitted, and perceived. In relation to organ pipes, acoustics plays a critical role in understanding how the length, shape, and openness of a pipe affect the sounds it produces.

Key concepts within acoustics relevant to organ pipes include:

• Wave Formation: Sound is transmitted as waves that reflect and resonate within the pipe. The pipe's dimensions determine the frequency and amplitude of these waves, directly influencing the sound's pitch and volume.
• Resonance: This occurs when the sound wave’s frequency matches the natural frequency of the pipe, amplifying sound. Different levels of resonance create distinct harmonics.

Understanding these concepts helps musicians and engineers design instruments capable of producing desired sounds. It also aids in optimizing the performance of musical instruments in various settings, whether in a grand concert hall or a modest living room.