Archimedes Principle

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Contact information
Ajay Singh Room No. 20/21, Physics Building,
basement In the Plamsa hallway Phone [phone] E
mail ajay.singh_at_usask.ca Office hours Thursday,
1- 4 PM, or any other time by appointment Plasma
hallway is generally locked from one side (closer
to my room), but other side is open for the time
being.
Please bring your calculator to the class, if you
dont already
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Chapter 16 WavesLecture 25
HAPPY AND SUCCESSFUL NEW YEAR TO ALL OF YOU
Most of the animations are courtesy Dr. Dan
Russel, Kettering University
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16.1. The Nature of Waves

• All waves
• are traveling disturbances
• carry energy from place to place

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Transverse Waves

• disturbance occurs PERPENDICULAR to the direction
  of travel of the wave

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Transverse Waves contd.

• disturbance occurs PERPENDICULAR to the direction
  of travel of the wave

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Longitudinal Waves

• disturbance occurs ALONG (PARALLEL) to the
  direction of travel of the wave

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Longitudinal Waves

• disturbance occurs ALONG (PARALLEL) to the
  direction of travel of the wave

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Water Waves

• combination of transverse longitudinal
• particles move in little circles

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Periodic Waves

• repetitive
• made up of cycles/patterns
• one cycle of wave
• one complete pattern within a wave

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snapshot
watching wave go by
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Properties of Waves

• Amplitude A (meters)
• max excursion of particle in medium (peak
  or trough of wave)
• Wavelength ? (meters)
• length of one wave cycle
• Period T (seconds)
• time taken to complete one wave cycle
• Frequency f (hertz, Hz)
• repetition rate of cycle

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Wave speed versus particle speed

• Each particle in the transverse wave moves with a
  speed vparticle
• Each particle does simple harmonic motion.
• This means that vparticle is not constant.

We can use the relationship (equation 10.7 from
chapter 10)
The properties A and ? of the source creating the
wave define the speed of the particle
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Wave Properties...

• Period The time T for a point on the wave to
  undergo one complete oscillation.

• Speed The wave moves one wavelength ? in one
  period T so its speed is v ??/ T.

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Wave Properties...
v ? / T

• The speed of a wave is a constant that depends
  only on the medium, not on amplitude, wavelength,
  or period. ? and T are related!
• ? v T or ? 2? v /?????(since?T 2? /
  ??? or ? v / f (since T
  1 / f )
• Recall f cycles/sec or revolutions/sec
• ? rad/sec 2?f

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Velocity of a Wave on a String

• velocity of wave depends on two quantities
• tension in the string (think adjacent pulling
  force)
• mass per unit length of string (think inertia)

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The properties of the medium that the wave passes
through determines the speed of the wave
Where, µm/L (kg/m), is mass per unit length of
the string, also called as the linear density of
a string.
Tension force and mass for the string determine
the velocity of the wave
The two speeds, vwave and vparticle are not the
same (unless by co-incidence)
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Ex.
Drop slinky
Rope

• A heavy rope hangs from the ceiling, and a small
  amplitude transverse wave is started by jiggling
  the rope at the bottom.
• As the wave travels up the rope, its speed will

v
(a) increase (b) decrease (c) stay the same
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Solution

• The speed at any point will be determined by

at that point
v

• The tension F in the rope
• near the top is greater than the tension near
• the bottom since it has to support the weight of
  the rope beneath it!
• The speed of the wave will be greater at the top!

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Example 2

• Q. Transverse waves travel on the strings of and
  electric guitar after the strings are plucked.
  The length of each string between its two fixed
  ends is 0.628 m. and the mass is 0.208 gm for the
  highest pitched E string and 3.32 g for the
  lowest pitched E string. Tension on each string
  is 226 N, find the speeds of the waves on the two
  strings.

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Solution

• The speed of a wave generated in a guitar string
  is given by

Eq. 16.2
It depends on the tension in the string and its
linear density m/L, since the tension is the
same for both the strings the one with smaller
linear density is expected to have waves with
greater speed. Soln. High pitched E
826 m/s
Low pitched E
207 m/s
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The Mathematical Description of a Wave

• A wave traveling in a medium displaces particles
  in the medium.
• We want an equation for the displacement of a
  particle
• moving x dir
• moving x dir

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Phase Angle of a Wave
x/v is the time taken by the wave to travel
distance x In other words SHM that occurs at
point x is delayed By the time interval x/v
compared to the motion at the origin

• F is in radians!
• this is very important
• dont forget to change your calculator to RAD
  mode, or you will get wrong numbers

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Problem 16.25

• A wave causes a displacement as follows (distance
  in meters, and time in seconds)
• Find amplitude, frequency, wavelength, and
  speed of wave.
• Is this wave moving in the x or x direction?

