PH2130C 2005 Exam Paperaaaaaaaaaaaaaaaaaaaaaaaaa

7/28/2019 PH2130C 2005 Exam Paperaaaaaaaaaaaaaaaaaaaaaaaaa

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UNIVERSITY O F LONDO N

BSc and MSci EXAM INATION 2005

For Internal Students ofRoyal Holloway

DO NOT TUR N OV ER UNTIL TOLD TO BEGIN

PH2130C: MATHEMATICAL METHO DSPH2130C R: MATH EMAT ICAL METHODS - PAPER FOR RESIT

CANDIDATESTime Allowed: TWO hours

Answer QUESTION ONE and TWO other questionsNo credit will be given for attempting any further questions

.Approximate part-marks for questions are given in the right-hand margin

Only CASIO fx85WA Calculators or CASIO fi85MS Calculators are permitted


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GENERAL PHYSICAL CONSTANTS

Permeability o f vacuumPermittivity of vacuum

Speed of light in vacuumElementary chargeElectron (rest) massUnified atomic mass constantProton rest massNeutron rest massRatio of electronic charge to massPlanck constant

h9114x9C

ememu

mpmnelm,hA = hl2n

Boltzmann constant kStefan-Boltzrnann constant 0Gas constant RAvogadro constant NAGravitational constant GAcceleration du e to gravity gVolume of one mole of an ideal gas at ST POne standard atmosphere PO

J K-'2 -4W m ' K

J mol-' K"mol-'N m2 kg-2m s-2m'N mm2

MATHEM ATICAL COMSTANTS


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- page I - PH2 130CPARTMARKS

ANSWER ONLY FIVE sections of Question One.You are advised not to spend more than 40 minutes answering Question One .

1. (a) Write down the diffusion equation and the wave equation. What is theessential difference between the solutions to these differential equations? 141

(b) Find the solid angle R subtended by a circular star of radius a that is adistance d away from the earth, where d >> a.[The solid angle d R subtended by the vector surface area d3 a distance

d3.1r away is d R =- where i is the unit vector in the radial direction .] 141r 2(c) When using the method of separation of variables to solve partialdifferential equations in cylindrical or spherical polar coordinates the

function @ ( p ) often takes the complex form exp(inp). Explain clearlywhy n takes integral values. 141

(d ) Give two integral properties of the Dirac delta h c t i o n . 141

(e) In the Fourier expansion of a square wave using cosines and sines, explainwith the aid of a sketch the basic characteristics of Gibb s phenomena. 141

(f) @ ( X ,, z) is a scalar potential. Show that in Cartesian coordinates theLaplacian v24 s given by

a24 a24 a24div (grad4 ) = +- ,ax ay2 azIs it meaningful to consider g rad (div 4) ?

TURN OVER


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- page 2 - PH2 130CPARTMARKS

N drunkards emerge at closing time from a pub at r = 0 and diffusethroughout town. The two-dimensional (2D) density of drunks u(r , t ) ata distance r from the pub at time t obeys the diffusion equation

where p is constant.r/(a) In plane polar coordinates, the Laplacian operator is

Verify that a suitable solution to the 2D diffusion equation is

where a is constant.

(b) Could the solution u(r , t ) given in (a) be obtained by separation ofvariables? I31

(c) By integrating the density u ( r , ) over all r, show thatN = 2nap.

Is this true for all t ? 151

(d) Show that the density of drunks at a distanceR from the pub neverexceeds

TURN OVER


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The time - frequency Fourier transform pair are defined as

(a ) Show that the spectrum of a finite sine wave rectangular pulseF ( t ) = ae-Ig' - r / 2 < t < r / 2

= 0 otherwiseis given by

(b) Ske tch F ( t ) and f (W ), and describe what happens as r becomeslarger. I41

(c) Using the result

and the transform definitions to show the power theorem

(d ) Apply the power theorem to the spectrum defined in part (a), and hencedetermin e the integral

TURN OVER


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- page 4 - PH 2 130CPARTMARKS

4. (a) Describe the Frobenius method for obtaining series solutions of anordinary differential equation. What advantages does this method haveover simple series solutions? 141d2y dy(b) Legendre's equation is given by ( I - 2)- - x- + l (l+ l ) = 0,dx 2 dxwhere I is a real constant. When the Frobenius method is applied to thisequation, the following recurrence relation is obtained:

Show that by starting with either a, or a,, he following two series can begenerated from the relation.

(c ) If I is an integer, show that one or other of the two series terminates.Use the two series to generate the first four Legendre polynomials:P o ( x ) , 4 ( x ) , P , ( x ) ,a nd P , ( x ) .In order to obtain the correct normalisation, use the condition that4 ( l ) = .

(d) The orthogonality integral for the Legendre polynomials4 X ) is

where b;, is the Kronecker delta symbol. In the range - 1 I l , thefunction f ( X ) may be expressed as a linear sum

Derive a formula to determine the coef icie nts b,.

TURN OVER


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5 . Euler's gam ma function is defined through the integral:

and its behaviour is shown in the graph.

(a) Show that r(1) 1 . 131(b ) Use integration by parts to derive the recurrence relation

I-(X) = ( X - i ) r ( x - l ) .

(c) The graph show s that T(x) diverges when X is 0 or a negative integer.Use the recurrence relation to explain this behaviour. 151(d) For integer n show that the gamma fbnction is connected with thefactorial function through

(C) Given that r(f)=& ind values for r(-i)nd I-(-+),

END

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