By sinusoidally vibrating one end of a long horizontal string up...

The speed of a transverse wave on a stretched string is given by the formula:

$v = \sqrt{\dfrac{T}{\mu}}$

Substituting the given values of tension $T = 90 \text{ N}$ and linear mass density $\mu = 0.10 \text{ kg m}^{-1}$:

$v = \sqrt{\dfrac{90}{0.10}} = \sqrt{900} = 30 \text{ m s}^{-1}$

In a sinusoidal wave, the displacement of any particle on the string becomes zero twice during each complete cycle. Since the displacement reaches zero $200$ times per second, the frequency $f$ of the wave is:

$f = \dfrac{200}{2} = 100 \text{ Hz}$

The wavelength $\lambda$ is related to the wave speed and frequency by the equation $v = f \lambda$.

$\lambda = \dfrac{v}{f} = \dfrac{30}{100} = 0.30 \text{ m} = 30 \text{ cm}$