A U-tube having uniform cross-section but unequal arm lengths ℓ1 and ℓ2(<ℓ1) has same liquid of

step 1 Hi friends, let the length of liquid, length of second liquid in A, B, Y, let the first liquid c

A U-tube having uniform cross-section but unequal arm...

A U-tube having uniform cross-section but unequal arm lengths $\ell_{1}$ and $\ell_{2}\left(<\ell_{1}\right)$ has same liquid of density $\rho_{1}$ filled in it up to a height $h$ as shown in Fig. $8.49$. Another liquid of density $\rho_{2}=\left(\rho_{1} / 2\right)$ is poured in $\operatorname{arm} A$. Both liquids are immiscible. What length of the second liquid should be poured in $A$ so that first overtone of $A$ is in unison with fundamental tone of B? (Take $\ell_{1}=5 \mathrm{~m}, \ell_{2}=1 \mathrm{~m}$ and $h=0.5 \mathrm{~m}$ )

Key Concepts

Standing Waves in Air Columns

Standing waves are patterns created when waves reflect back and forth within a confined space and interfere with each other. In tubes, these standing waves form under specific boundary conditions, such as open or closed ends, which determine the location of nodes (points of no motion) and antinodes (points of maximum motion). This concept is crucial for understanding the resonance frequencies of the tube.

Harmonics and Overtones

The fundamental frequency of a resonating system is the lowest frequency at which it naturally vibrates, and harmonics are integer multiples (or specific multiples, depending on the boundary conditions) of this fundamental frequency. The first overtone is the next higher mode of vibration after the fundamental and plays an essential role in tuning systems when matching different resonant modes.

Effective Length of a Resonating Column

The resonant frequency of an air column in a tube depends primarily on its effective length—the distance between the effective boundaries that define the vibrating air segment. Changes in physical conditions, such as the introduction of a liquid column that shifts the air-liquid interface, alter this effective length and, consequently, the natural frequency of the system.

Boundary Conditions in Tubular Resonators

Boundary conditions, determined by how a tube is terminated (open, closed, or interfaced with a liquid), dictate the phase of the reflected sound waves and the resulting positions of nodes and antinodes in the standing wave pattern. These conditions are vital for formulating the relationship between the physical dimensions of the resonator and its resonant frequencies.

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