Wave Equation Along Y-Axis Identification

To identify a wave travelling along the $y$-axis, examine the spatial variable within the wave's phase term. The standard form of a travelling wave depends on a spatial coordinate and time.

Wave Propagation Direction Analysis

A wave travelling along the $y$-axis progresses through space primarily in the $y$ direction. Its equation will reflect this dependence, typically appearing in the phase argument, like $(ky \pm \omega t)$.

Evaluating Wave Equations

• Options 1 & 2: $y = A \cos ky \sin \omega t$ and $y = A \sin ky \cos \omega t$. These represent displacements in $y$ and have spatial dependence $ky$. However, the structure suggests standing waves rather than a single travelling wave.
• Option 3: $x = A \sin (ky - \omega t)$. This equation describes displacement along the $x$-axis. Crucially, the phase depends on $y$ (spatial coordinate) and $t$. The term $(ky - \omega t)$ signifies wave motion where $y$ is the axis of propagation. Thus, this represents a wave travelling along the y-axis.
• Option 4: $y = A \sin (kx - \omega t)$. This describes displacement along the $y$-axis, but the phase depends on $x$ (spatial coordinate). The term $(kx - \omega t)$ signifies wave motion along the x-axis.

Final Determination

The equation where the spatial variable governing the phase evolution is $y$ represents a wave travelling along the $y$-axis.

Selected Equation: $x = A \sin (ky - \omega t)$