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. At rigid boundary, phase changes by π . Incident: y_i = 0.07 sin (25x - 50t) . Reflected: y_r = -0.07 sin (25x + 50t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = -0.07 sin (25x + 50t), illustrating frequency-length-speed interdependence and quantization by boundaries.