Show that the particle speed can never be equal to the wave speed in a sine wave if the amplitude is less than wavelength divided by $2\pi$.

Show that for a wave travelling on a string,

$\frac{y_{\max }}{v_{\max }}=\frac{v_{\max }}{a_{\max }}$,

where, the symbols have usual meanings. Can we use componendo and dividendo taught in algebra to write

$\frac{y_{\max }+v_{\max }}{y_{\max }-v_{\max }}=\frac{v_{\max }+a_{\max }}{v_{\max }-a_{\max }}$ ?