Question

Two identical half-open pipes each have a fundamental frequency of \(500 .\) Hz. What percentage change in the length of one of the pipes will cause a beat frequency of \(10.0 \mathrm{~Hz}\) when they are sounded simultaneously?

Short answer

Answer: The required percentage change in the length of one of the pipes is approximately -1.99%.

Step-by-step solution

Find the initial frequency and length of the pipes

For half-open pipes, the fundamental frequency (first harmonic) is given by \(f_1 = \cfrac{v}{4L}\), where \(f_1\) is the fundamental frequency, \(v\) is the speed of sound in air (approximately \(343 \, \mathrm{m/s}\)), and \(L\) is the length of the pipe. We are given that the fundamental frequency of each pipe is \(500 \, \mathrm{Hz}\). Therefore, $$500 \, \mathrm{Hz} = \cfrac{343 \, \mathrm{m/s}}{4L}$$ Solving for \(L\), we have: $$L = \cfrac{343 \, \mathrm{m/s}}{4 \times 500 \, \mathrm{Hz}} = 0.1715 \, \mathrm{m}$$. The initial length of each half-open pipe is \(0.1715 \, \mathrm{m}\).

Find the new frequency needed to create a beat frequency of 10 Hz

Beat frequency is the difference between the frequencies of the two half-open pipes. We are given that the beat frequency needs to be \(10 \, \mathrm{Hz}\). Let the new frequency of one of the pipes be \(f_2\). Therefore, $$f_2 = f_1 + 10 \, \mathrm{Hz} = 500 \, \mathrm{Hz} + 10 \, \mathrm{Hz} = 510 \, \mathrm{Hz}$$. The new frequency of one of the pipes should be \(510 \, \mathrm{Hz}\).

Find the new length of the pipe

Since we now know the new frequency of one of the pipes, we can find its new length \(L'\). We will use the same formula as before: $$510 \, \mathrm{Hz} = \cfrac{343 \, \mathrm{m/s}}{4L'}$$ Solving for \(L'\), we have: $$L' = \cfrac{343 \, \mathrm{m/s}}{4 \times 510 \, \mathrm{Hz}} = 0.1681 \, \mathrm{m}$$ The new length of the pipe is \(0.1681 \, \mathrm{m}\).

Calculate the percentage change in the length of the pipe

The percentage change in the length of the pipe is given by $$\% \, \mathrm{change} = \left(\cfrac{L' - L}{L}\right) \times 100$$ Plugging in the values for \(L'\) and \(L\), we have: $$\% \, \mathrm{change} = \left(\cfrac{0.1681 \, \mathrm{m} - 0.1715 \, \mathrm{m}}{0.1715 \, \mathrm{m}}\right) \times 100 = -1.9854\%$$ The required percentage change in the length of one of the pipes to create a beat frequency of \(10.0 \, \mathrm{Hz}\) when sounded simultaneously is approximately \(-1.99\%\).

Key concepts

Fundamental Frequency

The fundamental frequency is the lowest frequency produced by any vibrating object, and it is often referred to as the first harmonic. In the context of music and acoustics, fundamental frequency is particularly important as it determines the pitch of a note that we hear. For a half-open (or half-closed) pipe, like the one described in the exercise, the fundamental frequency is associated with the longest wavelength that can fit into the pipe with one end closed.

In simpler terms, when air is blown into such a pipe, it vibrates at a natural frequency. This frequency depends on the length of the pipe and the speed of sound within the pipe's medium, usually air. The equation connecting these quantities is given by the formula: \[f = \frac{v}{4L}\]where f is the fundamental frequency, v is the speed of sound, and L is the length of the pipe. When calculating the fundamental frequency for half-open pipes, it is essential to consider that the pipe supports a standing wave with a node at the closed end and an antinode at the open end, which is what mandates the use of the factor '4' in the formula.

Harmonics in Physics

Harmonics in physics refer to the frequencies of a vibrating object that are integral multiples of the fundamental frequency. These are also known as overtones and represent higher pitches compared to the fundamental frequency. Each harmonic has its characteristic wavelength and pattern of nodal (points of no displacement) and antinodal (points of maximum displacement) points along a vibrating medium.

If the fundamental frequency of a half-open pipe is represented by f, the second harmonic would be 3f, the third would be 5f, and so on, following the sequence of odd numbers. This is because half-open pipes support standing waves that fit an odd number of quarter wavelengths into the length of the pipe. Knowing about harmonics is crucial when understanding how changing the length of a pipe affects the frequencies it can produce.

Speed of Sound

The speed of sound is the pace at which sound waves propagate through a medium. It is not a constant value but changes with the properties of the medium, such as temperature, density, and the medium's molecular composition. In standard conditions which are defined as 20 degrees Celsius in dry air at sea level, the speed of sound is approximately 343 meters per second (m/s).

This value is critical for solving problems related to sound waves in air, like in the exercise with the half-open pipes. Changing environmental conditions can alter the speed at which sound travels through air, subsequently affecting the resonant frequencies of musical instruments and other sound-producing devices. The exercise assumes this standard value to calculate the fundamental frequencies and the resulting lengths of the pipes.

Percentage Change Calculation

To understand the impact of alterations in a system, it is sometimes necessary to calculate the percentage change. This quantifies the extent of change relative to the original situation and is frequently used in various fields like finance, science, and daily life. The formula to calculate the percentage change between two values is:\[% \, \text{change} = \left(\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}}\right) \times 100 \]In the context of the exercise, the percentage change represents how much the length of the half-open pipe needs to be altered to achieve the desired beat frequency. The formula helped us determine that shortening the pipe by approximately 1.99% would raise the fundamental frequency just enough to create a beat frequency of 10 Hz when played with an unaltered pipe. This illustrates how even small changes can have a measurable impact on physical systems.