MATH2115: Engineering Practice 3 - Mathematical Modelling for Engineers Practical Session Questions 5 - Part 1 (Weeks 11

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Math2115 Engineering Practice 3 Mathematical Modelling For Engineers Practical Session Questions 5 Part 1 Weeks 11 1 (220.13 KiB) Viewed 26 times

MATH2115: Engineering Practice 3 - Mathematical Modelling for Engineers Practical Session Questions 5 - Part 1 (Weeks 11 & 12 2022) Question 1: (1 +1+3 = 5 Points) Suppose I have a rope, tied at one end, and I give it a single flick so that it forms a perfect sine wave 2 metres in width, after which I hold my end at the same height as the tied end, and let the wave propagate though the rope (in other words, I watch the rope wobble). 1 0.5 04 (0,0) N (2,0) -0.5 -1 0 0.5 1 1.5 2 FIGURE 1. The rope's starting configuration We will see how how the wave propagates through the rope (i.e., solve the wave equation for this rope). • The starting time t = 0 is when the rope forms the perfect sine wave (see fig. 1). Assume for these calculations that the initial derivative W.r.t. time is zero. That is: (2,0) = 0.* (a) (1 Point) What are the boundary conditions for the wave equation describing this rope configuration? (b) (1 Point) What are the initial conditions for the wave equation describing this rope configuration? (c) (3 Points) Solve the wave equation for this rope configuration i.e., find u(x, t)). Show all working! Note: • You do not need to calculate a Fourier series for the initial condition for this problem. • You do need to think hard about the values of Fn. Many (perhaps even most) will be equal to 0, but not all of them. . In addition to your working: fill in the following table at the beginning of your answer for 3(c). Bn= En = F= *No, this isn't particularly realistic, but please suspend your disbelief for the sake's of the maths being easier to work with.