The amplitude of wave disturbance propagating in the positive x-direction is given by y = 1/(1 + x²) at time t = 0 and y = 1/(1 + (x - 2)²) at t = 1 s, where x and y are in metres. The shape of wave does not change during the propagation. The velocity of the wave will be ___________ m/s.

Solution

Physics sir · 3 steps

Follow the peak: it sits at x = 0 at the start and at x = 2 m one second later.

1. Shape just shifts. A wave moving in the +x direction without changing shape has the form y = f(x - vt). So the whole pulse slides along x by vt in time t.
2. Track the peak. At t = 0, y = 1/(1+x²) is largest where x = 0. At t = 1 s, y = 1/(1+(x-2)²) is largest where x - 2 = 0, that is at x = 2 m.

   v = (Δ x)/(Δ t) = (2 m - 0)/(1 s) = 2 m s⁻¹

3. So. Matching with y = 1/(1+(x - vt)²) at t = 1 s also gives v × 1 = 2. The wave speed is 2 m/s.

The idea to keep. For a pulse that keeps its shape, pick any easy point, like the peak, and see how far it moves. Distance moved divided by time taken is the wave speed.

Answer from the NTA final answer key. Solution by PrepWiser.