Applets in this eLecture: Moving Source Doppler Applet Doppler: Source Doppler: Observer Doppler. Reflection of Sound Applet

 

Longitudinal Waves  Disturbance propagates at a speed  v = 3000 m/s = 4500 miles per hour  Speed along slinky is much less  Particles vibrate along a line which is  parallel to the direction of travel of  the disturbance.

 A wave is a traveling disturbance, a self-sustaining disturbance of a supporting medium.

  Longitudinal Waves

Waves in sound are like waves in slinky:  a compression (a pulse) propagates away from the expanding speaker membrane. Speed of sound wave = 331 m/s at 0 C

Problem:  Air tube is 100 meters long.  How long will it take for the pulse to reach the other end?      t = 100 m / 331 m/s        = 0.302 s ------------------------------------------------ Problem:  Shout underwater.  How long will it take for the shout to be felt underwater 100 meters away?      t = 100 m / 1482 m/s        = 0.0675 s     (five times faster than through air)

 

Substance Speed   (m/s) Air 343 Helium 965 Water 1482 Lead 1960 Steel 5960 Granite 6000

  Wavelength, Frequency, and Speed

Vibrating speaker membrane sends  train of compression waves to ear.  Individual molecules oscillate about  fixed positions; they do not travel  with the wave.

l = wavelength = distance between maxima f  = frequency    = number of vibrations                               per second v = speed = l f

When a sound wave passes from air into water, what  happens to the speed, frequency, and wavelength? ---------------------------------------------------------------------------- Speed v decreases, but the frequency f doesn't change.

  Phase, Wave Fronts, and Plane Waves

Circular Wavefronts                     Spherical Wavefronts As spherical wave expands, its radius increases and the wavefronts flatten.  At very large distances, the wavefronts are quite flat and the disturbance resembles a plane wave. The stage or phase of the disturbance is the same at all points on a wavefront.   A wave front is a surface of constant phase.

   Transverse Waves

Particles vibrate along a line    which is perpendicular to the  direction of travel of the  disturbance.

Graph shows the vertical   position y of the particle   at point P.  (This is not   the shape of the rope.)

  Transverse Waves:        Wavelength      Frequency      Speed

Distance between consecutive maxima = l  Number of vibrations per second = f  m = mass of string     L = length of string    F = tension in string    v = [ F / ( m / L ) ] 1/2

v = lf               l = v / f --------------------------------------------------------- Example:  m = 2 kg           L = 3 m                            F= 100 N          f = 50 Hz   What is the distance between consecutive maxima? l = v / f = [ F / (m / L) ]1/2 / f                    = [100 / ( 2 / 3 ) ]1/2 / 50                    = 0.245 m

 Transverse Seismic Waves

Shear waves--called S-waves.            Molten core can't vibrate transversely        Primary waves --P-waves are longitudinal (not shown):  they arrive first

Compression Waves (Primary Waves: P-waves Transverse Waves (Shear Waves:  S-waves)

Compression waves arrive at seismic stations before the S-waves, so they're called "primary waves". P-waves predominate in underground nuclear explosions.

  Water Waves                                       Not tranverse, not longitudinal

Tidal waves are turbulent; we are  not studying this type of wave.

This is non-turbulent water-wave motion.

  Detail of Water-Wave Motion

  Water Waves are Circular

   Condensation and Rarefaction in Sound Waves

Speaker membrane expands, creating a region  where the air molecules are packed closely  together, a "condensation".  The air pressure  in a condensation is higher than normal.

As the membrance moves back, a region is left behind where few molecules are located, a "rarefaction".  Meanwhile, the condensation moves forward.

   Creating Condensations and Rarefactions

Push door open, force air to right.                      Close door, create region of low          Chain reaction is set up which                             pressure.  Chain reaction is set up          reaches curtains.                                                   which causes low-pressure region                                                                                                    (rarefaction) at curtains.

  Air Pressure in Sound Waves

The distance between consecutive maxima or consecutive minima is the wavelength of the sound. ----------------------------------------------- The sweep distance of the speaker membrane is very small compared to the wavelength. ----------------------------------------------- The larger the sweep of the speaker membrane, the greater will be the air pressure in the condensation, and the smaller it will be in the rarefaction.

   Reflection and Speed of Sound Applications

SONAR:   sound navigation ranging

Ultrasound is sound at a frequency which is outside of the range of human hearing.

At a height of 10 meters above the surface of a lake, a sound pulse is generated.  The echo from the bottom of the lake returns to the point of origin 0.140 second later.  The air and water temperatures are 20 C.  How deep is the lake?  ------------------------------------------------------------------------ Speed in air:  343 m/s      Speed in water: 1482 m/s Time in air      = 20/343 = 0.058 s Time in water = 0.140 - 0.058 = 0.082 s Time to hit bottom:  0.082 / 2 = 0.041 s d = 1482 (0.041) = 60.8 m

   Ultrasonic Scanner

Reflected waves show detail within the body.   Fetus reflects sound much better than the fluid in which it's immersed (amniotic fluid).

