To find the length of the pipe open at both ends, we use the relationship for the fundamental frequency of an open pipe:

$f = \frac{v}{2L}$

Where:

• $f$ is the fundamental frequency ($564 \, \text{Hz}$).
• $v$ is the speed of sound in air ($343 \, \text{m/s}$, at $20^\circ \, \text{C}$).
• $L$ is the length of the pipe.

Rearranging the formula to solve for $L$:

$L = \frac{v}{2f}$

Substitute the known values:

$L = \frac{343}{2 \times 564}$

$L = \frac{343}{1128} \approx 0.304 \, \text{m}$

Final Answer:

The length of the pipe is approximately 0.304 meters (30.4 cm).

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Do you want further clarification or have any questions? Here are 5 related questions you might consider exploring:

1. How does the speed of sound change with temperature, and how would it affect the length calculation?
2. What is the difference in behavior between an open and closed pipe with regard to harmonics?
3. How do you calculate the length for the second harmonic of a pipe open at both ends?
4. How would the calculation change if the pipe were closed at one end instead?
5. What are some real-world applications of fundamental frequency calculations in acoustics?

Tip: Always remember that the speed of sound in air is temperature-dependent, so adjust it if the environment differs significantly from ( 20^\circ , \text{C