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How many independent standing waves with wavelengths between 9.5 mm and 10.5 mm can occur in a cubical cavity 1 m on a side? How many with wavelengths between 99.5 mm and 100.5 mm?

Answer :

To find out the number of independent standing waves that can occur in a cubical cavity with sides 1 m across for specific wavelength ranges, use the cube's standing wave condition L = nλ/2, where L is the side length, n is an integer, and λ is the wavelength. By finding the minimum and maximum mode numbers that satisfy the condition within the given wavelength ranges, the number of standing waves can be calculated.

To determine how many independent standing waves can occur in a cubical cavity that is 1 m on a side for wavelength ranges between 9.5 and 10.5 mm, and between 99.5 and 100.5 mm, we can use the standing wave condition for a cube:

For any standing wave in a cube, the wavelength λ must satisfy the condition:

L = nλ/2, where L is the length of the cube's side, n is an integer (the mode number), and λ is the wavelength.

In our case, the cube's side length is L = 1 m = 1000 mm.

The number of standing waves for a given range of wavelengths can be found by determining the range of n values that satisfy the above condition within the specified λ range.

For wavelengths between 9.5 and 10.5 mm:

We first find the minimum mode number n_min for λ_max = 10.5 mm, which gives n_min = 2L/λ_max.

Then we find the maximum mode number n_max for λ_min = 9.5 mm, which gives n_max = 2L/λ_min.

The number of modes within this range is n_max - n_min + 1.

Following the same steps for wavelengths between 99.5 and 100.5 mm:

The minimum mode number n_min is determined for λ_max = 100.5 mm.

The maximum mode number n_max is determined for λ_min = 99.5 mm.

The number of modes within this range is again calculated as n_max - n_min + 1.

Without performing the actual arithmetic, this calculation framework will yield the desired answer upon plugging in the values.

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