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We are given the specific wave equation for this
wave
Compare this with our general equation for
travelling waves
The specific solution must be a solution of the
general solution
Find the Amplitude
We can see that amplitude, A 0.45 m
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Specific equation for this wave
General equation for wave
Solve for frequency
Solve for wavelength
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Now we have the wavelength ? and the frequency
f, we can calculate the velocity of the wave
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Finally, the direction the wave is travelling in
General wave equation
The sign here indicates the wave is moving in
the x direction
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The Nature of Sound

• Pressure waves
• longitudinal wave
• created by vibrating object (e.g., guitar
  string, speaker diaphragm)
• sound waves travel through a medium (carried by
  particles)
• no sound in a vacuum!

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The Nature of Sound contd.
speed of wave propagation

• condensations and rarefactions
• sound wave NOT mass movement of air like wind
  rather, it is local oscillations of molecules
• wave transfers energy and information
• ? wavelength of sound wave
• ? distance between condensations, or
• ? distance between rarefactions

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Pressure Amplitude of Sound Wave -Sound is a
pressure wave

• measureable
• regions of high and low pressure
• relative to background
• usually very small fluctuations
• e.g. person talking
• Psound 10-2 Pa
• Patm 105 Pa
• (million times smaller!)
• human ear is extremely sensitive to pressure
  changes

Loudness (perception) ? pressure amplitude
means ? loudness
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Frequency of Sound Wave

• same as source frequency
• e.g.
• 1000 Hz source produces 1000 condensations
  (followed by 1000 rarefactions) per second.
• frequency of sound wave is 1000 Hz
• one single frequency pure tone
• range of human hearing
• young person 20-20,000 Hz
• middle aged person 12-14 kHz

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Sounds outside the range of human hearing

• If the frequency is less than 20 Hz, the sound is
  INFRASONIC
• Rhinoceroses grunt to each other at 5Hz
• If the frequency is above 20000 Hz (20 kHz), the
  sound is ULTRASONIC
• Bats use frequencies of up to 100 kHz for
  communication and navigation
• Also used to clean equipments in laboratories

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Experimental determination of upper limit

• Use an audio generator to generate a pure tone of
  known frequency
• Test the hearing of the group in the lecture
  theatre (not a representative group from the
  whole population!)

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Humans and Sound

• frequency can be measured by instruments
• people can perceive sound BUT it is subjective,
  relative, not absolute
• Pitch
• high pitch ? high f
• low pitch ?low f

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Speed of Sound

• waves travel through medium
• solid, liquid, gas
• speed depends on properties of medium

Table 16.1
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Power of a wave

• Sound waves carry energy
• This can do work (for example, forcing the
  eardrum to vibrate)
• The power of the wave is the energy transported
  per second

SI units for power of wave Joules/sec Watt
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Intensity of a wave

• The sound intensity I is defined as the sound
  power P that passes perpendicularly through a
  surface divided by the area of that surface

SI unit of sound intensity watt/metre2
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Figure 16.21
Important to separate the concepts of sound power
and sound intensity
The loudspeaker puts out a fixed sound power (in
Watts) As the sound wave gets further from the
speaker, it spreads out, so the sound intensity
is less.
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Source emitting uniformly in alldirections

• If a source emits sound evenly in all directions
• The sound will move outwards with a spherical
  distribution
• The power of the wave will be distributed evenly
  over the whole area of the sphere, radius r

See figure 16.23, next slide
This is known as an inverse square law applies
to electromagnetic waves too
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Suppose the intensity is I1 at a distance R1 from
a sound emitter and the sound distribution is
spherically symmetric. Now you move to a new
position R2, where the intensity is half of I1.
What is the ratio R1/R2 ?
The basic equation
At position 1
At position 2
Relationship between the two intensities
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Review examples 6 and 7, pages 462-463 for
numerical examples
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Why is it so loud when you sing in the shower?

• I P/(4pR2) only works if there are no
  reflections (ceilings, walls, floors, etc.)
• reflected sound will also pass through the
  surface area, increasing the intensity

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Comparing Sound Intensities

• We want to have a scale which compares two sound
  intensities
• It needs to be appropriate for the human
  perception of loudness

The human ear does not measure sound intensity in
a linear fashion! Loudness is not the same as
Sound Intensity
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The decibel scale

• The intensity level in decibels ? is defined as

Where I and I0 are the intensities of the sounds
being compared
It is a ratio, so the dB is dimensionless
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Logarithms
Appendix D

• Definition
• Properties

The text book uses log(x) to mean logarithms to
base 10
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Remember an intensity level expressed in dB is
always a comparison between two sound levels
The value of I0 is often taken as the threshold
for human hearing I0 1.00 x 10-12
Watts/m2 However if you are using intensity
levels in dB, you should always check on what the
reference level is!
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• If a sound has an intensity of 8 W/m2. What is
  its intensity level?
• What if I I0 ?

