A standing wave is maintained in a homogeneous string of cross-sectional area s and density ρ . It is formed by the superposition of two waves travelling in opposite directions given by the equation y

Question

A standing wave is maintained in a homogeneous string of cross-sectional area s and density ρ . It is formed by the superposition of two waves travelling in opposite directions given by the equation y 1 ​ = asin ( ω t − kx ) and y 2 ​ = 2 asin ( ω t + kx ) . The total mechanical energy between the sections corresponding to the adjacent antinodes is

TestHub · Accepted Answer

Distance between two adjacent antinodes is $\frac{λ}{2}$ or $\frac{Π}{k}$ , volume of string between two adjacent antinodes $V = \frac{Π}{k} a$

$E = (V) (u_{1} + u_{2})$

Here, $u$ is the energy density (energy per unit volume) which is equal to $\frac{1}{2} ρ A^{2} ω^{2}$ $\Rightarrow E = (\frac{Π}{k}) (s) [\frac{1}{2} ρ a^{2} ω^{2} + \frac{1}{2} ρ (2 a)^{2} ω^{2}] = \frac{5 π s ρ a ^{2} ω ^{2}}{2 k}$

Physics - Wave on String Question with Solution | TestHub

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