Musical AcousticsSpring 2011Prof. Thomas

Acoustics - The Nature of SoundIts ProductionPropagationPerception

Everything should be as simple as possible, but no simpler.

Sound is a mechanical wave

“Mechanical” means “something moves” but be careful… the motion of the thing that moves and the motion of the wave are different!

Let’s look at different waves:

SlinkyWave Table

A mechanical wave is a disturbance (in a medium) that moves.

The “disturbance” of the medium (in a wave) …

may (in some media) be perpendicular to the wave motionwe call this a transverse wave

may be parallel to the wave motionwe call this a longitudinal wave

The wave on the wavetable is:

A] transverseB] longitudinal

But what properties of the medium allow a wave?

Pretty clearly, if there is no “restoring force,” there are no waves(If we try to send a wave through a putty-like material, it stops!)

The medium must “want” to return to its original state.

With the wavetable, you may also notice that the bars, in “trying to return” to their rest positions, overshoot!

This is because of inertia.

Properties that allow for mechanical waves:1) Restoring force2) Inertia

Sound (in this class) is a mechanical wave in air

Air is the medium

What is the motion? What is the restoring force?

Demonstration: Compressing air increases pressure

Sound is a wave involving compression of air.

Compressing the air increases the pressureThe air is pushed from high pressure regions to lower pressure regions -- a “restoring force”

Higher pressure occurs when there is a higher density of air molecules(in fact, P is proportional to the density of air molecules)

In order for there to be regions of high and low density, the air molecules must move (a little) back and forth.

Sound waves (in air) are

a) transverseb) longitudinal

Sound waves (in air) are longitudinal.

As an aside: not all waves are strictly longitudinal or transverse.

http://webphysics.davidson.edu/faculty/dmb/py115/waves.1.html

Ocean waves: water moves in circular motion

“Sound” waves in solids

The simplest waves are sine functionsThe next simplest waves are any repetitive function

The wavelength is the distance you have to go until the pattern repeats.

For a sine (or sinusoidal) wave, this is the distance from one peak to the next, or, for sound, from one compression to the next (or from one rarefaction to the next.)

The period P of a wave is the time until the pattern repeats.

The frequency of a wave is the number of repetitions that pass you in each second. f = 1/PIn music, frequency of sound is sensed as pitch.

A graph of a simple wave

Which length is the wavelength?

A graph of a not so simple wave

Which length is the wavelength?

The amplitude of a wave is the maximum extent of the disturbance. This may be measured by the maximum displacement of the medium from its rest position.

You notice that ocean waves passing a pier support have peaks at the 10 foot mark and troughs at the 4 foot mark. What is the amplitude of these waves?

A) 0 feet

B) 3 feet

C) 6 feet

D) 10 feet

More about pressure….

When there is a pressure difference (say between the inside and outside of a balloon) That pressure difference causes a force (or push) on the balloon wall.

The force on a part of the wall is proportional to the area of that part.

Specifically F = P x A

Or P = F / A

What is a “force”?

• A force is a push or a pull. • Your weight is a force… it’s the force the earths’ gravity pulls you

with• The metric unit of force is a Newton, named after Sir Isaac. It’s

abbreviated N.• 1 N is about equal to the weight of an apple!• 1 lb of weight is about 4 1/2 N.• Force also has a direction. (We call these vectors, but ‘nuff said.)

The force of gravity (your weight) pulls you down.The force of your chair on your *** pushes you up.So you don’t move!

Since force is in Newtons, and area (in metric system) is in meters2, pressure P = F/A is in N/m2.

The cartoon shows a pressure of 4 N/m2 acting on a wall with an area of 6 m2. What is the total force on the wall from this pressure?

A] 10 N

B] 24 N/m2

C] 24 N

D] 64 N

For no good reason (IMO), we also sometimes measure pressure in “atmospheres”.

At sea level, the pressure of the air is about 100,000 N/m2.

The area of your chest is about a quarter of a m2. At sea level, then, the air pushes down on your chest with a force of about 25,000 N > 5500 lbs!!

Why isn’t your chest crushed under this force?

