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Introduction to Acoustics and Simple Harmonic Oscillator in MATLAB

This course explores the fundamental concepts of acoustics through a hands-on approach using MATLAB for sound and signal manipulation. We will cover topics such as room acoustics, digital signal analysis, filtering, and correlations. You will engage in weekly classes focused on understanding the physical principles of acoustics, including the Simple Harmonic Oscillator (SHO) and Newton’s laws. Assignments will enhance your comprehension, and diligent participation will help you achieve a strong grade. Access course materials and homework solutions online.

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Introduction to Acoustics and Simple Harmonic Oscillator in MATLAB

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  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.

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