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This is PHYS 1240 - Sound and Music

Lecture 15 - review

Professor Patricia Rankin

Cell Phones silent

Clickers on

Physics 1240 Lecture 15

Today: Review

Next time: EXAM

Next Week : Scales, Intensity, Loudness

physicscourses.colorado.edu/phys1240

Canvas Site: assignments, administration, grades

Homework – HW6 Due Wed March11th 5pm

Homelabs – Hlab4 Due March 16th 5pm

Exam

Midterm – 1hr, Thursday March 5th – 3:30-4:30pm here

Accommodations – G135 March 5th – 3:30-5:00pm

(need to be on list)

Mix of 10 short questions (5pt), 5 longer ones (10pt)

Short – more clicker like (quick), just give answer

Long – more math, closer to homeworks

Based on first 12 lectures, first 6 homeworks and

relevant book chapters.

From page 1 of Hartmann….

Units

Base units:

meters [m] (3.3 ft), kilograms [kg] (2.2 lb), seconds (s)

Prefixes:milli (m) 0.001 1⨯10-3

centi (c) 0.01 1⨯10-2

deci (d) 0.1 1⨯10-1

kilo (k) 1000 1⨯103

mega (M)1,000,000 1⨯106

Unit Symbol Conversion

SI pascal Pa 1 Pa ≡ 1 N/m2

other atmosphere atm 1 atm = 101325 N/m2

otherpounds per

square inchpsi 14.7 psi = 1 atm

Pressure

Force per unit area (e.g. thumbtack, gas molecules hitting

wall, ears, lungs)

Sound is a mechanical disturbance of the pressure in a

medium that travels in the form of a longitudinal wave.

Wave Properties

• Speed (v=343 m/s for air at 20°C and 1 atm)

• Wavelength (λ in meters)

• Frequency (ƒ in hertz)

• 1 Hz = 1 s-1

v = λ ƒ

[m/s] = [m] [Hz]

V comes from velocity…

amplitude

Equilibrium

height

Key Formula

• frequency ∝stiffness

mass( 𝑓 =

1

2𝜋

𝑠

𝑚)

• Frequency, period 𝑓=1

𝑇

• Intuitive: trampoline, tight vs. loose string, tuba vs.

flute

Suppose you hang a 4.15 gram mass on a spring with a stiffness of 5.63 N/m. At what frequency (in Hz) will the mass oscillate? Enter only the number, not the number and the unit. Note that the SI unit for mass is kg, not g.

Answer:

5.87 Hz

𝑓 =1

2𝜋

5.63

0.00415=

1

2𝜋 1357 =

36.84

2𝜋

HW2, Prob 2

Transverse/Longitudinal Waves

The wave follows a path – the direction of

propagation

But the “medium” – the material in which the wave

propagates can go

“up and down” – perpendicular to the direction of

propagation

or “back and forth” – along the direction of propagation

Transverse wave – motion of the medium is

perpendicular to the direction of wave propagation

(peaks and troughs)

Longitudinal waves – motion of the medium is in the

direction of wave propagation (eg sound – compressions,

rarefactions)

𝜈 =𝜆

𝑇= 𝜆𝑓

speed of sound (m/s)

wavelength (m)

period (s)frequency (Hz)

HW5, Prob 5

Clicker 15.1

x(t) = A sin(360 t/T + ϕ)

What is the starting phase ϕ of the solid curve?

A) 0

B) 90

C) 180

D) 270

E) none of the above

3-4

Clicker 15.1 D

What is the starting phase ϕ of the solid curve?

A) 0

B) 90

C) 180

D) 270

E) none of the above

3-4

x(t)=A sin(360 t/T + ϕ) or x(t)=A sin(360 f t + ϕ)

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2 0.25

dis

pla

cem

ent

(cm

)

time (s)

sine wave function

HW2, Prob 4,5,6,7,8

x(t)=4*sin(360*20*t)

HW3, Prob 8

Then took basic system and

made more complex…..

