Question

Show that for a wave travelling on a string

$\frac{y_{max}}{{\nu }_{max}}=\frac{{\nu }_{max}}{{\alpha }_{max}},$

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

$\frac{y_{max}+{\nu }_{max}}{{\nu }_{max}-{\nu }_{max}}=\frac{{\nu }_{max}+{\alpha }_{max}}{{\nu }_{max}-{\alpha }_{max}}?$

Answer 1

$$\begin{gathered} \frac{y_{\max }}{v_{\max }}=\frac{v_{\max }}{a_{\max }} \\ y=A\sin \left(\omega t-kx\right) \\ y=A \\ v=\frac{dy}{dt}=A\omega \cos \left(\omega t-kx\right) \\ {v}_{\max }=A\omega \\ a=\frac{dv}{dt}=-A{\omega }^2\sin \left(\omega t-kx\right) \\ {a}_{\max }={\omega }^{2}A \end{gathered}$$

To prove,

$$\begin{gathered} \frac{y_{\max }}{v_{\max }}=\frac{v_{\max }}{a_{\max }} \\ \mathrm{LHS} \\ \frac{y_{\max }}{v_{\max }}=\frac{A}{A\omega }=\frac{1}{\omega } \\ \mathrm{RHS} \\ \frac{v_{\max }}{a_{\max }}=\frac{A\omega }{{\omega }^{2}A}=\frac{1}{\omega } \end{gathered}$$

No, componendo and dividendo is not applicable. We cannot add quantities of different dimensions.