Show that for a wave travelling on a string: y_max u_max = v_max …

Question

Show that for a wave travelling on a string:$\frac{\text{y}_\text{max}}{\text{u}_\text{max}}=\frac{\text{v}_\text{max}}{\text{a}_\text{max}},$
Where the symbols have usual meanings. Can we use componendo and dividendo taught in algebra to write,$\frac{\text{y}_\text{max}+\text{v}_\text{max}}{\text{y}_\text{max}-\text{v}_\text{max}}=\frac{\text{v}_\text{max}+\text{a}_\text{max}}{\text{v}_\text{max}-\text{a}_\text{max}}?$

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Answer

$\frac{\text{y}_\text{max}}{\text{u}_\text{max}}=\frac{\text{v}_\text{max}}{\text{a}_\text{max}}$$\text{y}=\text{A}\sin(\omega\text{t}-\text{kx})$
$\text{v}=\frac{\text{dy}}{\text{dt}}=\text{A}\cos(\omega\text{t}-\text{kx})$
$\text{v}_{\text{max}}=\text{A}\omega$
$\text{a}=\frac{\text{dv}}{\text{dt}}=-\text{A}\omega^2\sin(\omega\text{t}-\text{kx})$
$\text{a}_{\text{max}}=\omega^2\text{A}$
To prove,
$\frac{\text{y}_\text{max}}{\text{u}_\text{max}}=\frac{\text{v}_\text{max}}{\text{a}_\text{max}}$
LHS
$\frac{\text{y}_\text{max}}{\text{u}_\text{max}}=\frac{\text{A}}{\text{A}\omega}=\frac{1}{\omega}$
RHS
$\frac{\text{v}_\text{max}}{\text{a}_\text{max}}=\frac{\text{A}\omega}{\omega^2\text{A}}=\frac{1}{\omega}$
No, componendo and dividendo is not applicable. We cannot add quantities of different dimensions.