91Ó°ÊÓ

Using the given formula for the resonance frequencies and the relation between wavelength and frequency, we can derive the formula for the wavelength of the fundamental and overtones: \[\lambda = \frac{4L}{(2n+1)}\]

For the fundamental, n = 0: \[\lambda_1 = \frac{4L}{(2\cdot 0 + 1)} = \frac{4L}{1}= 4L\] For the first overtone, n = 1: \[\lambda_2 = \frac{4L}{(2\cdot 1 + 1)} = \frac{4L}{3}\] For the second overtone, n = 2: \[\lambda_3 = \frac{4L}{(2\cdot 2 + 1)} = \frac{4L}{5}\]

Now that we have the formulas for the wavelengths of the fundamental and overtones, we can find their ratio: \[\lambda_1:\lambda_2:\lambda_3 = 4L:\frac{4L}{3}:\frac{4L}{5}\] To find the ratio without including the common factor 4L, we can multiply each term by 15 (the least common multiple of the denominators 1, 3, and 5): \[15 \cdot (4L:\frac{4L}{3}:\frac{4L}{5}) = 60L:20L:12L\] Finally, simplify the ratio by dividing each term by their greatest common divisor, 4L: \[\frac{60L}{4L}:\frac{20L}{4L}:\frac{12L}{4L} = 15:5:3\] Substituting these simplified wavelengths back to the original ratio, we get: \[\lambda_1:\lambda_2:\lambda_3 = 1:\frac{1}{3}:\frac{1}{5}\] Thus, the correct answer is (B) \(1:\frac{1}{3}:\frac{1}{5}\).