A wave disturbance in a medium is described by \(y(x,t)=0.02\cos\left(50\pi t+\dfrac{\pi}{2}\right)\cos(10\pi x)\) where x, y are in meters and t is in second. Which statement(s) is/are correct?
The product of a cosine in time and a cosine in position describes a stationary wave. Nodes are where the position factor is zero, antinodes where its magnitude is one, and the speed of the component travelling waves is ω/k.
Step 1 — Reading off: \(k=10\pi\ \text{m}^{-1},\ \ \omega=50\pi\ \text{s}^{-1}\).
Step 2 — Wavelength: \(\lambda=\dfrac{2\pi}{k}=0.2\ \text{m}\), so the statement about a 0.2 m wavelength holds.
Step 3 — Nodes: \(\cos(10\pi x)=0\Rightarrow x=0.05,\ 0.15,\ 0.25,\ \ldots\ \text{m}\), so a node does occur at x = 0.15 m.
Step 4 — Antinodes: \(|\cos(10\pi x)|=1\Rightarrow x=0,\ 0.1,\ 0.2,\ 0.3,\ \ldots\ \text{m}\), so an antinode does occur at x = 0.3 m.
Step 5 — Wave speed: \(v=\dfrac{\omega}{k}=\dfrac{50\pi}{10\pi}=5\ \text{m s}^{-1}\), not 4 m/sec, so the speed statement fails.
A node occurs at x = 0.15 m, An antinode occurs at x = 0.3 m and The wavelength of the wave is 0.2 m options are correct.
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