## T iX

Because gravitational effects are being neglected, the potential energy of the string will consist of only strain energy caused by extension of the string. This can be expressed as

where, as before,

and the original length is ¿o- In order to pick up all of the linear terms in the generalized equations of motion, one must include all terms in the energy up through the second power of the unknowns. Taking the pertinent unknowns to be perturbations relative to the stretched but undeflected string, we can again write e(x, t) — e(x) + e(x, t), u(x, t) — u(x) + u(x, r), (2.86)

For E A equal to a constant, the strain energy is ea rl°

2 Jn

From Eqs. (2.6), we know that T — To and e — eo, where Tq and ct) are constants. Thus, the first term of P is a constant and can be ignored. Since Tq — EAeo, the strain energy simplifies to r'" „ ea re

Making use of Eqs. (2.86), one finds that the longitudinal strain becomes

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