BIG and small Numbers

• Since we sometimes have to deal with very big or very small numbers, we use “scientific notation”.

• There’s a handout on the class webpage that might be helpful.

• 1 atm ≈ 105 N/m2

• Examples on the board

Let’s do some “physics”!(This will not be on the exam!)

Consider a sound wave with wavelength 20 cm.

If, during a cycle, the air molecules moved a half wavelength, essentially all of them would end up in the compressions. (That’s far enough for a molecule to move all the way from a rarefaction to a compression.)

Let’s guess-timate that if they did this, they would double the density in a compression (over ambient density) (We moved all the molecules in the troughs to the peaks… doubling seems about right.)

What would be the pressure in the peaks and in the troughs?

Since pressure is proportional to density, doubling the density would double the pressure.

The peaks would have twice atmospheric pressure, and the troughs would have zero pressure!

Normal (pleasant) sound pressures are one-millionth of this!

Since 10 cm of motion gives a pressure change of a million times “normal” sound, we estimate that air motion for normal sound is about a million times less.

That’s about right!

We said that amplitude is the size of the disturbance (measured from the resting state)

For sound waves, we could measure the amplitude by how far air molecules moves from their rest positions.We call this the displacement amplitude.

Our calculations suggest that they move very little!(< millionths of a meter)

It’s easier to measure small pressure changes in a sound wave. So usually we describe the amplitude of a sound wave in terms of the maximum change in pressure. This is called the pressure amplitude.

Musical sounds have pressure amplitudes of about 0.01 - 1 N/m2 … about one-millionth of an atmosphere.

Prairiedogese:

http://www.npr.org/2011/01/20/132650631/new-language-discovered-prairiedogese

What is the wavelength?

A] about 1 m

B] about 2 m

C] about 3 m

D] about 4 m

E] There is no wavelength since it’s not a pure sine function

Speed of sound

In air at room temperature (20°C, 68°F), the speed of a sound wave is 344 m/s.

This is how fast a pressure peak moves. It is not how fast individual air molecules move, but it is related to that.

In order for the pressure peak to move, molecules must collide with their neighbors. The faster they do this, the faster the pressure peak can move.

How can you make individual air molecules move faster?

Add energy, i.e. heat them up!

The speed of sound increases 0.6 m/s for each 1°C above 20°C

Example: at 104°F, (40°C), the speed of sound is

344 + 0.6 x 20 = 356 m/s. €

V = 344 ms + 0.6ms°C (T −20°C)

We will see that the speed of sound is very important to the pitch of musical instruments, including the human voice.

The speed of sound in helium gas is 1000 m/s! What happens to the pitch of your voice if you inhale helium?

The distance sound (or anything else) goes in time t is

€

d = v × t

Notice the units (or dimensions) of each termDistance in metersSpeed (or velocity) in meters/secondTime in seconds

Also: the time it takes sound to go a distance d is

€

t = d /v

The speed of sound is the same for different wavelengths or different frequencies.

“Nondispersive”

This is a good thing! What would a marching band sound like if, for example, the high pitches (high frequencies) traveled

faster than the low pitches?

For the simplest waves, what is the relationship between frequency and wavelength?

Slinky Demo

The higher the frequency, the shorter the wavelength.

Since frequency is the number of wavelengths that pass by in a second, the wave speed must be

€

v = f ⋅λ

What is the wavelength of sound with a frequency of 344 Hz (waves/second)?(This is ~ F above middle C)

A] 0.1 mB] 1 mC] 10 mD] 100 m

The speed of sound is 344 m/s.

Sources of sound: “simple harmonic motion”

Sound is an wave / oscillation of air. We saw that wave oscillations require restoring force & inertia

An object (as opposed to a medium) can sustain oscillations if it also has restoring force & inertia.

The simplest example is a mass & spring.

If an object (in air) is set into oscillation at an audible frequency (between 20 and 20,000 Hz) it will make sound.

Let’s study the mass & spring a little bit - demoWe pretend our spring is “ideal”, meaning that the force it exerts is proportional to how much it deviates from its extension at rest. (In the actual demo, gravity contributes too, but the combined effect of gravity + spring is ideal : it pushes downward if the mass is above the rest point, and pulls upward if the mass is below the rest point.)

