Problem 5 (a) show that the following function satisfies the wave equation (this is a spherical wave). Show that its velocity is velocity v = ** f(r,t) = C = cos(kr - wt) The wave fronts are spherical shells at radius r that propagate outward from the origin; the amplitude of the disturbance decreases as the distance from the source. For this, use the Laplacian in spherical coordinates: 1 0 √²f= Acoustic Monopole af 1 მ r²sine de sinė 1 a²ƒ r²sin²0 04²
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- Legrende polynomials The amplitude of a stray wave is defined by: SO) =x (21+ 1) exp li8,] sen 8, P(cos 8). INO Here e is the scattering angle, / is the angular momentum and 6, is the phase shift produced by the central potential that performs the scattering. The total cross section is: Show that: 'É4+ 1)sen² 8, .2b. Consider that the input to the tank-pipe liquid system is formed of a source pressure pi, the atmospheric pressure pa, and the input flow rate q., whereas the output is the output flow rate q. Derive the corresponding transfer function matrix by means of complex impedances considering that known are the element properties Ci and R. Utilize the transfer function approach using a) matlab and b)Simulink to plot the output flow rate of the liquid system shown. Known are pa = 10 N/m, pi =2.2x10 N/m2, q=0.03 m3/s. d=1.5 m (diameter of the tank). p= 1000 kg/m', d, = 0.020 m (diameter of the pipe). / = 18 m (length of pipe), u=0.001N s/m?. P. Tank Pipe R. Pressure sourceSketch f(y) versus y. Show that y is increasing as a function of t for y < 1 and also for y > 1. The phase line has upward-pointing arrows both below and above y = 1. Thus solutions below the equilibrium solution approach it, and those above it grow farther away. Therefore, ϕ(t) = 1 is semistable.
- The following problem involves an equation of the form = f(y). dy dt Sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable or unstable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. dy = y(y − 1)(y — 2), yo ≥0 dt The function y(t) = 0 is no equilibrium solution at all. ▼ The function y(t) = 1 is Choose one The function y(t) = 2 is Choose onepls answer together with the sub questionsShow that a gaussian psi (x) = e ^(-ax^2) can be an eigenfunction of H(hat) for harmonic oscillator 1. Compute T(hat)*psi 2. Compute Vhat* psi - assume V operator is 1/2w^2x^2 3. Write out Hbar*psi and identify terms so Hber*psi=E*psi is true 4. From cancellation find a 5. insert back a to Schrodinger eq above and find E
- Respected sir, all parts pleaseA certain wavefunction is zero everywhere except between x = 0 and x = L. where it has the constant value A. Normalize the wavefunction.Calculate the reflection coefficient R and the transmission coefficient T of the scattering caused by a step potential given V(x) FLUX Vo V(x): SVo; x 0 ' = X as illustrated in the figure, when a particle stream moves from left to right.
- Consider the functions f(x) = x and g(x) = sin x on the interval (0, ). (a) Complete the table and make a conjecture about which is the greater function on the interval (0, ). (b) Use a graphing utility to graph the functions and use the graphs to make a conjecture about which is the greater function on the interval (0, ). (c) Prove that f(x) > g(x) on the interval (0, ). [Hint: Show that h′(x) > 0, where h = f − g.]Given an infinite well of length 0 to L, and an initial wavefunction which is atent shaped (triangle) with a value rising from zero at x=0 tosome maximum value at x=L/2 (midpoint) and then descending withequal, but opposite slope back to zero at x=L. The slope is positive a when0 < x < L/2 and negative a when L/2 < x < L. (A) write an equation (or more if you need to) for the wavefunction with a single normalization constant,A. (B) find A via normalization. (C) find the probability of a measurement of energy finding the value of the ground state energy. (D) find the probability of a measurement of energy finding the value of the first excidedstate energy. These eigenenergies are those of the infinite well,and you’ll need the corresponding eigenfunctions.What is y(x) (i.e., y as a function of x) given that dy dx X where you are given the boundary condition that y = yo at x = xo where yo and ro are constants.