Answer
$$D_{(y,t)}=({5.0\;\rm cm})\sin\left[ 4\pi y+{16\pi }t \right]$$
Work Step by Step
We know that the displacement equation for a sinusoidal wave
$$D_{(y,t)}=A\sin\left[ \dfrac{2\pi }{\lambda}y-{2\pi f}t+\phi_0 \right]$$
And according to the given data, $\lambda=0.5$ m, $A=5$ cm, $\phi=0$ rad.
Now we need to find the frequency which is given by
$$v=\lambda f$$
$$f=\dfrac{v}{\lambda}=\dfrac{4}{0.5}=8\;\rm Hz$$
Thus,
$$\boxed{D_{(y,t)}=({5.0\;\rm cm})\sin\left[ 4\pi y+{16\pi }t \right]}$$
The negative sign is due to moving in the $y$ negative direction.
