Mathematical Methods for Physics and Engineering - Matematica.NET

25.9 EXERCISES(b) Calculate F(s) on either side of the branch cut, evaluate the integral andhence determine f(t).(c) Confirm that the derivative with respect to s of the Laplace transformintegral of your answer is the same as that given by dF/ds.25.15 Use the contour in figure 25.7(c) to show that the function with Laplace transforms −1/2 is (πx) −1/2 .[ For an integrand of the form r −1/2 exp(−rx) change variable to t = r 1/2 .]25.16 Transverse vibrations of angular frequency ω on a string stretched with constanttension T are described by u(x, t) =y(x) e −iωt ,whered 2 ydx + ω2 m(x)y(x) =0.2 THere, m(x) =m 0 f(x) is the mass per unit length of the string and, in the generalcase, is a function of x. Find the first-order W.K.B. solution for y(x).Due to imperfections in its manufacturing process, a particular string hasa small periodic variation in its linear density of the form m(x) = m 0 [1 +ɛ sin(2πx/L)], where ɛ ≪ 1. A progressive wave (i.e. one in which no energy islost) travels in the positive x-direction along the string. Show that its amplitudefluctuates by ± 1 ɛ of its value A 4 0 at x = 0 and that, to first order in ɛ, the phaseof the wave is√ɛωL m0 πx2π Tsin2 Lahead of what it would be if the string were uniform, with m(x) =m 0 .25.17 The equation(d 2 ydz + ν + 1 2 2 − 1 )4 z2 y =0,sometimes called the Weber–Hermite equation, has solutions known as paraboliccylinder functions. Find, to within (possibly complex) multiplicative constants,the two W.K.B. solutions of this equation that are valid for large |z|. In each case,determine the leading term and show that the multiplicative correction factor isoftheform1+O(ν 2 /z 2 ).Identify the Stokes and anti-Stokes lines for the equation. On which of theStokes lines is the W.K.B. solution that tends to zero for z large, real and negative,the dominant solution?25.18 A W.K.B. solution of Bessel’s equation of order zero,d 2 ydz + 1 dy2 z dz + y =0, (∗)valid for large |z| and −π/2 < arg z < 3π/2, is y(z) =Az −1/2 e iz .Obtainanimprovement on this by finding a multiplier of y(z) in the form of an asymptoticexpansion in inverse powers of z as follows.(a) Substitute for y(z) in(∗) and show that the equation is satisfied to O(z −5/2 ).(b) Now replace the constant A by A(z) and find the equation that must besatisfied by A(z). Look for a solution of the form A(z) =z ∑ σ ∞n=0 a nz −n ,where a 0 = 1. Show that σ = 0 is the only acceptable solution to the indicialequation and obtain a recurrence relation for the a n .(c) To within a (complex) constant, the expression y(z) =A(z)z −1/2 e iz is theasymptotic expansion of the Hankel function H (1)0(z). Show that it is adivergent expansion for all values of z and estimate, in terms of z, the valueof N such that ∑ Nn=0 a nz −n−1/2 e iz gives the best estimate of H (1)0 (z).923