Consider the following problem for the wave equation with periodic boundary conditions at a = 0: Sozu = c*0u, (x, t) ER X R., u(0, t) = cos(wt), te R.. a) Find at least one solution. Hint: use the solution guess (*- ct) + n(x+ct) and find a choice of 5,n that fits the BCs. b) Is the solution unique or is there more than one?
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- 5C. Under suitable assumptions derive one dimensional wave equation.The graph of f (θ) = Acos θ + B sin θ is a sinusoidal wave for any constants A and B. Confirm this for (A,B) = (1, 1), (1, 2), and (3, 4) by plotting f .Fix c > 0. Show that for any constants a, ß, the function u(1, x) = sin(æct + B) sin(æx) satisfies the wave equation a?u ax2
- 3. Verify that u(r,l) = sin(x – at) satisfies the wave equation:Which of the following are parametric equations for the entire line y = x + 1? Choose all that apply. x(t) = cos(t), y(t) = cos(t) + 1 Ox(t)=t+2, y(t) = t + 3 x(t)=t, y(t) = t + 1 Ox(t)=t+1, y(t) = t Ox(t) = t1, y(t) = t x(t) =tan(t), y(t) = tan(t) + 1 x(t) = t², y(t) = ² + 1 Ox(t) = t³, y(t) = t³ + 1The wave equation describes the motion of a waveform: 0 u/ôt? – d²u/ðx² = 0. Which of the following functions does not satisfy the wave equation? u(x, t) = sin(x)sin(t) u(x, t) = sin(x – t) + cos(x + t) u(x, t) = sin(x – t) u(x, t) = sin(x – t) + cos(x – t) - None of the functions shown.
- Solve the following wave equation using finite difference method. 4fxx = ftt - Given: f(0, t) = 0 and f(1, t) = 0 f(x, 0) = ft(x, 0) = 0 sin(x) + sin(2x) (Ref: Hyperbolic Equation)Suppose that a particle follows the path r(t) = 2 sin(3t) i+4 cos(3t) j. Give an equation (in the form of a formula involving x and y set equal to 0) whose whose solutions consist of the path of the particle. = 0. (Answer in terms of x and y.) Determine the velocity vector of the particle when t = T : v(7) = (Answer in terms of t.) Determine the acceleration vector of the particle whent = t : a(7) = (Answer in terms of t.)The graph of f(θ) = A cos θ + B sinθ is a sinusoidal wave for any constants A and B. Confirm this for (A, B) = (1, 1), (1, 2), and (3, 4) by plotting f.