Question

A half-open pipe is constructed to produce a fundamental frequency of \(262 \mathrm{~Hz}\) when the air temperature is \(22^{\circ} \mathrm{C} .\) It is used in an overheated building when the temperature is \(35^{\circ} \mathrm{C} .\) Neglecting thermal expansion in the pipe, what frequency will be heard?

Short answer

Answer: To find the new frequency, follow these steps: 1. Calculate the speed of sound at 22°C: \(v_1 = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{22}{273.15}}\) 2. Calculate the speed of sound at 35°C: \(v_2 = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{35}{273.15}}\) 3. Find the ratio of the speeds of sound: \(ratio = \frac{v_2}{v_1}\) 4. Calculate the new frequency: \(new\_frequency = 262\mathrm{~Hz} \times ratio\) Compute the values and report the new frequency that will be heard in the overheated building.

Step-by-step solution

Calculate the speed of sound at 22°C

To find the speed of sound at the given temperature, we can use the formula: \(v = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{T}{273.15}}\) where \(v\) is the speed of sound and \(T\) is the temperature in Celsius. For \(T = 22^{\circ} \mathrm{C}\): \(v_1 = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{22}{273.15}}\) Calculate the value of \(v_1\).

Calculate the speed of sound at 35°C

Now, we will calculate the speed of sound at 35°C using the same formula. For \(T = 35^{\circ} \mathrm{C}\): \(v_2 = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{35}{273.15}}\) Calculate the value of \(v_2\).

Find the ratio of the speeds of sound

Find the ratio of the speeds of sound at the two temperatures: \(ratio = \frac{v_2}{v_1}\) Calculate the value of the ratio.

Calculate the new frequency

Now, we can find the new frequency by multiplying the original frequency with the ratio: \(new\_frequency = original\_frequency \times ratio\) \(new\_frequency = 262\mathrm{~Hz} \times ratio\) Calculate the value of the new frequency. This is the frequency that will be heard in the overheated building.

Key concepts

Speed of sound

The speed of sound is a fascinating topic that ties into many areas of physics and acoustics. To understand how it works, think of sound as a vibration moving through the air. The speed at which this vibration travels is what we refer to as the speed of sound. In air, this speed can be influenced by various factors, most notably the air temperature. To calculate the speed of sound, we can use the formula:\[ v = 331.4\mathrm{~m/s} \times \sqrt{1 + \frac{T}{273.15}} \]where \(v\) represents the speed of sound in meters per second, and \(T\) stands for the temperature in Celsius. The constant \(331.4\) m/s is the speed of sound at 0°C. This formula shows that as temperature increases, the speed of sound also increases, which is critical in understanding sound dynamics in various environments.

Fundamental frequency

When discussing musical instruments, such as a pipe, we often talk about its fundamental frequency—the lowest frequency at which the system naturally oscillates. In the context of a pipe, this is the longest wavelength that fits in the given length of the tube. For a half-open pipe, the fundamental frequency is based on the tone that resonates when the pipe's column of air vibrates in such a way that there is a node at the closed end and an antinode at the open end. The formula to find the fundamental frequency \(f\) is linked to the speed of sound \(v\) and the length \(L\) of the pipe:\[ f = \frac{v}{4L} \]This relationship reveals that changes in the speed of sound, due to temperature changes for example, will directly affect the fundamental frequency produced by the pipe.

Temperature effects on sound

Temperature holds a significant impact on the behavior of sound in the air. As the temperature of the air increases, its molecules move more rapidly. This increase in molecular movement allows sound waves to travel more swiftly through the medium. In practical applications, such as musical performances or acoustics in buildings, understanding and predicting how temperature impacts sound is critical. As the air temperature rises, so does the speed of sound, which affects frequencies produced by musical instruments:

• Higher temperatures result in a higher speed of sound.
• A higher speed of sound results in a higher frequency for a given length of a pipe.

These effects are particularly prominent in environments where temperature can fluctuate significantly, such as an overheated building.

Open and closed pipes

Pipes can be classified based on whether they are closed at one end (half-open) or open at both ends (open pipe). This classification is crucial because it determines the pipe's mode of vibration and, consequently, the frequency of sound it produces. In a pipe closed at one end (or half-open pipe like a flute), the fundamental frequency is set up so there is a node (point of no motion) at the closed end and an antinode (point of maximum motion) at the open end. On the other hand, in a fully open pipe, antinodes are present at both open ends. Key distinctions include:

• A fully open pipe supports vibration across full wavelengths that are multiples of half a wavelength.
• A half-open pipe supports vibration modes that result in odd multiples of quarter wavelengths.

Understanding these differences helps us comprehend the beautiful range of sounds produced by simple instruments like pipes.