No object smaller than the wavelength of the sound will reflect sound well.  The frequency required to see an object one mm (10-3 m) across, must be at least f = (1540 m/s) / 10-3 m   = 1.54 MHz This is a frequency about 77 times what a human ear can respond to.

   The Rule of Five for Lightning

Speed of sound = 343 m/s at 20 C Speed of light    = 3 x 108 m/s Rule :  See lightning, start counting seconds until sound is heard.  Divide by five to obtain distance of lightning Example:  10 / 5 = 2 miles Since sound was generated at essentially the same instant at which you see the lightning, the distance away is 10 x 343 m/s = 3430 m One mile =5280 feet = 1610 m Distance = 3430/1610 = 2.1 miles

 

Reflection of Sound Applet Sound wave strikes water

   Sound Refraction

Sound speed is greater in warm air.

Speed of sound in air depends on temperature:  v ~ T1/2 ---------------------------------------------------- During World War I drivers returning to the rear from the front encountered alternating regions of intense artillery sound/ no sound caused by refraction, then reflection, then refraction, etc. ---------------------------------------------------- On some evenings the whistle of trains miles away can be heard; no sound during the day.

  Sound Intensity

1000 Joules per second of sound energy is generated.  Spherical wave fronts spread outward from sound source at the center.

Power P of the source is 1000 watts.   The energy spreads outward through   the air.  At a distance R from the source   the area A of the spherical front is 4pR2    Intensity = number of joules per second                      per square meter                   = power/area = P/A                  Example:  R = 10 m                          I = 1000 watts /4p(10 m)2                         = 0.8 watts/m2    Is this a rock concert or a whisper?

 
   Typical Sound Intensities

Source	Intensity (watt/m2)	Source	Intensity (watt/m2)
Pin dropping	1.0 x 10-12	Loud car horn	0.003
Rustling leaves	1.0 x 10-11	Shout (1.5 meters)	0.010
Whisper	1.0 x 10-10	Rock concert	1
Conversation (one meter)	1.0 x 10-6	Yell into the ear (20 cm)	1
Loud music	1.0 x 10-4	Jet engine	100

 Comparing sound intensities to the threshold of hearing  (a pin dropping) is more
 useful. Thus, a jet engine has 10¹⁴ times the intensity (not 10¹⁴ times the loudness)
 of the threshold of hearing, and a loud car horn has 3.0 x 10⁹ times the intensity.

  Relative Intensity    

Source	Intensity (watt/m2)	I / I0 I0 = 1 x 10-12 watt/m2
Rustling leaves	1 x 10-11	10
Whisper	1 x 10-10	100
Conversation	1 x 10-6	106
Loud car horn	0.003	3 x 109 = 109.5
Shout	0.010	1010
Rock concert	1	1012
Jet engine	100	1014

  Review Logarithms 
    Logarithms are exponents.  The logarithms in the table below are the exponents to
    which 10 must be raised to equal the number.)

log (10-12) = -12 10-12 = 10-12	log (10-3) = -3 10-3 = 10-3	log (1/100) = -2 1/100 = 10-2	log (1/10) = -1 1/10 = 10-1	log (1) = 0 1 = 100
log (10) =1 10 = 101	log (100)  = 2 100 = 102	log (106) = 6 106 = 106	log (3 x 109) = 9.5 3 x 109 = 109.5	log (1010) = 10  1010  = 1010

  Needed later:  log (A/B) = log A - log B

 Sound Intensity Level   (Loudness) 
  b = 10 log (I/I₀)                  I₀ = 1 x 10^-12 watts/m²

Source	Intensity     watt/m2	Relative Intensity         I / I0	Intensity Level b   decibels (dB)	Intensity Level      bels
Rustling leaves	1 x 10-11	10	10	1
Conversation	1 x 10-6	106	60	6
Loud car horn	0.003	3 x 109 = 109.5	95	9.5
Shout	0.010	1010	100	10
Rock concert	1	1012	120	12
Jet engine	100	1014	140	14

   To calculate intensity, multiply threshold intensity by ten raised
   to the power b/10:  I = I₀  x 10^b/10

 

Doubling the intensity: Need to use property:  log (A/B) = log A - log --------------------------------------------------------------     I2 = 2 I1         (iintensity is doubled)       b2 = 10 log (I2/I0) = 10 (log I2 - log I0)       b1 = 10 log (I1/I0) = 10 (log I1 - log I0)     b2 - b1 = 10 (log I2 - log I1)                   = 10 log(I2/I1)                   = 10 log (2)                   = 10 (0.301)                   = 3.01 A 100 dB horn outputs twice the power as a 97 dB horn

One dB change is smallest change in loudness noticeable by humans: To human ears, a change of 10 dB corresponds to a doubling of loudness.

