Q.Which of the following equations represents a wave travelling along the y-axis? (A) $x=A\sin(ky-\omega t)$
(B) $y=A\sin(kx-\omega t)$
(C) $y=A\sin(ky)\cos(\omega t)$
(D) $y=A\cos(ky)\sin(\omega t)$

[!TLDR] A progressive wave travelling along the y-axis must have its ENTIRE argument as (ky $\mp$ $\omega$t), with the disturbance in a transverse (here x) direction: $x=A\sin(ky-\omega t)$. Options (B) has x and y swapped (wave along x-axis), while (C) and (D) are products of separa …

From section 6.2, a genuine PROGRESSIVE wave must be expressible as a single function of the combined argument $(kx\mp\omega t)$ (or, for propagation along y, $(ky\mp\omega t)$) -- this is exactly what makes the whole disturbance shape slide along bodily with time, as required by $y(x,t)=f(x-vt)$. Option (B), $y=A\sin(kx-\omega t)$, is a valid progressive wave, but its argument is $(kx-\omega t)$, so it travels along the x-axis, not the y-axis -- ruled out by the question's requirement. Option (A), $x=A\sin(ky-\omega t)$, has exactly the right combined-argument form $(ky-\omega t)$, but with the disturbance now written as x (the transverse displacement) as a function of position y and time t -- this correctly represents a wave whose disturbance travels along the y-axis, transversely displacing particles in the x-direction (exactly analogous to $y=A\sin(kx-\omega t)$ but with the roles of the two coordinates swapped). Options (C) $y=A\sin(ky)\cos(\omega t)$ and (D) $y=A\co …

Check which option has its space and time variables combined into a single argument $(ky\mp\omega t)$, which is the defining signature of a progressive wave travelling along the y-axis; reject any option whose argument is along x inste …

Picking option (B) because it 'looks like' the standard wave equation, without checking that its argument variable is x (travelling along the x-axis) rather than y as the question requir …