Question

Show that for a wave traveling on a string yₘₐₓ/vₘₐₓ = vₘₐₓ/aₘₐₓ

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

(yₘₐₓ+vₘₐₓ)/(yₘₐₓ-vₘₐₓ) = (vₘₐₓ+aₘₐₓ)/(vₘₐₓ-aₘₐₓ)?

Answer 1

y = A.sin2π(t/T-x/λ) ....... (i)

yₘₐₓ = A. {Because maximum value of sine = 1}

v = ∂y/∂t = A.cos 2π(t/T-x/λ)*2π/T

→vₘₐₓ = 2πA/T. {} Because also maximum value of cosine =1}

a = ∂²y/∂t² = A.sin2π(t/T-x/λ)*4π²/T²

→aₘₐₓ = 4π²A/T²

Now yₘₐₓ/vₘₐₓ =A/(2πA/T) = T/2π and vₘₐₓ/aₘₐₓ = (2πA/T)/(4π²A/T²) =T/2π

Hence, yₘₐₓ/vₘₐₓ = vₘₐₓ/aₘₐₓ.

We cannot use Componendo and Dividendo because addition and subtraction are allowed only between quantities having same unit or dimension. Here yₘₐₓ, vₘₐₓ and aₘₐₓ all have different units and dimensions.