Damping

All real oscillators have some damping

Natural Mode/Normal Mode

Most things have a natural vibration mode or modes and can oscillate or vibrate in more than one way

Resonances or Periodically Driven Oscillators

Drive frequency = mode frequency → big response

Doppler Effect

Source and receiver are moving relative to each other

Superposition

Dealing with more than one source of waves

Damping/Resonance/Estimates

Normal or natural modes

Q – Damping

number of cycles before amplitude

drops to 4.32% of starting value

Q – Resonance – peak frequency/bandwidth that excites

motion at least half peak amplitude

Estimating distance using sound – echoes, lighteningAfter you see lightning, start counting to 30 (30s). If you hear thunder before you reach 30, go indoors. Suspend activities for at least 30 minutes after the last clap of thunder.

𝜈 =𝜆

𝑇= 𝜆𝑓

Wavelength

PeriodFrequency

• Decreasing amplitude: damping

• Increasing amplitude: resonance

• Damping:

• All oscillations eventually decay away, unless driven

• What causes sound to decay?

• (resistance, loss of energy)

• What happens to the frequency?

• (nothing – amplitude not freq)

Resonance – Tacoma Narrows Bridge

Clicker 15.2

On a cool summer evening when the air temperature is 20°C, you see a

flash of lightning and hear the sound of thunder 2 seconds later. Assuming

you saw the light at the same time the sound was produced, how far away

was the bolt?

A) 172 m

B) 343 m

C) 686 m

D) 1.7 km

E) 20 km

Clicker 15.2 C

On a cool summer evening when the air temperature is 20°C, you see a

flash of lightning and hear the sound of thunder 2 seconds later. Assuming

you saw the light at the same time the sound was produced, how far away

was the bolt?

A) 172 m

B) 343 m

C) 686 m

D) 1.7 km

E) 20 km

(343 m/s)⨯(2 s) = 686 m

HW3, Prob 3

A (normal) mode is a motion where every point moves with the same frequency

𝑳

A node is a place where a mode has no motion

An anti-node is a place where a mode has maximum motion

Harmonics

3rd harmonic,3f

4th harmonic, 4f

5th harmonic, 5f

2nd harmonic, 2f

1st harmonic, f

Doppler Effect

• Doppler effect: the shift in frequency of a wave where

the source and the observer are moving relative to one

another (higher frequency if moving toward each other)

∆𝑣

𝑣sound≅ 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 change in 𝑓 =

𝑓1−𝑓0

𝑓0

∆𝑣 = source velocity – observer velocity

𝑓0=emitted frequency

𝑓1=received frequency

HW4, Prob 8

Refraction

Light

Why straws look bent in water

Refraction:

bending due to a change in the

speed of sound (change in medium)

Diffraction

Allows sound to get around corners

Depends on wavelength of wave/size of gap

peaktrough

Adding/Superimposing

Constructive Interference

• For ∆𝐿 = 𝐿2 − 𝐿1 = difference between your distance from one source

and your distance from a second source:

• Constructive: ∆𝐿 = 𝑛λ (where 𝑛 is an integer), n=1 longest wavelength

Destructive Interference

• For ∆𝐿 = 𝐿2 − 𝐿1 = difference between your distance from one source

and your distance from a second source:

• Destructive: ∆𝐿 = (𝑛 + 1/2)λ2

(where 𝑛 is an integer)

HW4, Prob 1

HW5, Prob 7

Beats

0 0.2 0.4

=

0 0.05 0.1

100 Hz105 Hz

= 5 Hz

P = 0.2 s

f = 1 / P = 5 Hzfbeats = f2 – f1

Waveform

• Waveform: the shape that forms the repeating pattern of a wave

Timbre

• Timbre: the musical quality of a sound wave that isn’t encompassed by its

pitch or loudness

Below is a picture of a standing wave on a 30 meter

long string.

What is the wavelength of running waves that

the standing wave is made from?

L = 30 mA.30 m

B.60 m

C.15 m

D.Impossible to tell

Clicker 15.3

Below is a picture of a standing wave on a 30 meter

long string.

What is the wavelength of running waves that

the standing wave is made from?

L = 30 mA.30 m

B.60 m

C.15 m

D.Impossible to tell

Clicker 15.3 B

A (normal) mode is a motion where every point moves with the same frequency

𝑳 = λ/2 ;

= 2L

A node is a place where a mode has no motion (interior nodes – one less than harmonic)

An anti-node is a place where a mode has maximum motion (antinodes = harmonic)

(count to get # of mode)

𝑳 = λ

𝑳 = 3λ/2 ; = 2L/3

𝑳 = nλ/2 ; = 2L/n

Can we summarize relations as an equation? Yes !

