The displacement of a plane progressive wave in a medium, travelling towards positive x-axis with velocity 4 m/s at t = 0 is given by \(y = 3\sin 2\pi\left(-\dfrac{x}{3}\right)\). Then the expression for the displacement at a later time t = 4 sec will be
\(y = 3\sin 2\pi\left(-\dfrac{x-16}{3}\right)\)
A wave travelling along the positive x-direction with speed v keeps its shape and simply shifts forward, so its displacement is a function of \(x - vt\). The profile at any later time is obtained by replacing x with \(x - vt\) in the profile at t = 0.
Step 1 — At t = 0 the profile is \(y = 3\sin 2\pi\left(-\dfrac{x}{3}\right)\) and the speed is \(v = 4\ \text{m s}^{-1}\).
Step 2 — General form: \(y(x,t) = 3\sin 2\pi\left(-\dfrac{x - vt}{3}\right)\).
Step 3 — At t = 4 s the shift is \(vt = 4 \times 4 = 16\ \text{m}\), giving \(y = 3\sin 2\pi\left(-\dfrac{x-16}{3}\right)\).
The expression with \(\dfrac{-x-16}{3}\) depends on \(x + 16\), i.e. a profile shifted 16 m towards the negative x-axis. The expressions with \(\dfrac{-x+1}{3}\) and \(\dfrac{-x-1}{3}\) correspond to a shift of only 1 m, which is not the distance covered in 4 s at 4 m/s.
\(y = 3\sin 2\pi\left(-\dfrac{x-16}{3}\right)\) option is correct.
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