Chapter 7.2, Problem 17P

Show that equation (2.10) for a wave can be written in all these forms: $\begin{array}{l}y=A\sin \frac{2\pi }{\lambda }(x-vt)=A\sin 2\pi \left(\frac{x}{\lambda }-\frac{t}{T}\right)\\ =A\sin \omega \left(\frac{x}{v}-t\right)=A\sin \left(\frac{2\pi x}{\lambda }-2\pi ft\right)=A\sin \frac{2\pi }{T}\left(\frac{x}{v}-t\right).\end{array}$

Here $\lambda$ is the wavelength, $f$ is the frequency, $v$ is the wave velocity, $T$ is the period, and $\omega =2\pi f$ is called the angular frequency. Hint: Show that $v=\lambda f$.