Answer
See the detailed answer below.
Work Step by Step
a) The wave is moving in the negative $y$-direction since it is $D_{(y,t)}$ and since the sign in front of $t$ is positive.
b) We know that sound waves are longitudinal waves, and this wave is moving in the $ y$-direction, so the air is oscillating back and forth in the $ y$-direction.
c) Knowing that
$$D_{(y,t)}=A\sin\left( ky-\omega t+\phi_0\right)=A\sin\left( \frac{2\pi}{\lambda}y-\frac{2\pi }{T} t+\phi_0\right)$$
Hence,
$$A\sin\left( \frac{2\pi}{\lambda}y-\frac{2\pi }{T}t+\phi_0\right)=0.02\times 10^{-3}\sin\left( 8.96y+3140t+\frac{\pi}{4}\right)$$
Thus the period is given by
$$3140=\frac{2\pi }{T}$$
$$T=\dfrac{2\pi}{3140}=\color{red}{\bf 2.00\times 10^{-3}}\;\rm s$$
The wavelength
$$\dfrac{2\pi}{\lambda}=8.96$$
so
$$\lambda=\dfrac{2\pi}{8.96}=\color
{red}{\bf 0.701}\;\rm m$$
The speed of the wave is given by
$$v=\lambda f=\dfrac{\lambda}{T}$$
$$v=\dfrac{2\pi }{8.96\times 2\times 10^{-3}}=\color
{red}{\bf 351}\;\rm m/s$$
