1. March 10, 2016Introduction1 Important Notations

2. March 10, 2016Introduction2 Notations

3. March 10, 2016March 10, 2016March 10, 2016 Introduction3 Complex Numbers We will extensively use complex numbers throughout this course. We will extensively use complex numbers throughout this course. Their use is NOT mandatory. Their use is NOT mandatory. But the use gives tremendouse convenience in classical wave physics. But the use gives tremendouse convenience in classical wave physics.

4. Definition : And also satisfied by -i Complex Numbers

5. x & y : real numbers i : unit imaginary number iy : pure imaginary number Now Every algebraic equation can be solved!! Representation & Notation : Complex Numbers

6. Imaginary Exponent

7. Suppose ThereforeFormulation

8. In general The conjugate Complex Conjugate

9. Phasor = Rotating Arrow + Associated Phase AnglePhasor

10. Representation of a complex number in terms of real and imaginary components Im Complex Plane r sin  r cos  Re z  ^ Argand/Phasor Diagram

11. © SPK Complex Plane

12. Taylor series is a series expansion of a function about a pointseries expansionfunction 1-d Taylor series expansion of a real function about a point isreal function Taylor series Maclaurin series is a Taylor series expansion of a function about 0Taylor series i.e., if a = 0, the expansion is known Maclaurin series Maclaurin series

13. Consider the following Maclaurin series expansions Expansions are valid for complex arguments x too Function :

14. Series expansionSeries expansion of Series expansionSeries expansion of

15. Series expansionSeries expansion of

16. Complex Numbers & Simple Harmonic Oscillations

17. March 10, 2016March 10, 2016March 10, 2016 Introduction17 Why Exponential Form? o Both sine & cosine are available in one form, take real or imaginary part o Periodic nature of displacement is reproducible o Leaves the form under differentiation and integration o Algebraic manipulations are quite easy

18. March 10, 2016Introduction18 Exponential solution: Real and imaginary parts of z(t) satisfy simple harmonic equation of motion A=Complex amplitude x(t)=Re z(t)

19. March 10, 2016Introduction19 Additions of two SHMs become convenient For, [] +

20. © Hecht x=Asin  0 t x=Asin(  0 t+  /3) x=Asin(  0 t+  /2) tt tt tt SHM Example

21. © SB Complex Representation

22. The real part of the complex number. Represents the oscillating quantity © SBMeaning/Significance

23. Complex Velocity

24. © SBVelocity

25. Time Average

26. Average of Oscillations

27. Time Average

28. Time Average of KE & PE in SHM

29. © SB Root Mean Square (RMS)

30. The natural frequency of a simple harmonic oscillator is 1/  sec -1. Initially (at t=0), the displacement of the oscillator from its equilibrium position is 0.3 m and velocity 0.7 m.sec -1. Use complex notation to determine the amplitude and phase of the motion: An Example

31. Solution

32. Solution

33. 1. FEYNMAN LECTURES ON PHYSICS VOL I Author : RICHARD P FEYNMAN IIT KGP Central Library : Class no. 530.4