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Typical Sound Intensities(Table 16.2)
note power of 10 intensity increase ? 10 dB
intensity level increase
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Why use the decibel scale?

• Because it is convenient, since it follows the
  way the human ear perceives loudness
• A 1 dB change in the intensity level between two
  sounds corresponds approximately to the smallest
  change in loudness that the average listener with
  normal hearing can detect

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The Bel scale

• The original unit for determining loudness was
  the Bel (1 Bel 10 decibels)
• Named after Alexander Graham Bell
• Devised to measure the drop-off of signal
  strength in early telephone systems.
• Too big to be of practical use for human hearing!

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Comparing intensities

• Audio system 1 produces an intensity level of ?1
  90.0 dB
• Audio system 2 produces an intensity level of ?2
  93.0 dB
• Corresponding intensities are I1 and I2
• Determine the ratio I1/I2

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Now take the antilog (10x) of both sides
Double the intensity produces only a small
increase in the loudness
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Loudness and intensity

• Experiment shows that if the intensity level
  increases by 10 dB, the new sound seems to be
  about twice as loud as the original sound

e.g. 70 dB sounds twice as loud as 60 dB
(remember this fact it often appears in Section
A questions!)
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Loudness of audio systems

• Usually audio systems advertise their power in
  terms of watts
• How much louder is a 200W sound system than a 20W
  sound system?
• Assume both are set on maximum volume, and power
  outputs are 20W and 200W respectively.
• Assume the observer is the same distance from
  each set of speakers

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We have the power of the speakers, not the
intensity
Intensity I is in (watts/metre2), but the
observer is the same distance from both sources
(r r0)
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We can use the decibel equation to compare two
intensities I and I0 directly, and we have just
shown that in this case, the power and the
intensity are proportional
There is a difference of 10 dB between the 2
systems so the more powerful one (200W) only
sounds twice as loud as the 20W system
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Loudness human perception

• From dB equation
• 10 times intensity ? 10 dB increase (of intensity
  level)
• double intensity ? 3 dB increase
• halve intensity ? 3 dB decrease
• From experiments
• 1 dB minimum change average person can
  detect
• 10 dB increase ? sound seems twice as loud

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Human Ear

• Sound is channeled by the outer ear to arrive
  perpendicularly on the ear drum
• Pressure wave causes the ear drum to vibrate
• Pressure is transmitted through the middle ear by
  the bones of the ear (malleus, incus, stapes)
  which act as levers to magnify the force
• Resultant pressure stimulates the oval window
  separating middle and inner ears
• The mechanosensitive hair bundle in the cochlea
  sends a signal to the brain each time it senses a
  movement.

A mechanosensitive hair bundle in the cochlea of
the ear. Each hair bundle is made up of 30-300
stereocilia (tiny hairs).
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• Area of oval window is smaller than that of the
  ear drum. This magnifies the force transmitted
• Receptor cells on the cochlea are sensitive to
  different frequencies, depending on the position
  on the cochlea

http//www.bcm.tmc.edu/oto/research/cochlea/Volta/
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Sensitivity of Human Ear- perception again -

• Fletcher-Munson curves
• surveyed a large number of people
• asked them when different frequencies had the
  same loudness
• did this for a variety of levels of loudness

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Fletcher-Munson Curves
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from Fletcher-Munson

• Human ear
• less sensitive to low and high frequencies
• most sensitive 1-5 kHz (middle range)
• loud noises flat line, so sensitivity similar
  at all frequencies
• quiet noises large range of sensitivities
  (least sensitive at low and high f)

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Doppler Effect

• Change in frequency (pitch) detected by observer
  when sound source and/or observer are moving
  relative to the sound wave
• sound approaching
• high pitch (small ?, large f)
• sound moving away
• low pitch (large ?, small f)

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• stationary source
• stationary observer
• sound ? of source same as ? at observer
• observer hears true frequency

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Doppler Effect, The Movie

• moving sound source

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• truck moves forward
• condensations in front of truck squashed together
  -- shorter ?
• condensations behind truck more widely spaced --
  longer ?

• observer in front hears higher frequency
• observer behind hears lower frequency
• as truck passes observer, observer hears true
  frequency

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Source Moving Toward Observer
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Observer Moving Toward Source

• ? does not actually change in this case, even
  though it seems so to the observer!

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Doppler General Formula

• /- and -/ signs top ? toward
• Difference in Two Cases
• Moving Source ? actually changes
• Moving Observer ? does NOT change it only
  seems to.