1. March 19, 2016Introduction1 Important Notations

2. March 19, 2016Introduction2 Notations

3. March 19, 2016March 19, 2016March 19, 2016 Introduction3 Complex Numbers We will extensively use complex numbers throughout this course. Their use is NOT mandatory. 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 Euler's formula

7. e = 2.71828

8. Suppose ThereforeFormulation

9. In general The conjugate Complex Conjugate

10. Phasor = Rotating Arrow + Associated Phase AnglePhasor

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

12. © SPK Complex Plane

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

14. Complex Numbers & Simple Harmonic Oscillations

15. March 19, 2016March 19, 2016March 19, 2016 Introduction15 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

16. March 19, 2016Introduction16 Exponential solution: Real and imaginary parts of z(t) satisfy simple harmonic equation of motion A=Complex amplitude x(t)=Re z(t)

17. March 19, 2016Introduction17 Additions of two SHMs become convenient For, [] +

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

19. © SB Complex Representation

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

21. Complex Velocity

22. © SBVelocity

23. Time Average

24. Average of Oscillations

25. Time Average

26. Time Average of KE & PE in SHM

27. © SB Root Mean Square (RMS)

28. 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 initial phase of the motion: An Example

29. Solution

30. Solution