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