Problem 34 - Physics

Question

An organ pipe is open at both ends. It is producing sound at its third harmonic, the frequency of which is \(262 \mathrm{~Hz}\). The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). What is the length of the pipe?

Step-by-Step Solution

Answer

The length of the pipe is approximately 1.965 meters.

Step 1: Understanding Harmonics for Open Pipes

For an organ pipe open at both ends, the harmonics are defined by integer multiples of the fundamental frequency. The nth harmonic for such a pipe corresponds to n/2 wavelengths fitting along the length of the pipe. Therefore, for the third harmonic, 3/2 wavelengths will fit in the length of the pipe.

Step 2: Determine Wavelength from Frequency and Speed of Sound

Use the relationship between speed, frequency, and wavelength, given by the formula \(v = f \lambda\) where \(v = 343 \text{ m/s}\) is the speed of sound and \(f = 262 \text{ Hz}\) is the frequency of the third harmonic. Solving for the wavelength \(\lambda\), we have \(\lambda = \frac{v}{f} = \frac{343}{262} \approx 1.31 \text{ m}\).

Step 3: Calculate the Length of the Pipe

Since the pipe is open at both ends and we are dealing with the third harmonic, we know that 3/2 of a wavelength fits into the length of the pipe. Therefore, the length of the pipe \(L\) can be found by the equation \(L = \frac{3\lambda}{2}\). Substituting \(\lambda = 1.31 \text{ m}\), we get \(L = \frac{3 \times 1.31}{2} \approx 1.965 \text{ m}\).

Key Concepts

Third Harmonic

In the context of an open pipe, understanding harmonics is essential. Harmonics are essentially the different modes in which a standing wave can resonate inside a pipe. When a pipe is open at both ends, it supports a specific pattern of air vibrations. For such pipes, the fundamental frequency allows exactly one half-wave to fit inside the pipe length.

The third harmonic, as the name indicates, is the third resonant mode possible in the pipe. This means that there are three half-waves vibrating between its two open ends. In simpler terms, three halves of the wave's length fit within the pipe. Visualizing this can help: imagine a wave that looks like the letter 'S' crammed into the length of the pipe. Therefore, understanding that the third harmonic allows for three halves is crucial when solving sound and wave-related problems in open pipes.

Speed of Sound

The speed of sound is a critical aspect when dealing with wave equations in physics. In the exercise, it's provided as 343 m/s. But what does it really mean? This value indicates how quickly sound waves travel through the medium, in this case, air. Multiple factors can affect the speed of sound, like temperature and pressure, but for simplicity, this exercise uses a standard value applicable at room temperature.

Sound waves travel when air molecules collide with each other and pass the sound energy from one to the next. Thus, the speed of sound essentially measures how fast these energy transfers happen in a given medium. Knowing this speed allows you to relate it to the frequency (how often the wave cycles occur) and the wavelength (the distance between successive crests or troughs of the wave), giving you a complete picture of the wave's behavior within the pipe.

Wavelength Calculation

Calculating the wavelength is straightforward when you know both the speed of sound and the vibration frequency. The exercise involves finding the wavelength associated with the third harmonic of the open pipe. With the formula \(v = f \lambda\), where \(v\) is speed of sound, \(f\) is the frequency of the third harmonic, and \(\lambda\) is the wavelength, you can rearrange it to solve for wavelength: \(\lambda = \frac{v}{f}\).

For the organ pipe scenario, inserting the values \(v = 343\; \mathrm{m/s}\) and \(f = 262\; \mathrm{Hz}\), the calculation becomes \(\lambda = \frac{343}{262} \approx 1.31\; \mathrm{m}\). This result represents the distance over which the wave completes one full cycle. It is crucial because it leads to further understanding how these standing waves fit within the constraints of the pipe length in harmonic scenarios.