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\begin{document}


{\bf Question}

In analysing the linear-free vibrations of a uniform clamped-beam,
one encounters the eigenvalue problem

$$\cos\lambda\cosh\lambda=-1$$

Show that for $n$ integer,

$$\lambda \sin
\left(n-\ds\frac{1}{2}\right)\pi+2e^{-(n-\frac{1}{2})\pi}\sin\left(n-\ds\frac{1}{2}\right)\pi,\
n\to+\infty$$

\vspace{.5in}

{\bf Answer}

$\cosh \lambda\to\infty$ as $\lambda\to\infty$ so

$$\cos\lambda=-\ds\frac{1}{\cosh\lambda}\to 0\ \rm{as}\
\lambda\to\infty$$

Therefore we look for large $\lambda$ solutions $\lambda \approx
\left(n-\ds\frac{1}{2}\right)\pi\ \ n$ integer, being the zeros of
$\cos\lambda$.

So try $\lambda=\left(n-\ds\frac{1}{2}\right)\pi+\delta,\
\delta=o(1)$.

Substitute back into full equation.

$$\cos\left(\left(n-\ds\frac{1}{2}\right)\pi+\delta\right)
=-\ds\frac{1}{\cosh((n-\frac{1}{2})\pi+\delta)}$$

\end{document}
