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------------------------------------------------ A standing wave with $ n $ antinodes is observed to have a frequency of 66.9 Hz. In the same medium, another standing wave with $ n+2 $ antinodes has a frequency of 117 Hz. What is the frequency of the standing wave with the longest wavelength in this medium?

The frequency of the standing wave with the longest wavelength (one antinode) in the medium is approximately 9.56 Hz when the wave speed is constant. This is calculated using the ratio of frequencies for different numbers of antinodes given in the problem.The frequency of the standing wave with the longest wavelength in the medium can be determined by understanding the relationship between frequency, the number of antinodes, and the wavelength.

Explanation:

For standing waves on a string with fixed ends, the wavelength (λ) is related to the length of the string (L) and the number of antinodes (n) by the equation

λ = 2L/n. The frequency (f) is related to the wave speed (v) and the wavelength by

f = v/λ.

Given that a standing wave with n antinodes has a frequency of 66.9 Hz and a wave with n+2 antinodes has a frequency of 117 Hz, we can deduce that the frequency is proportional to the number of antinodes since the wave speed is constant for a particular medium. To find the frequency of the standing wave with the longest wavelength, we look for the lowest mode of vibration, which occurs when there is just one antinode (n=1). Using the ratio of the frequencies given, we find that the frequency with n+2 antinodes is ≈ 1.75 times the frequency with n antinodes. Therefore, we can calculate the frequency with one antinode by dividing the given frequency of the standing wave with n antinodes (66.9 Hz) by n and then multiply by 1 (since we want the frequency for n=1).

The frequency of the wave with the longest wavelength (one antinode) can be found as follows:

Let's denote the frequency with n antinodes as Fn.

Fn+2 (117 Hz) = Fn * (n+2)/n

We find the ratio of n+2 to n by dividing 117 Hz by 66.9 Hz, which gives us ≈ 1.75.

So, F1 = Fn / n

= 66.9 Hz / n

Since we know the ratio of (n+2)/n is approximately 1.75, we can say n/1.75

= n+2, which solves to

n=7.

Therefore, F1 = 66.9 Hz / n

= 66.9 Hz / 7

≈ 9.56 Hz

Therefore, the frequency of the standing wave with the longest wavelength in this medium is approximately 9.56 Hz.