Mathematics 2 the First and Second Lectures Second week 19 - 15/ 5/ 1438 هـ أ / سمر السلمي

Outline for today Office Hours Syllabus References Chapter One Grades Office Hours Syllabus References Chapter One Fourier Series Simple harmonic motion and wave motion; periodic functions Applications of Fourier Series Average Value of a Function Fourier Coefficients Solving examples of Fourier Coefficients

Grades 6 % Attendance and participation % 20 % 34 % 40 --------------100% Attendance and participation Worksheets, Homework & Online quizzes Periodic Exam ( 2 exams ) Final Exam:

Office Hours Time of Periodic Exams Sunday, Tuesday and Thursday from 11 to 12 p.m. you can put any paper or homework in my mailbox in Faculty of Physics Department I will put any announcement or apology in my website (https://uqu.edu.sa/smsolamy) , so please check it my email is smsolamy@uqu.edu.sa for any question. every Wednesday a homework will be submit at my mailbox (or email if you did not came to university ) every week a worksheet will be submit in class Time of Periodic Exams The first periodic exam in 20- 21 -22 / 6 / 1438 h every in her group The second periodic exam in 11-12-13 / 8 / 1438 h every in her group

Syllabus Chapter One, Fourier Series & Transforms Chapter Two: Dirac Delta Function Chapter Three , Special Function: Gamma Function , Beta Function & the Error Function Chapter Four, Series Solutions of Differential Equations: Legender , Bessel , Hermite & laguerre Functions

References Mathematical Methods in the Physical Sciences by Mary Boas (main book) we will study from it ch 7 , ch15 section )4, 7 ( ch11 , ch 12) , There is a electronic copy in my website in the university (extra books) Mathematical Methods for Physicists by Arfken and Weber Mathematical Physics, World Student Series by Butkov Advanced Engineering Mathematics by Zill & Cullen Mathematical Handbook of Formulas and Tables by Schaum’s

Lectures will be from From the book Mathematical Methods in the Physical Sciences By Mary Boas: I will put electronic copy in my website in the university. we will study from it (ch 7 , ch15 section )4, 7 ( ch11 , ch 12) , Respectively. From the class: I will write examples & solving problems of homework in board.

From the electronic copy of the book Mathematical Methods in the Physical Sciences By Mary Boas Chapter One : Ch 7, pg. 297 Fourier Series Simple harmonic motion and wave motion; periodic functions Section 2, pg 297 -302 Applications of Fourier Series Section 3, pg 302 -304 Average Value of a Function Section 4, pg 304 -307 Fourier Coefficients Section 5, pg 307 – 312

Simple harmonic motion and wave motion; periodic functions Let particle P move at constant speed around a circle of radius A. At the same time, let particle ṕ move up and down along the straight line segment RS in such a way that the y coordinates of P & ṕ are always equal. Here w is the angular velocity of P in radians per second. θ = wt Then in the y coordinates y = A sin θ = A sin wt here we can say that particle ṕ move Simple Harmonic Motion R S

Z = x + iy = A(cos wt + i sin wt) = A eiwt Simple harmonic motion and wave motion; periodic functions If we look at particle P the y coordinates of it y = A sin θ = A sin wt & the x coordinates of it x = A cos wt where A is amplitude of the vibration or the function or the wave & T= 2π /w is the periodic time If we want look at this particle in complex plane, we write Z = x + iy = A(cos wt + i sin wt) = A eiwt also if we want to take about the wave motion, we should remember φ phase difference y = A sin (wt + φ)

Simple harmonic motion and wave motion; periodic functions In the Wave Motion or Wave Properties, we should remember The frequency w=2π f & the wavelength λ

Simple harmonic motion and wave motion; periodic functions What is the difference here amplitude A the angular velocity w phase difference φ

Simple harmonic motion and wave motion; periodic functions the function f(x) called periodic function with period of T and we write f ( x ± T) = f(x) Where T is constant called period of function Example : sinx sin (x + 2π) = sin (x) Where T= 2π -∏ -∏/2 ∏/2 ∏

