Lecture 3 Damping, Transients, Envelopes The Principle of Superposition Wave Reflection Instructor: David Kirkby (firstname.lastname@example.org)
Physics of Music, Lecture 3, D. Kirkby2 Miscellaneous I have added links to the PowerPoint presentation and a condensed printable version of each lecture to the course web site.course web site My office hours are 9-11am Wednesdays in FRH 3182. Problem Set #1 is due at the beginning of Thursdays class.
Physics of Music, Lecture 3, D. Kirkby3 Review of Lecture 2 We did a quick survey of most of freshman physics: Force and acceleration Vector addition Newtons Second Law Simple Harmonic Motion Waves We were able to cover so much material by focusing on the concepts rather than how to perform calculations and solve problems. You already know most of the material intuitively (eg, Crown Victoria vs. Mustang acceleration)
Physics of Music, Lecture 3, D. Kirkby4 We studied 3 examples of Simple Harmonic Motion(SHM): Mass on a horizontal spring Pendulum (mass on a vertical string) Uniform circular motion (eg, a mass on a horizontal string or the Earth circling the Sun) SHM is a common feature of many different physical processes. The SHM of a solid body generates sound waves with the same frequency and amplitude in the air surrounding it. Sound propagates as a longitudinal wave at about 345m/s.
Physics of Music, Lecture 3, D. Kirkby5 Is Simple Harmonic Motion Music ?
Physics of Music, Lecture 3, D. Kirkby6 Listening to Simple Harmonic Motion Listen to this example of a sound created by SHM. Does it remind you of any musical instruments? What aspects of the sound are unnatural or unmusical?
Physics of Music, Lecture 3, D. Kirkby7 Towards Musical Sound Here is a graphical representation of the sound you just heard: This sound is lacking in two general areas: it starts and stops abruptly with no shape to it the tone is pure but unnatural (and uninteresting)
Physics of Music, Lecture 3, D. Kirkby8 During todays lecture we will learn what is needed to improve the sound in both of these areas: Damping and transients give a sound its shape. Vibrations that are more complex that SHM give a sound its musical tone. The Principle of Superposition is a powerful tool for understanding these more complex vibrations in terms of SHM.
Physics of Music, Lecture 3, D. Kirkby9 Sound Envelope We call the shape of a sound its envelope. An envelope fits snugly around the maximum motion of the air particles (or a solid object undergoing SHM) and so tracks how the motions amplitude changes with time. The SHM sound we heard earlier has a box-like envelope: silence abrupt startabrupt stop …no change in volume…
Physics of Music, Lecture 3, D. Kirkby10 Envelope: Example 1 Here are some examples envelopes for musical sounds:
Physics of Music, Lecture 3, D. Kirkby11 Envelope: Example 2
Physics of Music, Lecture 3, D. Kirkby12 Envelope: Example 3
Physics of Music, Lecture 3, D. Kirkby13 What general features do these envelopes have in common? How are they different from SHM? gongmarimba piano
Physics of Music, Lecture 3, D. Kirkby14 Damping and Dissipation The main feature missing from the SHM envelope is a gradual decay of the amplitude as the sound dies out. This decay process is called damping. Damping is present in most examples of SHM. It is usually the result of friction taking energy permanently away from the vibrations (dissipation). Dissipation occurs in all physical processes. It is the reason why a ball will not bounce forever and why you have to keep pumping to keep a swing going.
Physics of Music, Lecture 3, D. Kirkby15 Damping Examples These online demonstrations compare the damped and undamped motions of:online demonstrations a mass on a horizontal spring a pendulum
Physics of Music, Lecture 3, D. Kirkby16 Damping = Exponential Decay Damping generally means that the amplitude of vibrations (ie, how loud the sound is) decreases by a fixed fraction per unit time: Amplitude(now) = Fraction x Amplitude(before) In this case, how long does it take for the sound to completely disappear? If a quantity decreases by a constant fraction per unit time, then we say it obeys an exponential decay law. The quantity can be anything (not just sound amplitude).
Physics of Music, Lecture 3, D. Kirkby17 Exponential Decay: Example 1 How do you make a candy bar last a long time? Try eating 10% of the candy every minute… 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 100% 90% 81% 73% 59% 48% 66% Mins: 1 2 3 4 5 6 7 8 9 10 11… How many minutes until you finish the candy? 53% 43% 39% 35% 31%
Physics of Music, Lecture 3, D. Kirkby18 Exponential Decay Law The progression of an exponential decay is described by a single number which we can choose to be the fraction that disappears per unit of time (eg, 10%/minute). Another number we could choose is how long it takes for half to disappear. This was about 6.5 minutes in our example. Is it obvious that we will loose another half every 6.5 minutes? The time taken for another half to disappear is called the half life of the decay.
