Step 1: Note down what is given.
The wave is $Y = Y_0 \sin 2\pi \left( nt - \frac{x}{\lambda} \right)$. We are told the wave speed is one eighth of the largest speed a single particle can have. We must find the wavelength.
Step 2: Write the wave speed.
For this wave the speed of the wave itself is $v = n\lambda$, where $n$ is the frequency and $\lambda$ is the wavelength.
Step 3: Write the biggest particle speed.
A particle of the medium moves up and down. Its fastest speed is $v_{p,\max} = 2\pi n Y_0$. Here $Y_0$ is the amplitude.
Step 4: Put the given condition into symbols.
Wave speed is one eighth of the top particle speed, so $n\lambda = \frac{1}{8}\left( 2\pi n Y_0 \right)$.
Step 5: Cancel and clean up.
The frequency $n$ sits on both sides, so we cancel it. We get $\lambda = \frac{1}{8}\left( 2\pi Y_0 \right) = \frac{2\pi Y_0}{8}$.
Step 6: Simplify the fraction.
$\frac{2\pi Y_0}{8} = \frac{\pi Y_0}{4}$. So the wavelength is $\frac{\pi Y_0}{4}$.
\[ \boxed{\lambda = \frac{\pi Y_0}{4}} \]