(20) Find the displacement u(x, t) for a piece of thin flexible string of length 1, of negligible weight. Suppose the two ends of the string are firmly secured ("clamped") at some supports so they will not move. Assume the set-up has no damping. Then, the vertical displacement of the string, 0 < x < 1, and at any time t > 0, is given by the displacement function u(x, t). It satisfies the homogeneous one-dimensional undamped wave equation: მ2 მე2 where the constant coefficient c² is given by the formula c² = T/p=1, such that c = horizontal propagation speed (also known as phase velocity) of the wave motion, T = force of tension exerted on the string, p = mass density (mass per unit length). It is subjected to the homogeneous boundary conditions u(0,t) = 0 u(1,t) = 0 t> 0. The two boundary conditions reflect that the two ends of the string are clamped in fixed positions. Therefore, they are held motionless at all time. The equation comes with 2 initial conditions, due to the fact that it contains the second partial derivative of time, utt. The two initial conditions are the initial (vertical) displacement u(x, 0), and the initial (vertical) velocity u(x, 0), both are arbitrary functions of x alone. u(x, 0) = √2 u₁(x, 0) = 0.

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