Physics - Sound Wave Question with Solution | TestHub

$\frac{\lambda }{2}=L$

$\lambda =2L$

$\tau =\mu V^2$

The amplitude at a distance x from the origin is given by
$\mathrm{A} = \mathrm{a}\mathrm{sinkx}$
consider an element of mass dm and length dx of string at a distance x from the origin
The total energy of this element = its maximum KE

$=\frac{\text{1}}{\text{2}}\left(\text{d}\text{m}\right){\text{ω}}^2{\text{A}}^2$

$=\frac{\text{1}}{\text{2}}\text{μ}\text{d}x4{\text{π}}^2{\text{f}}^2{a}^2{\text{sin}}^{2}kx$

(Because$\text{μ}=\frac{\text{dm}}{\text{dx}}$and$\text{ω}=\text{2πf}$and it is given that$\text{A}=\text{a sinkx )}$

$\text{total energy of string}={\int }_0^{\text{L}}2{\text{π}}^2{\text{μ}\text{f}}^2{a}^2{\text{sin}}^{2}kx\text{d}x$

$={\text{π}}^2{\text{μ}\text{f}}^2{a}^2[x-\frac{\text{sin}2kx}{2}{]}_0^{\mathrm{L}}$

$={\text{π}}^2{\text{μ}\text{f}}^2{a}^2\text{L}$

$={\text{π}}^2\text{μ}\frac{{\text{V}}^2}{{\text{λ}}^2}{a}^2\text{L}=\frac{{\text{π}}^2}{4{\text{L}}^2}\text{T}{a}^2\text{L}=\frac{{\text{π}}^2{a}^2\text{T}}{4\text{L}}$