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

BMET2960_Week6_Tutorial.pdf - BMET2960 Week 6 Tutorial...

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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( ) , ( n n n L x π n t λ B t x y where 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 λ n = cn π / L , frequency = λ n /2 π ) (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, where L =0.65 m, the tension T = 71 N and the mass per unit length is ρ = 0.401 x 10 -3 kg/m. The deflection, y , of the string is given by the PDE equation: ) 2 ( 2 xx tt y c y = where c 2 = 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: < < < < = L x L L x L k L x L kx x y 2 / for / ) ( 2 2 / 0 for / 2 ) 0 , ( where k = 5 mm is the initial deflection. Given the analytical solution in (1), you will now:
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