Solve the 1-Dim wave equation using D'Alembert and separation of variables method, given the following initial conditions for a string of length L = 1 m, c = 1 m/s h(x)=1-4(x -0.5) 1 0
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- Q.4) Find the position of the hammered string of a piano at any time t, the length of the string is 2 meter and it is fixed at both ends. The wave constant c² = 2, the initial displacement of the string is zero and its initial velocity is given by function: u₂(x,0) = sin (xx)Evaluate L{t° cosh4t} 1 1 A. (s – 1) (s+1)' 3. (s-1)* (s+1)* 1 C. (s-4)' (s+4) D. (s -4)* (s+4)* 1 1 (s –6) (s+6)* 1. 3. 3. 3. B. E.Consider the forced linear harmonic oscillator y" + why = cos wt where the frequencies wo = 9.0 W = 6 Find a particular solution Yp = A cos wt and enter the amplitude factor A rounded to 4 decimal places. Your Answer:
- Q1 (A): Find the deflection equation of simply supported beam which shown in figure by using the Fourier half range sine expansion. (EI=constant). (B): Find the general solution of O.D.E y" - y = sinh(x) + cosh(x) 2.0 kN/m 2m2m - 2m -2. Use the substitution technique to find S 5t sin(5) dt exactly (not an approximation).2. Solve the equation 3x² = e*, by applying 4 iterations of: a. Bisection Method (use the interval [0,2]) b. Secant Method (use the initial guess 0 and 1) c. Newton's Method (use 1 as initial guess)
- B-Solve the wave equation subject to the initial conditions (a) u(x, 0) = sinx (all x) (b) 및(x, 0)%3D0 (all x) du Use both the d'Alembert solution and the separation of variables method and show that they both give the same result.Q.6 Use eigenfunction expantion methodto solve: d у 1 -(xy')- dx У(1) %3 е, у(е) %3D0 :Evaluate the following: a. L {e3t + cos6t - e3t cos6t} b. L-1 {6s/(s2 + 25) + 3/(s2 + 25)} c. L-1 {se-4s/[(3s + 2)(s - 2)]} Solve: y'' - 6y' + 15y = 2sin3t ; y(0)=-1, y'(0) = -4
- Repeat the derivation in this section when the vertical motion of the string is retarded by a damping force proportional to the velocity of the string. Obtain the damped wave equation Utt = co(x)²upx kuz, 2 kut, - where k is the constant damping coefficient.Solve the equation siny(X+siny)Dy+2x²cosydy=0Solve the wave equation for the following cases: ii) For free free ends beam with following condition: du Boundry conditions: For x = 0 and x = 1 => əx Inetial conditions: inetial desplacement: u(x, 0) = f(x): du(x, 0) inetial velocity at = g(x). ||