A lightly stretched homogeneous string of length $$ L $$, with its fixed ends at $$ x=0 $$ and $$ x=L $$ executes transverse vibration. Motion is started with zero initial velocity by displacing the string into the form $$ f(x)=k\left(x^{2}-x^{3}\right) $$. Find the deflection $$ u(x, t) $$

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Mathis Schofield · Accepted Answer

The deflection $$ u(x, t) $$ of the string is given by the Fourier sine series: $$ u(x, t) = \sum_{n=1}^{\infty} A_n \sin\left(\frac{n \pi x}{L}\right) \cos\left(\frac{n \pi c t}{L}\right) $$ where $$ A_n $$ are coefficients determined by the initial displacement $$ f(x) = k(x^2 - x^3) $$ and the wave speed $$ c $$.

A lightly stretched homogeneous string of length \( L \), with its fixed ends at \( x=0 \) and \( x=L \) executes transverse vibration. Motion is started with zero initial velocity by displacing the string into the form \( f(x)=k\left(x^{2}-x^{3}\right) \). Find the deflection \( u(x, t) \)

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A lightly stretched homogeneous string of length \( L \), with its fixed ends at \( x=0 \) and \( x=L \) executes transverse vibration. Motion is started with zero initial velocity by displacing the string into the form \( f(x)=k\left(x^{2}-x^{3}\right) \). Find the deflection \( u(x, t) \)

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Solución paso a paso

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