A wave pulse travels on a string towards the positive X -axis with...

The displacement of the string $g(x)$ has the dimension of length, so $[g(x)] = [L]$.

From the given equation $g(x) = A \sin(x/a)$, the amplitude $A$ must have the same dimension as $g(x)$.

Therefore, $[A] = [L]$.

The argument of the trigonometric function must be dimensionless. Thus, $[x/a] = [M^0 L^0 T^0]$.

Since $x$ represents position, $[x] = [L]$.

Therefore, $[a] = [x] = [L]$.

The dimensions of $A$ and $a$ are $L$ and $L$ respectively.