Syllabus overview

1. Syllabus overview • No text. Because no one has written one for the spread of topics that we will cover. • MATLAB. There will be a hands-on component where we use MATLAB programming language to create, analyze, manipulate sounds and signals. Probably 1 class per week (in computer lab at end of hall WPS211).

2. Grading • Participation is key! • Attempt all the work that is assigned. • Ask for help if you have trouble with the homework. • If you make a good faith effort, don’t miss quizzes, hand in all homework on time, etc. you should end up with an A or a B.

3. Web page • Lecture Powerpoints are on the web as are homeworks, and (after the due date) the solutions. • MATLAB exercises are also on the web page http://physics.mtsu.edu/~wmr/Phys3000home.htm

4. Objectives • Physical understanding of acoustics effects and how that can translate to quantitative measurements and predictions. • Understanding of digital signals and spectral analysis allows you to manipulate signals without understanding the detailed underlying mathematics.

5. Areas of emphasis • Room and auditorium acoustics • Modeling and simulation of acoustics effects • Digital signal analysis • Filtering • Correlation and convolution • Forensic acoustics examples

6. The Simple Harmonic Oscillator I’m pickin’ up good vibrations… The Beach Boys

7. Simple Harmonic Oscillator (SHO) • SHO is the most simple, and hence the most fundamental, form of vibrating system. • SHO is also a great starting point to understand more complex vibrations and waves because the math is easy. (Honest!) • As part of our study of SHOs we will have to explore a bunch of physics concepts such as: Force, acceleration, velocity, speed, amplitude, phase…

8. Ingredients for SHO • A mass (that is subject to) • A linear restoring force • We have some terms to define and understand • Mass • Force • Linear • Restoring

9. Mass • Boy, this sounds like the easy one to start with; but you’ll be amazed at how confusing it can get! • Gravitational mass and inertial mass. Say what! • What is the difference between mass and weight?

10. Force • What does a force do to an object? • Why is the idea of vectors important? • What is a vector? • What is the difference between acceleration, velocity, and speed? • Acceleration, velocity, and calculus…aargh

11. Calculus review? • What does a derivative mean? • Example:

12. Sin and Cos curves

13. Position versus time graph-what does the slope mean?

14. Velocity versus time graph—what does slope mean?

15. Summarize • Position (a vector quantity) • Velocity (slope of position versus time graph) • Acceleration (slope of velocity versus time graph). Same as the second derivative of position versus time. • Key: If I know the math function that relates position to time I can find the functions for velocity and acceleration.

16. Newton’s Second Law • Relation between force mass and acceleration

17. Apply Newton’s second law to mass on a spring • Linear restoring force—one that gets larger as the displacement from equilibrium is increased • For a spring the force is • K is the spring constant measured in Newtons per meter.

18. Newton's second law • Substitute spring force relation • Write acceleration as second derivative of position versus time

19. Final result • Every example of simple harmonic oscillation can be written in this same basic form.

20. Solution • The solution to the SHO equation is always of the form • To show that this is a solution differentiate and substitute into formula. • Note: A and w are constants; x,t are variables

21. Dust off those old calculus skills • First differential • Second differential

22. Put it all together • Substitute parts into the equation • Conclusion (after cancellations)

23. General form of SHO

24. Why is this solution useful? • We can predict the location of the mass at any time. • We can calculate the velocity at any time. • We can calculate the acceleration at any time.

25. Example • What is the amplitude, A? • How can we find the angular frequency, w? • At which point in the oscillation is the velocity a maximum? What is the value of this maximum velocity? • At which point in the oscillation is the acceleration a maximum? Value of amax?

26. One other item: phase • The solution as written is not complete. The oscillator always is at x=0 at t=0. We could use the solution x=Acos(wt) but that means that the oscillator is at x=A at t=0. The general solution has another component –PHASE

27. Example • To find the phase angle look at where the mass starts out at the beginning of the oscillation, i.e. at t=0. • Spring stretched to –A and released. • Spring stretched to +A and released • Mass moving fast through x=0 at t=0.

28. What is the phase here?

29. Helmholtz Resonator • Trapped air acts as a spring • Air in the neck acts as the mass. (vs is the speed of sound)

30. Helmholtz resonator II • Where is the air oscillation the largest? • Why does the sound die away? Damping • Real length l versus effective length l’. • End correction 0.85 x radius of opening. • Example guitar 1.7 x r.

31. SHO : relation to circular motion • Picture that makes SHO a little bit clearer.

32. Complex exponential notation • Complex exponential notation is the more common way of writing the solution of simple harmonic motion or of wave phenomena. • Two necessary concepts: • Series representation of ex, sin(x) and cos(x) • Square root of -1 = i

33. Exponential function • Very common relation in nature • Number used for natural logarithms • Define (for our purposes) by the infinite series

34. Sin and cos can be described by infinite series • Sin(x) • Cos(x)

35. Imaginary numbers • Concept of √-1 = i • i2 = -1, i3 = -i, i4 = ? • Not a “real” number—called an imaginary number. • Cannot add real and imaginary numbers—must keep separate. Example 3+4i • Argand diagram—plot real numbers on the x-axis and imaginary numbers on the y-axis.

36. Argand diagram

37. Two ways of writing complex numbers • 3+4i = 5[cos(0.93) + i sin(0.93)]

38. Can we put sin and cos series together to get ex series? Not if x is real. But if ix.

39. eix series

40. Complex exponential solution for simple harmonic oscillator • Note: We only take the real part of the solution (or the imaginary part). • Complex exponential is just a sine or cosine function in disguise! • Why use this? Math with exponential functions is much easier than combining sines and cosines.

41. Relation to circular motion. • Simple harmonic motion is equivalent to circular motion in the Argand plane. Reality is the projection of this circular motion onto the real axis.