March 10, 2016Introduction1 Important Notations

March 10, 2016Introduction2 Notations

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.

Definition : And also satisfied by -i Complex Numbers

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

Imaginary Exponent

Suppose ThereforeFormulation

In general The conjugate Complex Conjugate

Phasor = Rotating Arrow + Associated Phase AnglePhasor

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

© SPK Complex Plane

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

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

Series expansionSeries expansion of Series expansionSeries expansion of

Series expansionSeries expansion of

Complex Numbers & Simple Harmonic Oscillations

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

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)

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

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

© SB Complex Representation

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

Complex Velocity

© SBVelocity

Time Average

Average of Oscillations

Time Average

Time Average of KE & PE in SHM

© SB Root Mean Square (RMS)

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

Solution

Solution

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