How much "louder" is the 2000-watt stereo?

If the difference in decibel level (i.e.,   b) is10 dB, the sound is twice as loud.       b2 = 10 log (I2 / I0) = 10 (log I2 - log I0)       b1 = 10 log (I1 / I0) = 10 (log I1 - log I0)     b2 - b1 = 10 (log I2 - log I1)                   = 10 log(I2 / I1)                   = 10 log (2000 / 20)                   = 10 log 100                   =  10 (2) = 20 dB Thus, the output power is 100 times greater, but it's perceived as being only four times as loud by the human ear.

   More Decibel Math     Using I = I₀  x 10   ^b/10

Example:  By how much must the output sound power of a sound system be increased to raise the loudness of a speaker from I1 = 70 dB to I2 = 71 dB?  Assume a point source and that the listener is 20 meters from the source. ----------------------------------------------------------------------------------- I1 = 107.0 (1.0 x 10-12 watt/m2)  = 1.0 x 10-5 watts/m2 I2 = 107.1 (1.0 x 10-12 watt/m2)  = 1.0 x 10-4.9 watts/m2                                                                   = 1.26 x 10-5  watts/m2 ------------------------------------------------------------------------------------ Increase per square meter = .26 x 10-5  watts/m2 Number of square meters = 4  pR2 = 5000 m2 Added power = .26 x 10-5 (5000) = 0.013 watt Compared to I1 x 4  pR2 = 0.050 watt (not very much of a stereo's power is given up as sound)

   10,000,000 people talking at the same time produce sound energy equal to the energy
   needed to light a small light bulb.  Hearing is possible because our ears are extraordinarily
   sensitive.  Very few microphones can detect sounds softer than we can hear.     

If two people talk simultaneously and each creates an intensity level of 65 dB at a certain point, does the total intensity level at that point equal 130 dB? ------------------------------------------------------------------------------- Intensities add, but not intensity levels.  The total intensity is 2 x I0 x 106.5 or, I0 x (2 x 106.5) = I0 x 106.8.  The intensity level is 68, not 130. Rule:  a 3 dB gain in loudness represents a doubling of intensity.

 
   Sensitivity of the Human Ear:  Part I

Note:  threshold is much lower at 3000 Hz than it is at 15,000 Hz.

     Sensitivity of the Human Ear:  Part II

Optimum frequency is about 3000 Hz                  Dogs:  44,000 Hz        Rats:   70,000 Hz         Infrasound: < 20 Hz               Ultrasound: > 20 KHz

Outer ear collects sound energy, bones in middle ear transmit vibrations to fluid in canals (the inner ear).   Canals in cochlea separated by a flexible partition which flexs at different points depending on the frequency; nerve hairs in canals send information to the brain.

 Moving Observer Doppler Effect

f '  =     f (1 + v0 / v)             ---------------------------------------------- If the observer were moving away:  f ' = f (1 - v0 / v)

 
   The Doppler Effect:  Moving Source

Small vibrating sphere is dragged across  water.

v = lf      T = 1 / f     l = vT    (1)             (2)            (3) --------------------------------------- T = distance/speed of sound     = l/v Source has velocity vs New distance = l - vsT  (4) T' = (l   - vsT) / v             (5)     = (vT - vsT) / v             (6)     = (1 - vs / v) T              (7) f ' = 1 / T'                         (8) f ' = f  / (1 - vs / v)           (9)

 

Moving Source Doppler Applet (also shows shock waves) Doppler: Source Doppler: Observer Doppler. Good

  Doppler Effect: Moving Source    

f ' = f /(1 - vs / v)     (Moving toward observer) A train moving at a speed of 60 m/s sounds its whistle, which has a frequency f = 600 Hz.  What is the frequency f ' heard as the train approaches an observer? f ' = 600 / [1 - (60 / 343)] = 727 Hz --------------------------------------------------------------------------------------------- What is the frequency heard after the train has passed the observer? f ' = f /(1 + vs / v)      (Moving away from observer) f ' = 600 / [1 + (60 / 343)] = 511 Hz

  Doppler Flow Meter

Used to locate narrowed blood vessels where the speed is greater. Typical blood speed:  0.10 m/s Transmitter frequency:  f = 5 megahertz                                      = 5 x 106 Hz ------------------------------------------------- Use moving source Doppler equation: f ' = f / (1 + vs / v) f ' = (5 x 106) / (1 + 0.10 / 343)    =  4.99854 x 106 Hz Change =1460 Hz