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									                               Fundamentals of Acoustics and
                                    Fourier Transforms

                                               Reid McCargar

                              Graduate Research Assistant, NW Electromagnetics
                               and Acoustics Research Laboratory (NEAR Lab)

11/18/2010, Reid McCargar 1

                 Lesson 1 – a heavy dose of math

                 • A review of sinusoidal functions – non propagating functions
                 • Relationship between oscillation and rotation
                 • Radians
                 • Calculus and oscillators
                 • Complex exponentials, the uber-brief version
                 • Real oscillating systems
                 • Resonance and frequency response

                 Lesson 2 – a lighter dose of math, more conceptual focus

                 • Propagating waves
                 • Time and frequency domain representations, frequency
                 • Adding waves, adding phasors
                 • Conceptual Fourier synthesis and decomposition
                 • Impulse response and transfer functions
                 • Basics of performing transfer function measurements

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                                There is a point to this

             Do you love sound enough that you
             would want to do it for a living?
             Without needing a day job?

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                              Our stated purpose, before we dive in

                 To design and implement sound systems in which:

                              1. All frequencies within the audio band have
                                 nearly equal magnitude.

                              2. All frequencies arrive synchronously.

                              3. Goals 1 & 2 are achieved as well as possible
                                 for as many listeners as possible.

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                         A review (hopefully) of sinusoidal functions

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                              Visualizing oscillation with rotation

                                      Here t = θ

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                                      Oscillation/rotation math and

                          Period: T – The time it takes to complete a full cycle [sec]

                          Frequency: f = The number cycles in a given length of time [1/sec]

                          Phase: φ – More specifically the initial phase, specifies an offset in
                                     angle (θ ≠ 0 at t = 0)

                          Phase: θ – Everything inside the argument of the cosine

                          DC offset: A0 – The average y-value. Usually taken to be zero, even
                                     when it’s not.

                      Non-propagating because function depends only on time.
                      There is no spatial dependence. Example – voltage output
                                  of a microphone in a fixed location.

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                              Important things—If you remember
                                 nothing else, remember this

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                   Angular frequency…try to remember this, too

                               It’ll make life easier later.

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                               Take 5

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                  Position, Velocity and Acceleration…and force

                     We want to fully describe the motion, x (t) or y (t),
                     of an object.

                     If an object has constant velocity, the position is
                     just v X t, and the velocity can be found from the
                     change in position over a period of time.

                     If not, we rely on some calculus to describe the
                     instantaneous velocity and position. Although
                     calculus scares most people (probably for good
                     reason), it is very simple for sinusoids.

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                               Calculus for the very mathematically

              Derivative : The rate of change of a function – pick a
              point on the function, the slope of the function is the
              derivative at that point.
                                                      f (t)
                                 1.5                  slope







                                        T   2T   3T       4T

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                               Calculus for the very mathematically

                   Velocity is the rate of change of position– it’s the derivative of position.

                     Acceleration is the rate of change of velocity – it’s the 2nd derivative of

                      One last thing: Force = mass X acceleration

                       We can now fully describe the motion of many vibratory systems. All
                            we need to know is the forces and the initial conditions.

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                                      Derivatives of sinusoids

                   You now know 30% of the calculus you need, and you can even forget
                   this part because there’s an easier way. Try to hold on to it for the next
                                                 10 minutes.

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                                      Solution for a simple oscillator

                               Block of mass m, attached to a rigidly-anchored spring.

                               The spring exerts a restoring force proportional to the
                               amount it is stretched, y times a spring stiffness constant,

                               Although this seems really simplistic, loudspeakers and
                               microphones have a lot in common with this vibratory
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                                   Solution for a simple oscillator

                                            First, we use F = ma

                               How do we express this in terms of a derivative of y?

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                                     Solution for a simple oscillator

                                              First, we use F = ma

                               So the equation describing the motion of the object is?

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                                     Solution for a simple oscillator

                                              First, we use F = ma

                               So the equation describing the motion of the object is

                          For now, don’t sweat the m and k. They’re just numbers
                          that don’t change the form of the answer. The important
                          thing is that we have a situation where the 2nd derivative
                          of a function equals a negative constant times the
                          function itself. Can you think of a type of function that
                          satisfies this equation?

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                               Solution for a simple oscillator

                                 Substituted into our equation of motion

                                               We get

                                                                           We call this our
                                                                           natural frequency

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                                  Solution for a simple oscillator









                          T      2T   3T   4T

           So our spring system’s response         Which can be represented in the
           is a sinusoid of a single frequency     frequency domain as a single spike

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                               Break Time

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                               Real oscillators, imaginary numbers

                               Imaginary numbers—it’s like the
                                    Easter bunny, but not

     Although it may seem ridiculous to use numbers that don’t exist, the mathematics of
     sound become absurdly complicated without them.

