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

\[\frac{y_\max}{v_\max} = \frac{v_\max}{a_\max}\] 

\[y = A\sin\left( \omega t - kx \right)\] 

\[v = \frac{dy}{dt} = A\cos\left( \omega t - kx \right)\] 

\[ v_\max  = A\omega\] 

\[a = \frac{dv}{dt} =  - A \omega^2 \sin\left( \omega t - kx \right)\] 

\[ a_\max  =  \omega^2 A\]
To prove,

\[\frac{y_\max}{v_\max} = \frac{v_\max}{a_\max}\] 

\[LHS\] 

\[\frac{y_\max}{v_\max} = \frac{A}{A\omega} = \frac{1}{\omega}\] 

\[RHS\] 

\[\frac{v_\max}{a_\max} = \frac{A\omega}{\omega^2 A} = \frac{1}{\omega}\]
No, componendo and dividendo is not applicable. We cannot add quantities of different dimensions.