Physics of Music, Lecture 3, D. Kirkby19 Exponential Decay: Example 2 Exponential decay is more general than dissipation, which is already very general. An example that has nothing to do with dissipation is radioactive decay where the number of radioactive particles in a sample follows an exponential decay. This progression allows us to measure how long ago a living organism died. This is the principle behind Carbon-14 dating (your body maintains a tiny constant amount of radioactive carbon until you die). C-14 has a half life of about 6000 years.
Physics of Music, Lecture 3, D. Kirkby20 Sidebar on the Exponential Function The mathematical function that describes exponential decay is: N(t) = N(0) exp(- c t) 100%90%81%73%59%48%66%53%43%39%35%31% The exponential function is not a straight line!
Physics of Music, Lecture 3, D. Kirkby21 Transients A transient is a brief burst of motion. Transients in physical systems are often the result of an explosive external disturbance. Examples: A hammer used to start a horizontal spring in motion The initial force required to start a trumpet players lips vibrating Transients are not easy to describe mathematically, unlike Simple Harmonic Motion (cos,sin) or Damping (exp).
Physics of Music, Lecture 3, D. Kirkby22 Transients in Musical Sounds Look for transients in a musical sound at the beginning of its envelope: Transients!
Physics of Music, Lecture 3, D. Kirkby23 Complex Vibrations If we zoom in to examine a short segment of a musical sound, we discover that its vibrations are more complex than we expect from Simple Harmonic Motion:
Physics of Music, Lecture 3, D. Kirkby24 The Principle of Superposition In order to study these more complex vibrations, we will use a very powerful characteristic of all linear systems known as the Principle of Superposition (PoS): The combined effect on a systems motion of applying two disturbances at the same time is just the sum of the individual motions from each disturbance applied separately.
Physics of Music, Lecture 3, D. Kirkby25 We can also express this mathematically (t is time): Result of disturbance A Result of disturbance B Combined result of disturbances A+B F A+B (t) = F A (t) + F B (t) We say that the function F(t) describing the result of the disturbance is a linear function.
Physics of Music, Lecture 3, D. Kirkby26 Sidebar on Linear Functions Linear functions all have the same general form: F(t) = a + b t Where a and b are two arbitrary parameters (numbers). Most mathematical functions are non-linear. E.g., sin(2 ft), exp(-ct) exp(1+1) = 7.389exp(1) + exp(1) = 5.437 Therefore, exp(A+B) exp(A) + exp(B)
Physics of Music, Lecture 3, D. Kirkby27 Superposition: Example 1 A glass of water is a trivial example of a linear system. Disturbance A = pour in 1/4 of a glass Disturbance B = pour in 1/2 of a glass What is the result of the combined A+B disturbances? The glass is 3/4 full (!)
Physics of Music, Lecture 3, D. Kirkby28 Most interesting physical systems are not exactly linear. Examples: A water glass filled beyond its top A spring compressed almost to zero A spring stretched beyond its breaking point These systems are described by non-linear functions. However, all systems are approximately linear for small enough disturbances! (Transients are usually due to momentary excursions beyond the range of linear disturbances.)
Physics of Music, Lecture 3, D. Kirkby29 Sidebar on Non-Linear Functions Any reasonable mathematical function (including non-linear ones) is approximately linear when considered over a small enough region: y(x) x
Physics of Music, Lecture 3, D. Kirkby30 Superposition: Example 2 A long rope provides a medium for transverse waves. To keep things simple, we can send individual pulses down the rope and just focus on the middle of the rope (ie, ignore what happens when a pulse reaches the end). Disturbance A = pulse sent from left end of rope. Disturbance B = pulse sent from right end of rope. What does the PoS tell us will happen when pulses are sent from both ends of the rope at once (Disturbance A+B) ?
