Question

A metallic rod of length 2 m is rigidly clamped at its midpoint. Longitudinal stationary waves are set up in the rod in such a way that there are 2 nodes on either side of the midpoint. The amplitude of an antinode is 4 × 10^-6 m. Write an equation of motion of the constituent waves in the rod. (Young’s modulus = 2×10¹¹Nm^-2 and density = 8000 kg m^-3 )

Answer 1

General equation of a standing wave is y = 2A sink x cos ωt where

Which is obtained by adding 2 identical progressive waves travelling in opposite directions

i.e., y = y₁ + y₂ where

y₁ = A sin (kx – ωt)

y₂ = A sin (kx + ωt)

Let l be the length of the rod: l = 2 m

From the figure below, we see that

l = 5λ/2

We know that velocity of longitudinal wave is given by

∴ Equation of standing wave :

y = 2(4 × 10^-6 ) sin (2.5 πx) cos (12.5 × 10³π) t

Equation of constituent waves

y₁ = (4 ×10^-6 ) sin (2.5πx – 12.5 × 10³ πt)

y₂= (4 × 10^-6 ) sin (2.57πx + 12.5 × 10³ πt)