     The concept of imaginary numbers was opposed for many years in the scientific
     community because obviously, they don’t make sense. Does π make sense?
     Perhaps in some twisted way, but you really can’t run from it. Numbers like these are
     fundamental to nature.

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                               Some basics of complex numbers

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                               The complex plane

                                           A complex number
                                           describes a right triangle
                                           in the complex plane

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                               Life, death, and another nonsense number

                                                      Most things in nature grow and
                                                      die in this fashion.

                                                      To math geeks, it’s about as
                                                      exciting as π, or Optimus

                                                      e gets to be more fun for
                                                      everybody else when you see
                                                      how much easier it makes
                                                      calculus and vibrational system

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                               Complex exponentials
                                      Proving this requires some acrobatics that
                                      aren’t relevant. Just accept it or talk to me

                                  Ok, but how does this make our lives any easier?

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                               Derivatives of complex exponentials

                                          So far it looks pretty simple. Anyone remember
                                          what the derivative was for cosine? sine?

     For the 1st derivative, we just multiplied by . For the second derivative, we just
     did it twice. Is calculus with complex exponentials as simple as multiplying by
     every time you see a derivative? Actually yes.

     You’re now 2/3rds done with calculus, and the remaining bit is even easier.

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                       Antiderivatives with complex exponentials

                      Quite often we need to do derivatives in reverse. With complex
                      exponentials, we multiply by      to differentiate. To undo
                      differentiation (an operation called integration) we … divide by   .

                                                                  Alright, that’s all the calculus,
                                                                  and it’s as easy as multiplying
                                                                  or dividing by       .

                                                                  The integral over some range
                                                                  a to b is just the area under the

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                               Take 5

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                                 Solution for an unforced oscillator
                               Let’s take a look at a more realistic oscillator—one in which
                               there is also a damping force proportional to the velocity:

                               Let’s take a look at a more realistic oscillator—one in which there
                               is also a resistive damping force proportional to the velocity:

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                                        Solving the damped oscillator
                         So the 2 forces acting on our object are the spring restoring
                         force, and the resistive damping force. F = ma gives

                               I won’t solve this now so we can get to resonance, but I’ll
                               give you a hint so you can solve it at home. If we assume
                               the solution is a complex exponential, rearrange this and
                               divide everything by m, we get something in the form of a
                               quadratic equation:

                                You can cancel y, then solve for the angular frequency.

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                                     Damped oscillator solution

                               If you go through the math, you’ll find that

                                          Remember what the natural
                                          frequency of our undamped
                                          oscillator was?

   So our damped oscillator oscillates at a lower frequency than the undamped system.

                               Using this, we find the complete solution is:

        What’s this?

                                                                               I know you remember

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                               Damped oscillator solution

                               Let’s define a decay coefficient:

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                                    Forced oscillations

     There are quite a few systems that behave in (somewhat) principle like
     unforced, damped oscillators:

     • cymbals & vibrating drumheads
     • plucked or struck strings
     • systems in which there is no energy input into the system after being set in motion

     And many that don’t:

     • loudspeakers driven with an amplified electric signal
     • microphones, which are driven by pressure fluctuations
     • bowed strings
     • systems with forces that persist after initial excitation

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                                Solution for a forced oscillator

           Our equation of motion is now:

           If we suppose a forcing function:

                               After a bunch of math, we find that


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                                             Mechanical impedance

           It makes sense to define a quantity called impedance:

           For this system, the impedance is:

                               The smallest this can ever be is   , which happens when

                This condition is known as resonance. Designing and
                implementing good systems is largely a game of working with,
                and working around resonance.

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                               Resonance of a simple oscillator

                                              A 1 kHz resonance frequency was
                                              chosen to make these plots. There is
                                              a mathematical definition of Q, which
                                              I’ll spare you. Higher Q results from
                                              low damping.

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            What just happened?

            • We reviewed sinusoidal functions & rotation
            • We learned about radians, phase, frequency, and period
            • We learned calculus for sinusoids, which we forgot
            • We learned about an idealized system with a sinusoidal solution
            • We learned about complex exponentials in order to forget about sinusoids
            • We learned the calculus of complex exponentials
            • We dealt with real oscillating systems and resonance

            In short, you’ve had a lot of difficult concepts thrown at you. Don’t
            expect to understand or remember it all right away. Think about it the
            next time you wait for the bus or would otherwise zone out.

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            Next time we’ll be doing more of the same,
           but dealing with functions of time and space.

                  The good news is that we’ve covered the
                            worst of the math.

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