Physics of Music, Lecture 3, D. Kirkby31 Try the pulse examples in this online demonstration…online demonstration What did we learn? Two pulses can pass right through each other without disturbing each other Two positive pulses combine momentarily to make a single pulse that is twice as big: constructive interference A positive pulse and a negative pulse can exactly cancel one another momentarily: destructive interference
Physics of Music, Lecture 3, D. Kirkby32 Superposition: Example 3 What if, instead of pulses, we apply a periodic SHM-like disturbance to the rope? Again, we ignore the ends of the rope. Disturbance A = SHM wave traveling left to right Disturbance B = SHM wave traveling right to left Disturbances A and B could have different frequencies and amplitudes. What does the PoS tell us now?
Physics of Music, Lecture 3, D. Kirkby33 Try the wave examples in this online demonstration…online demonstration The resulting wave goes back and forth between constructive and destructive interference. The resulting wave also moves to the right. (Why not to the left?)
Physics of Music, Lecture 3, D. Kirkby34 Superposition of >2 Disturbances We have focused on examples of the superposition of 2 disturbances, but superposition holds for an arbitrary number of disturbances: F A+B+C+D+… (t) = F A (t) + F B (t) + F C (t) + F D (t) + … We can even imagine (and often do!) an infinite number of disturbances combined together. This is fine as long as the disturbances get smaller and smaller sufficiently fast, eg, 1/2 + 1/4 + 1/8 + 1/16 + … = 1
Physics of Music, Lecture 3, D. Kirkby35 Back to Complex Waves We learned about the Principle of Superposition in order to help us understand the complex waves that characterize musical sound. How is it helping us? A musical sound is the result of a complex vibration. The PoS tell us we can understand a complex vibration as the combined result of many simpler vibrations.
Physics of Music, Lecture 3, D. Kirkby36 Armed with the PoS, we study a musical sound by asking: What are the basic simple vibrations (modes) that combine to make the final complex vibration? What determines the contribution of each mode to the final sound?
Physics of Music, Lecture 3, D. Kirkby37 Reflection of Pulses So far we have ignored what happens at the end of a long rope when a pulse reaches it. What does happen? Try this online demonstration to find out.online demonstration The reflected pulse is flipped: a positive pulse becomes a negative pulse and vice versa. Why? Because the fixed end of the rope cannot move and this is the only way to do it.
Physics of Music, Lecture 3, D. Kirkby38 Reflection and the PoS Reflection is mathematically the same as sending a negative pulse from the opposite end of a long rope that meets the positive pulse where the fixed end would be. Compare the earlier demonstrations of reflections and superposition to convince yourself.
Physics of Music, Lecture 3, D. Kirkby39 Boundary Conditions The fact that the end of the rope is fixed is an example of a boundary condition. Another possible boundary condition is that the end of the rope is free to move. What is the motion in this case? Try this online demonstration to find out.online demonstration With this new boundary condition, the reflected pulse is no longer flipped!
Physics of Music, Lecture 3, D. Kirkby40 Reflection of Waves What if we send a periodic wave down the rope instead of a pulse? Back to the online demonstrations to find out.online demonstrations
Physics of Music, Lecture 3, D. Kirkby41 Reflection: Example 1 Sound waves that hit a smooth solid object (eg, a large building) are reflected back. This is just an echo. Did you know that the reflected sound is the negative of the original sound? The reflections of sound waves against the walls of a room have a big influence on the sound we hear.
Physics of Music, Lecture 3, D. Kirkby42 Reflection: Example 2 Light shining on a smooth shiny surface (eg, a mirror) is reflected back.
Physics of Music, Lecture 3, D. Kirkby43 How Smooth? Reflection occurs from any surface, but the reflection from a rough surface is incoherent (bounces off in all directions) which the reflection from a smooth surface is coherent (bounces all in the same direction). How smooth is smooth enough for coherent reflections? The wavelength sets the scale. Remember that sound has wavelengths about 1 meter while light has wavelengths about 1 micrometer.
Physics of Music, Lecture 3, D. Kirkby44 Summary Although SHM of an object generates sound, it is not a musical sound. Two key features missing from the sound generated by SHM are envelope and complex vibrations. Damping is the result of dissipation and leads to an exponential decay of a sounds envelope.
Physics of Music, Lecture 3, D. Kirkby45 The PoS is a powerful tool for understanding a complex vibrations in terms of the simple modes that contribute to it. Reflection is a universal feature of any wave that passes from one medium into another.
Physics of Music, Lecture 3, D. Kirkby46 Review Questions What does the envelope of SHM look like? How long does an exponentially decaying sound last for? Can a string vibrate at two different frequencies at the same time? Why does sound reflect off a brick building but light does not?