BMET2960_Week6_Tutorial.pdf - BMET2960 Week 6 Tutorial Analytical Solution to the Wave Equation In this week\u2019s tutorial we will look at the analytical

1 BMET2960 Week 6 Tutorial: Analytical Solution to the Wave Equation In this week’s tutorial we will look at the analytical solution to the wave equation. For stationary initial conditions, the general solution for a wave problem which has an initial displacement but zero initial velocity can be stated as a Fourier series in space and time as follows: ∑∞==1)1()/sin()cos(),(nnnLxπntλBtxywhere y(x, t) is the displacement as a function of space and time. First we are going to apply this to the guitar string example given in the lecture notes and then to a biological membrane oscillating under the influence of hydrodynamic pressure. In these examples we want to: (i)determine the fundamental frequencies (eigenvalues λn= cnπ/L, frequency = λn/2π) (ii)determine the exact solution (iii)given an initial disturbance, plot the disturbance behaviour in time and space 1.1 Analytical solution for a vibrating guitar string In the lectures we consider the motion of the first string on a Fender guitar, whereL=0.65 m, the tension T= 71 N and the mass per unit length is ρ= 0.401 x 10-3kg/m. The deflection, y, of the string is given by the PDE equation: )2(2xxttycy=where c2= T/ρ. The guitar player pulls the string into an initial triangular shape as given in the lecture notes and then releases it. This initial condition is stated mathematically as: <<−<<=LxLLxLkLxLkxxy2/for/)(22/0for/2)0,(where k= 5 mm is the initial deflection. Given the analytical solution in (1), you will now: