Lecture 3 Damping, Transients, Envelopes The Principle of Superposition Wave Reflection Instructor: David Kirkby (dkirkby@uci.edu)

Lecture 3Damping, Transients,

EnvelopesThe Principle of

SuperpositionWave Reflection

Instructor: David Kirkby (dkirkby@uci.edu)

Physics of Music, Lecture 3, D. Kirkby 2

MiscellaneousI have added links to the PowerPoint presentation and a condensed printable version of each lecture to the course web site.

My office hours are 9-11am Wednesdays in FRH 3182.

Problem Set #1 is due at the beginning of Thursday’s class.

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Review of Lecture 2We did a quick survey of most of freshman physics:

•Force and acceleration•Vector addition•Newton’s 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)

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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.

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Is Simple Harmonic Motion

Music ?

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Listening to Simple Harmonic MotionListen 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”?

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Towards Musical SoundHere 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)

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During today’s 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.

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Sound EnvelopeWe 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 motion’s amplitude changes with time.

The SHM sound we heard earlier has a box-like envelope:

silence silence

abrupt start abrupt stop

…no change in volume…

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Envelope: Example 1Here are some examples envelopes for musical sounds:

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Envelope: Example 2

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Envelope: Example 3

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What general features do these envelopes have in common? How are they different from SHM?

gong marimba

piano

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Damping and DissipationThe 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.

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Damping ExamplesThese online demonstrations compare the damped and undamped motions of:

•a mass on a horizontal spring•a pendulum

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Damping = Exponential DecayDamping 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).

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Exponential Decay: Example 1 How do you make a candy bar last a long time? Try eating

10% of the candy every minute…

12345678910

123456789

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%

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Exponential Decay LawThe 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.

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Exponential Decay: Example 2Exponential decay is more general than dissipation, which is already very general.

An example that has nothing to do withdissipation is radioactive decay where thenumber of radioactive particles in a samplefollows an exponential decay.

This progression allows us to measure howlong ago a living organism died. This is theprinciple behind Carbon-14 dating (yourbody maintains a tiny constant amount ofradioactive carbon until you die). C-14 hasa half life of about 6000 years.

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Sidebar on the Exponential FunctionThe 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!

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TransientsA 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 player’s lips vibrating

Transients are not easy to describe mathematically, unlike Simple Harmonic Motion (cos,sin) or Damping (exp).

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Transients in Musical SoundsLook for transients in a musical sound at the beginning of its envelope:

Transients!

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Complex VibrationsIf 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:

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The Principle of SuperpositionIn 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 system’s motion of applying two disturbances at the same time is just the sum of the individual motions from each disturbance applied separately.

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We can also express this mathematically (t is time):

Result of disturbance A

Result of disturbance B

Combined resultof disturbances A+B

FA+B(t) = FA(t) + FB(t)

We say that the function F(t) describing the result of thedisturbance is a linear function.

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Sidebar on Linear FunctionsLinear 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(2ft) , exp(-ct)

exp(1+1) = 7.389 exp(1) + exp(1) = 5.437Therefore, exp(A+B) exp(A) + exp(B)

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Superposition: Example 1A 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 (!)

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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.)

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Sidebar on Non-Linear FunctionsAny reasonable mathematical function (including non-linear ones) is approximately linear when considered over a small enough region:

y(x)

x

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Superposition: Example 2A 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) ?

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Try the pulse examples in this 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

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Superposition: Example 3What 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?

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Try the wave examples in this 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?)

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Superposition of >2 DisturbancesWe have focused on examples of the superposition of 2 disturbances, but superposition holds for an arbitrary number of disturbances:

FA+B+C+D+…(t) = FA(t) + FB(t) + FC(t) + FD(t) + …

We can even imagine (and often do!) an infinite number of disturbances combined together. This is fine as longas the disturbances get smaller and smaller sufficiently fast, eg,

1/2 + 1/4 + 1/8 + 1/16 + … = 1

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Back to Complex WavesWe 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.

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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?

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Reflection of PulsesSo 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.

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.

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Reflection and the PoSReflection 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.

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Boundary ConditionsThe 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.

With this new boundary condition, the reflected pulse is no longer flipped!

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Reflection of WavesWhat if we send a periodic wave down the rope instead of a pulse?

Back to the online demonstrations to find out.

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Reflection: Example 1Sound 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.

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Reflection: Example 2Light shining on a smooth shiny surface (eg, a mirror) is reflected back.

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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.

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SummaryAlthough 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 sound’s envelope.

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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.

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Review QuestionsWhat 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?