Waves in Semiclosed Tubes: Calculating Frequencies and Lengths

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In summary, Aleph found that a semiclosed tube can be sounded at room temperature and that it can oscillate at several frequencies, including 1000, 1400, and 1800 Hz. He also found that the length of the tube is proportional to the fundamental frequency and the harmonics. Finally, he was able to calculate the length of the tube using his formula.
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CioCio
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I'm doing a self-guidance type of study, trying to learn my way through a book based on physics and the sound of music. I haven't dabbled with waves since high school.

A semiclosed tube is sounded at room temperature. It is observed that the tube can be made to oscillate rather easily at several frequencies that include 1000, 1400, and 1800 Hz consecutively. What is the length of the tube?

I know that these are sequential harmonics. Therefore, given that it's a semiclosed tube and L = length of tube in meters:

λ1 = 4L
λ3 = 4L/3
λ5 = 4L/5
λ7 = 4L/7​

I'm having trouble figuring out how to determine exactly which harmonics these are. Converting the frequencies into their respective periods gives me:


Tn = [itex]\frac{1}{1000 Hz}[/itex] = 0.001000 s
Tn+2 = [itex]\frac{1}{1400 Hz}[/itex] = 0.0007143 s
Tn+4 = [itex]\frac{1}{1800 Hz}[/itex] = 0.0005555 s​

(where n is the nth harmonic)​

Looking at the basis for harmonics (in semiclosed tubes), I know that if a wave travels a distance of 4L, or an odd-numbered integral quotient of 4L, in a time equal to its own period, a standing wave is formed. In other words, the 1000 Hz wave listed above must travel 4L/n meters in 0.001000 second for a standing wave to form.

But how can I determine what n is? One cannot simply assume that 1000 Hz is the fundamental frequency. Likewise, as shown by the basic wave formula of v = ƒλ, one cannot calculate wavelength from frequency/period alone.

I thought about an approach based around velocity but, again, I only know the time component of one oscillation.

Assuming room temperature (23 °C; 296 K), I know that:

ƒn = [itex]\frac{86n}{L}[/itex]

The calculation would be simple, but I don't know how I can calculate n. I apologize for not having more work to show, and I've tried to make up for that by posting my logic, but I'm at a standstill.
 
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  • #2
Converting into wavelengths and time periods is not the easiest way to do this.

If the fundamental frequency of a closed tube is f, what are the frequencies of the harmonics?

Try to match up that information with the observed frequencies of 1000, 1400, 1800. (Hint: think about the difference between successive frequencies).
 
  • #3
Thanks, Aleph. You're right; I was making it too complicated. :approve:

AlephZero said:
If the fundamental frequency of a closed tube is f, what are the frequencies of the harmonics?

For closed tubes, ƒn = nƒ1

So, in response to your question: ƒ2 = 2f, ƒ3 = 3f, ƒ4 = 4f, and so on.

Try to match up that information with the observed frequencies of 1000, 1400, 1800. (Hint: think about the difference between successive frequencies).

With the 400 Hz difference between successive frequencies, and knowing that even harmonics are skipped in semiclosed tubes, that must mean the fundamental frequency is 200 Hz. Which means that 1000 Hz is the fifth harmonic, 1400 Hz is the seventh, and 1800 Hz is the ninth.

Now, given my formula ƒn = 86n/L, I can calculate the length of the tube. - 0.43 m

Thanks!
 

Related to Waves in Semiclosed Tubes: Calculating Frequencies and Lengths

1. What are waves in semiclosed tubes?

Waves in semiclosed tubes refer to the movement of fluids or gases in partially enclosed spaces, such as pipes or tubes. These waves can be created by external sources, such as pumps or fans, or by changes in pressure within the tube.

2. How do waves in semiclosed tubes affect fluid flow?

Waves in semiclosed tubes can significantly impact fluid flow by creating disturbances and turbulence. This can lead to changes in flow rate, pressure, and other flow characteristics. It is important to understand and control these waves in order to optimize fluid flow processes.

3. What causes waves in semiclosed tubes?

Waves in semiclosed tubes can be caused by a variety of factors, including changes in fluid velocity, changes in fluid density, and changes in tube geometry. These changes can be caused by external influences, such as pumps or fans, or by changes in the fluid itself, such as temperature or pressure variations.

4. How can waves in semiclosed tubes be controlled?

There are several methods for controlling waves in semiclosed tubes, including using dampeners or baffles to reduce turbulence, adjusting flow rates to minimize pressure fluctuations, and optimizing tube geometry to reduce wave reflections. It is important to carefully consider these control methods in order to achieve desired flow characteristics.

5. What are some applications of studying waves in semiclosed tubes?

Studying waves in semiclosed tubes has many practical applications, including improving fluid flow processes in industrial settings, optimizing ventilation systems in buildings, and designing more efficient transportation systems. Understanding the behavior of waves in semiclosed tubes can also help with predicting and preventing potential problems, such as pipe bursts or flow disruptions.

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