Simple harmonic motion and wave motion; periodic functions f ( x± T) = f(x) where T period of function Other examples cot x , tan x tan (x + π) = tan (x) ، cot (x + π) = cot (x) where T= π sin 2π x sin 2π(x +1) =sin (2πx + 2π) = sin 2πx where T= 1 sin (πx / l) sin (π/ l)(x +2l) =sin (πx / l + 2π) = sin πx / l where T= 2l Note: f ( x± nT) = f(x) where n is integer & T period of function

Applications of Fourier Series Fourier series deals with problems involving vibrations or oscillations or repeating motion. Examples: a vibrating tuning fork, a pendulum, a weight attached to a spring, water waves, sound waves, etc.. All to these examples are most probably periodic i.e. it repeats itself with time. These motions that repeat themselves regularly with time are also called harmonic motion. Fourier series is based on the idea that it approximates the function under study to sines and cosines.

Applications of Fourier Series As the tuning fork vibrates it pushes against the air molecules, creating alternately regions of high and low pressure. If we measure the pressure as the function of x and t from the tuning fork to us, we find the pressure is of the form of If we measure the pressure where we are as a function of t as the wave passes, we find the pressure is a periodic function of t. The sound wave is a pure sine wave of a definite frequency. Now suppose that several pure tones are heard simultaneously. In the resultant sound wave, the pressure will not be a single sine function but sum of several sine functions

Applications of Fourier Series There several types or shapes of waves ( Sine , square , Triangle , … elc)

Average Value of a Function For the average value for number example (the average value of homeworks for Mathematical Methods 2 = (HW1 +HW2 + HW3) / 3 In the same way , we can find the average value of a function f(x) on interval from a = x1 to b = xn . Where is the length of the interval, When & ,\ the numerator is , then we get

Average Value of a Function The average value of function f(x) on interval from a = x1 to b = x2 Examples: the average value of 1- sin x on interval (-π, π) (note : the average value of sinx in any periodic interval is zero) 2- sin2nx on interval (-π, π) (note : the average value of sin2nx & cos2nx in any periodic interval is half) 3- sin mx cos nx on interval (-π, π) 4- x - cos26x on interval ( 0, π/6 ) (Worksheet )

Average Value of a Function We will use the average value for below on interval (-π, π)

Fourier Coefficients The general formula for Fourier Series We want to find the formula for coefficients a0 , an & bn

Fourier Coefficients To find the value of coefficient a0 We want all the integrals on the right – hand side are zero except the first term (zero term), so we use the two equations below : We assume that n=0 و m ≠ 0 & m = 1,2 , 3, …..

Continue finding the value of coefficient a0 Fourier Coefficients Continue finding the value of coefficient a0 all the integrals on the right – hand side are zero except the first term after we take the average value for all terms on interval (-π, π)

To find the value of coefficient an Fourier Coefficients To find the value of coefficient an The same way, we first Multiply all terms with cos nx, then take the average value for all terms on the same interval (-π, π)

Fourier Coefficients Continue finding the value of coefficient an we see that all terms will be zero except the nth term for cos nx if we use the two equations below the nth term

Fourier Coefficients Continue finding the value of coefficient an all terms in right – hand will be zero except the nth term for cos nx if , then Thus, Because we now know the nth term for cos nx , we will know the other values

Fourier Coefficients To find the value of coefficient bn The same way, we first Multiply all terms with sin nx, then take the average value for all terms on the same interval (-π, π)

Fourier Coefficients Continue finding the value of coefficient bn we see that all terms will be zero except the nth term for sin nx if we use the two equations below the nth term

Fourier Coefficients Continue finding the value of coefficient bn all terms in right – hand will be zero except the nth term for sin nx if , then Thus, Because we now know the nth term for sin nx , we will know the other values

Solving Fourier Series’ problems Fourier Coefficients Solving Fourier Series’ problems First, we find the value coefficients from 3 eq. Second, we put them in The general formula for Fourier Series

Fourier Coefficients Expand the periodic function f(x) in a sine-cosine Fourier Series? 1} 2}

Next class review Complex form of Fourier Series Other Intervals