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**worc3247** A string with displacement y(x,t), where $\displaystyle a \leq x \leq b$ is fixed at each end so that y(a,t) = 0 = y(b,t) for all time t. The string's energy is give by:

$\displaystyle E(t) = \int^b_a {[\frac{1}{2} T (\frac{\partial y}{\partial x})^2 + \frac{1}{2} p (\frac{\partial y}{\partial t})^2}]\,dx $ where the first term is the tensile energy and the second term represents ths string's kinetic energy. Given the wave equation holds, show that the string's energy is constant througout the motion. Hint: You may assume the integrand is sufficiently smooth to allow the use of the Leibniz integral rule.

I can't see how to start this question. So far I have got:

$\displaystyle (\frac{\partial y}{\partial x})^2 = (X'(x)T(t))^2$ and

$\displaystyle (\frac{\partial y}{\partial t})^2 = (X(x)T'(t))^2$ but I can't see how this will help me. Any thoughts on where to start?