Key points

1. The motion is “sinusoidal”. (The position as a function of time is the same shape as the sine (or cosine) function.)

2. Adding mass slows the oscillation. Think “inertia resists change”

3. Using a stiffer spring (more force at the same extension) speeds the oscillation. Think “pushing harder moves things faster”

4. The frequency of the oscillation is independent of the amplitudeThere’s a formula for the frequency that captures 2&3

€

f =1

2π

k

mk is the spring stiffness

m is the mass

Interpreting graphs of SHM

At what point is the mass at its rest position?(i.e. where it rests when not in oscillation)

Or choose C, at neither point.

Interpreting graphs of SHM

At what point is the mass moving upward (toward +y)

Or choose C, at neither A nor B is it moving upward

Interpreting graphs of SHM

At what point is the mass “at rest”? (i.e. at what point is the mass not moving?)

Or choose C, it’s always moving

Interpreting graphs of SHM

Each tick on the t axis is 1 second. What is the period of oscillation?

A] 1/2 sB] 1 sC] 2 s

D] 4 s

0 1 2 3 4

The same spring is used with different masses, and graphs of the resulting SHM are made. Which graph corresponds to the larger mass?

The same mass is used with different springs, and graphs of the resulting SHM are made. Which graph corresponds to the softer (or weaker) spring?

Energy & “Work”

In physics, “work” has a special (and non-intuitive) meaning.A force does work on an object if it acts while the object moves through some distance.

Work = Force x distance

Example: I push a block with 10 N of force through 2 m across the desk. I do 20 Nm of work.

Energy & “Work”

If I hold books weighing 20 N at a height of 3 m for an hour, how much (physics) work do I do?

A] 0

B] 20 Nm

C] 60 Nm

D] 60 Nm-hr

Energy & “Work”

If I slowly raise books weighing 20 N from the floor to a height of 3 m, how much (physics) work do I do?

A] 0

B] 20 Nm

C] 60 Nm

D] 60 Nm-hr

Why did physicists invent such a silly concept?(It’s damned hard work to hold 100 N of books over

your head for an hour, no matter what physicists say!!)

It turns out that doing work (physics-type work) on an object increases its energy.

Holding books overhead for an hour may be “hard work”, but it doesn’t change the energy of the books.

What is “energy”? Easy…

It’s the ability to do work!Does this seem circular?

When I lift books up, I do work on them. I could let them turn a generator as they drop back down, making electrical energy (by doing work on the generator).

Work can be positive or negative!If the object moves in the direction in which it is being pushed by my hand, I do positive work on it.

If it moves in the opposite direction, I do negative work on it, and it does positive work on my hand.

A block sliding on a frictionless surface is brought to a stop by a (non-ideal) spring.

The spring exerts a constant force of 10 N on the block.The block, after first touching the spring, moves 0.2 m before stopping.

(After it stops, of course, it will “bounce back”, but we don’t care about that now.)

How much energy did the block have, when it was moving full speed?

A] 0 NmB] 1 NmC] 2 NmD] 10 Nm

The spring did -10 x 0.2 Nm of work on the block. The block did + 10 x 0.2 Nm of work on the spring.So 2 Nm of energy was transferred from the block to the spring.

Blocks in motion can do work, so they have energy.“Kinetic Energy”

After the block stops (but before it has rebounded), it has used all its kinetic energy to do work on the spring.

What has happened to that energy?

A] It is gone foreverB] It is in the etherC] It is now stored in the spring

Blocks in motion can do work, so they have energy.“Kinetic Energy”

Compressed springs can do work, so they have energy.“Potential Energy”

I need to CMA here for those of you who are physics afficianados.When there is friction in a system, it sometimes appears that energy is lost… in other words, we do work on a system but can’t get that energy back as work. (Consider pushing a block across the desk, for example.)

James Prescott Joule (1818-1889) showed that heat is a form of energy. In other words, although we can’t get all the energy back in the form of work, we can still “conserve energy” when we count the heat energy correctly.

To honor Joule’s contribution, we also call unit of work (the Nm) a “joule”,Abbreviated J