USU PHYS 3750 - Lecture 21 Separation of Variables in Cylindrical Coordinates

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Lecture 21 Phys 3750 D M Riffe -1- 3/18/2013 Separation of Variables in Cylindrical Coordinates Overview and Motivation: Today we look at separable solutions to the wave equation in cylindrical coordinates. Three of the resulting ordinary differential equations are again harmonic-oscillator equations, but the fourth equation is our first foray into the world of special functions, in this case Bessel functions. We then graphically look at some of these separable solutions. Key Mathematics: More separation of variables; Bessel functions. I. Cylindrical-Coordinates Separable Solutions Last time we assumed a product solution ()()()()()tTzZRtzqφρφρΦ=,,, to the cylindrical-coordinate wave equation 2222222222111zqqqqtqc ∂∂+∂∂+∂∂+∂∂=∂∂φρρρρ, (1) which allowed us to transform Eq. (1) into ZZRRRTTc′′+ΦΦ′′+′+′′=′′221111ρρ. (2) We now go through a separation-of-variable procedure similar to that which we carried out using Cartesian coordinates in Lecture 19. The procedure here is a bit more complicated than with Cartesian coordinates because the variable ρ appears in the ΦΦ′′ term. However, as we shall see, the equation is still separable. A. Dependence on Time As with Cartesian coordinates, we can again make the argument that the rhs of Eq. (2) is independent of t , and so the lhs of this equation must be constant. Cognizant of the fact that we are interested in solutions that oscillate in time, we call this constant 2k− (where we are thinking of k as real) and so we have 221kTTc−=′′, (3) which can be rearranged as 022=+′′TkcT . (4)Lecture 21 Phys 3750 D M Riffe -2- 3/18/2013 By now we should recognize this as the harmonic oscillator equation, which has two linearly independent solutions, ()ikctkeTtT±±=0. (5) And so again we have harmonic time dependence to the separable solution.1 For the present case it is simplest to assume that k can be positive or negative; thus we really only need one of these solutions. So we simply go with ()ikctkeTtT0= . (6) B. Dependence on z Using Eq. (3), the lhs of Eq. (2) can replaced with 2k−, which gives, after a small bit of rearranging, ΦΦ′′+′+′′=′′−−22111ρρRRRZZk . (7) Notice that the rhs of Eq. (7) is independent of z . Thus the lhs is constant, which we call 2a− .2 We thus have 22aZZk −=′′−− , (8) which can be rearranged as ()022=−+′′ZakZ , (9) which, yet again, is the harmonic oscillator equation! The solutions are zakiakeZZ220,−±±= . (10) Although k and a can be any complex numbers, let's consider the case where both 2k and 2a are real and positive. Then these solutions oscillate if 22ak > , exponentially grow and decay if 22ak<, and are constant if 22ak=. Note that here, because the square root is always taken as positive, we need to keep both the positive-sign and negative-sign solutions. We can, however, assume that 0≥a . 1 It should perhaps be obvious that this will always be the case for separable solutions to the wave equation. 2 Why not? We can call the constant anything we want. It just so happens that 2a− is convenient.Lecture 21 Phys 3750 D M Riffe -3- 3/18/2013 C. Dependence on φ Using Eq. (8), the lhs of Eq. (7) can be replaced with 2a−, which gives us ΦΦ′′+′+′′=−22111ρρRRRa. (11) To separate the ρ and φ dependence this equation can be rearranged as 2221aRRRρρρ+′+′′=ΦΦ′′−. (12) Because each side only depends on one independent variable, both sides of this equation must be constant. This gives us our third separation constant, which we call 2n . The equation for Φ we can then write as 02=Φ+Φ′′n , (13) which is again the harmonic oscillator equation. The solutions to Eq. (13) are φinne±±Φ=Φ0. (14) Now we need to use a little physics. Because we expect any physical solution to have the same value for 0=φ and πφ2= we must have (for the +Φn solution) π20 inee = , (15) or π21ine= (16) Using Euler's relation, it is easy to see that Eq. (16) requires that n be an integer. Now because n can be a positive or negative integer, we do not need to explicitly keep up with the −Φn solution, and so we write for the φ dependence φinne0Φ=Φ , K,2,1,0 ±±=n (17) D. Dependence on ρ We are now left with one last equation. Also setting the rhs of Eq. (12) equal to 2n and doing a bit of rearranging yieldsLecture 21 Phys 3750 D M Riffe -4- 3/18/2013 ()02222=−+′+′′RnaRRρρρ (18) This is definitely not the harmonic oscillator equation! Its solutions are well know functions, but to see what the solutions to Eq. (18) are we need to put it in "standard" form, which is a form that we can look up in a book such as Handbook of Mathematical Functions by Abramowitz and Stegun (the definitive, concise book on special functions, which has been updated an is available online as the NIST Digital Library of Mathematical Functions). To put Eq. (18) in standard form, we make the substitution ρas = . Now the substitution would be trivial, except that because Eq. (18) is a differential equation in the independent variable ρ, we need to change the derivatives in Eq. (18) to derivatives in s. As usual we do this using the chain rule. If we now think of ()ρR as a function of ρ through the new independent variable s , i.e., as ()[]ρsR , then using adsdRddsdsdRddR==ρρ (19a) and 2222222222adsRddsddsdRddsdsRdddsdsdRdddRd=+==ρρρρρ. (19b) we can rewrite Eq. (18) as () ()()()02222=−+′+′′sRnsasRasasRas. (20) where we now emphasize that ()sRR=. Equation (20) obviously simplifies to () ()()()0222=−+′+′′sRnssRssRs. (21) This equation is know as Bessel's equation. It's two linearly independent solutions are known as Bessel functions ()sJn and Neumann functions ()sYn. Sometimes the functions ()sJn and ()sYn are called Bessel functions of the first and second kind, respectively. The subscript n is know as the order of the Bessel function Although one can define Bessel functions of non-integer order, one outcome of the Φ equation is that n is an integer, so we only need deal with integer-order Bessel functions for this problem.Lecture 21 Phys 3750 D M


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USU PHYS 3750 - Lecture 21 Separation of Variables in Cylindrical Coordinates

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