A sinusoidal wave travelling in the positive direction on stretched string has amplitude 2.0 cm, wavelength 1.0 m and wave velocity 5.0 m/s. At x = 0 and t = 0 it is given that y = 0 and $\mathrm{\partial }\mathrm{y}/\mathrm{\partial }\mathrm{t}<0$. Find the wave function y(x, t).

We start with a general form for a rightward moving wave,

 $\mathrm{y}(\mathrm{x},\mathrm{t})=\mathrm{Asin}(\mathrm{kx}-\mathrm{\omega t}+\mathrm{\varphi })$

The amplitude given is $A=2.0\mathrm{cm}=0.02\mathrm{m}$.
The wavelength is given as, $\lambda =1.0\mathrm{m}$
Wave number $=\mathrm{k}=2\mathrm{\pi }/\mathrm{\lambda }=2{\mathrm{\pi m}}^{-1}$
Angular frequency, $\omega =\mathrm{kv}=2\pi \times 5=10\mathrm{\pi } \mathrm{rad}/\mathrm{s}$
$\mathrm{y}(\mathrm{x},\mathrm{t})=(0.02)\sin \left[2\mathrm{\pi }\right(\mathrm{x}-5.0\mathrm{t})+\mathrm{\varphi }]$

$\begin{array}{l} \mathrm{y}=0\text{ and }\frac{\mathrm{\partial }\mathrm{y}}{\mathrm{\partial }\mathrm{t}}<0\\ \mathrm{i}.\mathrm{e}., 0.02\sin \mathrm{\varphi }=0 (\mathrm{as} \mathrm{y}=0)\\ \mathrm{and} -0.2\mathrm{\pi cos}\mathrm{\varphi }<0\end{array}$

From these conditions, we may conclude that

$\mathrm{\varphi }=2\mathrm{n\pi }\text{ where }\mathrm{n}=0,2,4,6,\dots$

Therefore, $\mathrm{y}(\mathrm{x},\mathrm{t})=(0.02\mathrm{m})\sin \left[\left(2{\mathrm{\pi m}}^{-1}\right)\mathrm{x}-\left(10{\mathrm{\pi s}}^{-1}\right)\mathrm{t}\right]\mathrm{m}$