𝑣𝑡 = 𝑓𝑛𝑛=𝑓𝑛2𝐿

𝑛

𝑓𝑛 = 𝑛 ∙𝑣𝑡2𝐿

𝑛 = 1, 2, 3, 4, …

𝑳 = nλ/2 ; = 2L/n

Vibrating Strings (handy formulae)

• For the 𝑛th harmonic,

𝐿 = 𝑛λ

2

• Recall: 𝑣 = 𝜆𝑓

⇒ 𝑓𝑛 = 𝑛𝑣𝑡2𝐿

• New formula: 𝑣𝑡 =𝑇

𝑚/𝐿=

𝐹

μ

𝑓𝑛 =𝑛

2𝐿

𝑇

𝑚/𝐿

tension

mass per

unit length

Octaves

When you increase frequencies by an octave you

double the frequency

Two octaves higher means an increase by a factor

of four

Three octaves higher is a factor of 8 (= 2x2x2 = 23 )

When things increase in this way (according to a

power) we have an example of an exponential

growth

Waves in Pipes

Open-open n=1,2,3,4

Closed-open n = 1,3,5

Pressure nodes at open ends, pressure antinodes at closed ends

Displacement nodes/antinodes opposite to pressure ones.

Open-open 𝑓𝑛=𝑛∙𝑣𝑠/2𝐿

Closed-open 𝑓𝑛=𝑛∙𝑣𝑠/4𝐿

So, fundamental doesn’t depend just on length of pipe

p

L

Tube open at both ends

first mode

L

Tube open at both ends

second mode

p

Modes of Strings and Tubes 𝒇 =𝒗

𝝀

𝒇𝒏 = 𝒏 ∙𝒗𝒕𝟐𝑳 𝒏 = 𝟏, 𝟐, 𝟑, 𝟒, …

𝑳

open-open tube

closed-open

tube

string

𝒇𝒏 = 𝒏 ∙𝒗𝒔𝟐𝑳 𝒏 = 𝟏, 𝟐, 𝟑, 𝟒, …

𝒇𝒏 = 𝒏 ∙𝒗𝒔𝟒𝑳 𝒏 = 𝟏, 𝟑, 𝟓, …

Blind test: panpipe versus flute, same pitch

Superposition

We can add/superimpose sine waves to get a more

complex wave profile

Overall shape (Timbre) depends on the frequency spectra

(the frequencies of waves added together), and the

amplitudes of the waves

Ohm's acoustic law, sometimes called the acoustic phase

law or simply Ohm's law (but another Ohm’s law in Electricity

and Magnetism), states that a musical sound is perceived by

the ear as the fundamental note of a set of a number of

constituent pure harmonic tones. The law was proposed by

physicist Georg Ohm in 1843.

1st Harmonic

3rd Harmonic

5th Harmonic

Sum

All of the harmonics meet each other at the

fundamental frequency!

(This is the pitch that you hear)

Fourier Synthesis and the

Harmonic Series

f1

2f1

3f1

sum

fundamental

2nd harmonic

3rd harmonic

•

•

•

overtones

Sum of pure tones

gives a complex waveform

CD Quality

Nyquist Frequency = sampling rate / 2

Sample Rate is 44,100 Hz

Stereo so two channels

Largest possible amplitude = 2(bit depth)/2, smallest

amplitude is 1

Bit depth 16

Storage depends also on length of recording.

Time * 44,110 samples/sec * 2channels*16 bits*

(1byte/8bits)

Clicker 15.4

On a question asking you to determine

the file size of a digital sample, what

quantities should be given?

A)Bit depth, sample rate, loudness, time

B)Bit depth, bits per byte, time, number of

channels, pitchC)Bit depth, sample rate, bits per

byte, number of channels,

frequency of tone, loudness

D) Bit depth, sample rate, time, number of

channels

Clicker 15.4 D

On a question asking you to determine

the file size of a digital sample, what

quantities should be given?

A)Bit depth, sample rate, loudness, time

B)Bit depth, bits per byte, time, number of

channels, pitchC)Bit depth, sample rate, bits per

byte, number of channels,

frequency of tone, loudness

D) Bit depth, sample rate, time, number of

channels

Fourier Synthesis