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Answer :
Final answer:
The number of independent standing waves with wavelengths between 9.5 and 10.5 mm that can occur in a cubical cavity 1 m on a side is 8, while the number with wavelengths between 99.5 and 100.5 mm is 9.
Explanation:
To calculate the number of independent standing waves with wavelengths between 9.5 mm and 10.5 mm that can occur in a cubical cavity with sides 1 m in length, we need to consider the possible wavelengths that can resonate in the cavity.
Since maximum air displacements are possible at the open end and none at the closed end, the standing wave in the cavity will have three-fourths of its wavelength inside the cavity. Therefore, the wavelength inside the cavity is 4/3 times the side length of the cube.
For the given range of wavelengths, we have 9.5 mm ≤ (4/3)L ≤ 10.5 mm, where L is the side length of the cube. Solving this inequality, we find that the possible side lengths of the cube are 7.125 mm ≤ L ≤ 7.875 mm. Since the side length of the cube is 1 m or 1000 mm, there are (7.875 mm - 7.125 mm) / (1 mm) + 1 = 8 possible side lengths within the given range.
Therefore, the number of independent standing waves with wavelengths between 9.5 and 10.5 mm that can occur in a cubical cavity 1 m on a side is 8.
Following the same process, for the range of wavelengths between 99.5 and 100.5 mm, we have (4/3)L = 99.5 mm and (4/3)L = 100.5 mm. Solving these equations, we find that the possible side lengths of the cube are approximately 74.625 mm and 75.375 mm. Therefore, there are approximately (75.375 mm - 74.625 mm) / (1 mm) + 1 = 9 possible side lengths within the given range.
Therefore, the number of independent standing waves with wavelengths between 99.5 and 100.5 mm that can occur in a cubical cavity 1 m on a side is 9.
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