A tunning fork of frequency 340 Hz resonates in the fundamental mode with an air column of length 125 cm in a cylindrical tube closed at one end. When water is slowly poured in it, the minimum height of water required for observing resonance once again is ______ cm.
(Velocity of sound in air is 340 ms$$^{-1}$$)

The frequency of the tuning fork is $$f = 340$$ Hz, the length of the air column for the fundamental mode is $$L = 125$$ cm, and the velocity of sound is $$v = 340$$ m/s. First, find the wavelength of sound: $$\lambda = \frac{v}{f} = \frac{340}{340} = 1 \text{ m} = 100 \text{ cm}$$.

For a closed pipe, resonance occurs when the air column length satisfies $$L = \frac{(2n-1)\lambda}{4}, \quad n = 1, 2, 3, \ldots$$. The possible resonant lengths are: for $$n = 1$$, $$L_1 = \frac{\lambda}{4} = 25$$ cm; for $$n = 2$$, $$L_2 = \frac{3\lambda}{4} = 75$$ cm; and for $$n = 3$$, $$L_3 = \frac{5\lambda}{4} = 125$$ cm.

The tube resonates in the fundamental mode at 125 cm. Note that this “fundamental mode” refers to the first resonance observed with the given tube length of 125 cm, which corresponds to $$n = 3$$ (the 3rd harmonic of a closed pipe with $$L = 125$$ cm).

When water is poured in, the effective air column length decreases, and the next resonance will occur at $$L_2 = 75 \text{ cm}$$. The minimum height of water required is $$h = L - L_2 = 125 - 75 = 50 \text{ cm}$$. Hence, the answer is 50 cm.