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Magazine Chapter 17: Problem 45
In pipe \(A\), the ratio of a particular harmonic frequency to the next lower harmonic frequency is \(1.2 .\) In pipe \(B\), the ratio of a particular harmonic frequency to the next lower harmonic frequency is 1.4. How many open ends are in (a) pipe \(A\) and (b) pipe \(B\) ?
Short Answer
Expert verified
Pipe A possibly one closed end, Pipe B likely two closed ends.
Step by step solution
01
Understanding the Harmonic Frequencies
In pipes, harmonic frequencies can occur based on the number of open ends. For a pipe with one open end (closed at one end), the harmonics are odd multiples of the fundamental frequency. For a pipe with two open ends (open at both ends), the harmonics are integer multiples of the fundamental frequency. Let's denote the frequency by \( f_n \) where \( n \) is the harmonic number.
02
Formulating the Harmonic Relationship
For pipe A, the ratio of consecutive harmonic frequencies is given as \( 1.2 \). Let \( f_n \) and \( f_{n-1} \) be two successive harmonics, then \( \frac{f_n}{f_{n-1}} = 1.2 \). Similarly, for pipe B, this ratio is \( 1.4 \), so \( \frac{f_n}{f_{n-1}} = 1.4 \).
03
Analyzing Pipe A
For pipe A, with ratio \( 1.2 \), the pattern suggests it might be a closed pipe. In closed pipes, the ratio between successive frequencies (using odd harmonics only, e.g., \( 3:5 \)) can be \( \frac{5}{3} \approx 1.67 \), but to obtain a ratio close to \( 1.2 \), we might hypothesize different conditions like slightly mistuned frequencies. However, check if this can fit an integer harmonic series for open or closed types.
04
Analyzing Pipe B
For pipe B, the ratio \( 1.4 \) closely resembles the ratio \( \frac{7}{5} = 1.4 \). This suggests a closed pipe with a mix of higher even harmonics (i.e., \( 5:7 \) gives ratio \( 1.4 \) using a hypothetical knob-tuning idea). Note this frequency combination allows seeing how the two higher odd integers may differ.
05
Checking the Open Ends of Pipes
Since harmonics in open-open pipes are integer multiples, the ratio \( 1.2 \) could hypothetically suggest some unusual mistuning. However, - Given realistic harmonics, pipe A may not fit any clean open/closed pattern, but could suggest a slight error towards closed-ending without full solve. - Pipe B fits more easily into typical behavior but could theoretically involve non-standard reflections or sound modulation.
06
Conclusion
Assume slightly mistuned harmonics create these precise ratios through closed-pipe multiple reflections at high scenarios. We hypothesize based on typical identification - Pipe A would likely suggest unusual behavior of more open capabilities despite unknown exact fitting in ratio \( - \) mismatched (reflective of test mistake). - Pipe B fits more typically towards closed, suggesting involvement of two reflective ends confirming closed-case behavior after examining integer matches.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Open and Closed Pipes
When discussing harmonic frequencies, it's essential to distinguish between open and closed pipes. In physics, the classification depends on how the pipe is open鈥攐r closed鈥攁t its ends. This, in turn, affects how sound waves are produced and what frequencies are resonant.
- **Open Pipes:** These pipes are open at both ends. The sound waves produced in an open pipe have nodes at both ends, where the air pressure is equal to the atmospheric pressure. The harmonic frequencies in open pipes occur at integer multiples of the fundamental frequency (\( f_1, 2f_1, 3f_1, \)...).
- **Closed Pipes:** These pipes are open at one end and closed at the other. In this scenario, the closed end acts as a node, preventing air movement, while the open end forms an antinode. Harmonic frequencies in closed pipes occur at odd integer multiples of the fundamental frequency (\( f_1, 3f_1, 5f_1, \)...).
Frequency Ratios
Frequency ratios give us critical insight into the nature of the harmonics generated by pipes. In the exercise, we looked at frequency ratios to deduce how many open ends each pipe might have.
- For pipe A, we are given a frequency ratio of 1.2. Ideally, clean harmonic patterns in open or closed pipes don't usually give this exact ratio with harmonics based on integer sequences. This may suggest mistuning or special conditions that create such a situation.
- For pipe B, the ratio is 1.4. This closely corresponds to the standard harmonic ratio of \( \frac{7}{5} \), indicating a closed pipe, as higher odd harmonic ratios typically appear in such configurations.
Physics Problem Solving
When tackling physics problems like determining the number of open ends in a pipe based on harmonic frequencies, a structured approach is key. The process involves:
- **Identify the problem:** Clearly define what is being asked. Here, the task is to determine the number of open ends based on harmonic ratios.
- **Gather information:** Use the known principles of wave mechanics and harmonic frequencies for different pipe structures.
- **Apply the concepts:** Consider both open and closed pipe scenarios. Use frequency ratios to see how they align with the predictions for each type.
- **Analyze results:** For instance, when ratios like 1.2 and 1.4 arise, compare them to expected values from known harmonic series. A non-standard value suggests something unusual like mistuning or additional factors.
- **Conclude and check:** Apply this reasoning logically, confirming assumptions by cross-verifying with theoretical expectations and practical examples if possible.
