Waves: PYQ Walkthrough (4)

Question

A transverse wave on a string is described by $y(x, t) = 0.05\sin(20\pi x - 100\pi t)$ in SI units. Find the amplitude, wavelength, frequency, wave speed, and direction of propagation. (Adapted from NEET 2022.)

Solution — Step by Step

Why This Works

Reading off $k$ and $\omega$ from a wave equation is the most heavily tested skill in waves PYQs. Once you have those two numbers, every other quantity follows from the relations $\lambda = 2\pi/k$, $f = \omega/2\pi$, and $v = \omega/k$.

The sign convention for direction is worth memorising: $(kx - \omega t)$ moves right, $(kx + \omega t)$ moves left. A simple way to remember — at fixed phase, increasing $t$ requires increasing $x$ in the first case, so the wave shape shifts right.

Alternative Method

Plot the wave at $t = 0$ and at a slightly later time, say $t = 0.001$ s. You will see the entire pattern shifts to the right. This visual check is useful when the equation is given in a non-standard form like $\cos(\omega t - kx)$.

Common Mistake