3. For a vibrating string of length with fixed ends, each mode of vibration can be written as where wk ux(x, t) = M* sin(wxt + k) sin(x) and Mk, Ok are determined by initial conditions. For all k > 1, sin(x) has zeros in [0, 1], at say points x = z, so uk(z, t) = 0 for all t. Points where the deflection u of a mode is zero (u(z, t) = 0) are called nodes. (a) What is the distance between nodes for each mode k > 1? (b) Where should you place your finger so that it is at a node of the second mode? This will prevent any mode that doesn't have a node at this point, such as the first mode, from vibrating. Express your answer in terms of fraction of the distance from the bridge, so that an answer 1 would be an open string, 0 would be no string at all. (c) Where should you place your finger so that it is at a node of the third mode? This will prevent any mode that doesn't have a node at this point, such as the first and second modes, from vibrating. Again give your distance in terms of a fraction of the distance from the bridge.