Practice question

**Longitudinal vibrations in rods** clamped at middle have fundamental with node at clamp and antinodes at ends, f₁ = v/(2L), v speed of sound in material (m/s). This relation allows v extraction from measured f₁ and length L, e.g., v = 2L·f₁. Speed of sound in a gas is v = √((gamma P/rho)) , and since P/rho ∝ T (ideal gas law), higher temperature increases molecular velocity, thus increasing v . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Molecular velocity increases, illustrating frequency-length-speed interdependence and quantization by boundaries.