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What parameters does the speed of sound depend on? ... Sound is a form of energy that moves. ... that vibrations are involved in sound, we can try the wave ... – PowerPoint PPT presentation

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Title: Sound

  • What is sound?
  • How do we make sound?
  • Why does sound move that fast? What parameters
    does the speed of sound depend on?
  • How do we work with the pitch and the volume of

Sound a form of energy
  • Sound is a form of energy that moves.
  • Is this energy carried by particles (that we call
    phonons), or is it carried by waves?
  • The fact that we can call particles of sound
    phonons doesnt necessarily mean that they
  • Since we note that vibrations are involved in
    sound, we can try the wave idea.

Sound what does it go through?
  • Sound is transmitted through air.
  • Is sound transmitted through water (and other
    liquids)? Can you pipe sound into a swimming
    pool? YES! In the navy, they use sound in sonar
    to listen for and find things.
  • Is sound transmitted through solids (like a knock
    on a door)? YES! Geologists use this to look
    for oil!
  • Is sound transmitted through vacuum? No!

Sound what is waving?
  • In waves on a string, the pieces of the string
    pulled each other via the tension in the string.
  • In sound, the molecules of the gas, liquid, or
    solid will pull on each other via the pressure in
    the material.
  • What is waving (or oscillating)? Both the
    pressure and the molecules positions!

Sound a travelling wave
  • We have already considered waves on a string. We
    were able to work with Newtons 2nd law to get a
    wave equation for this. Can we do the same for
  • YES! We use the fluid equivalent of Newtons
    Second Law to get a wave equation. With this we
    have two adjustments we need a Bulk Modulus (B
    in Nt/m2) instead of a Tension (Nt), and we have
    a volume density (r in kg/m3) instead of a linear
    density (kg/m).

Speed of sound
  • Newtons Second law in fluid form gives us the
    wave equation for sound. From this, we get for
    the molecular displacement, y
  • y(x,t) A sin(kx /- wt)
  • where v w/k (2pf / 2p/l) lf
  • and from the wave equation, v B/r1/2
  • (this is just like v T/m1/2 for a string).

Volume and pitch
  • Note that the speed of sound depends on B and r,
    and that it relates l and f. Thus, changing the
    frequency does NOT change the speed, v instead
    it will change the wavelength. Changing the
    material (changing B and/or r) will change the
  • For sound, then, we see that the pitch is
    related to frequency (f, or equivalently, l),
    while the volume is related to the amplitude, A.

Energy, Power and Intensity
  • In oscillations, we saw that the energy of a mass
    (piece of the string) was related to w2A2.
  • The power (Energy/time) of the wave down the
    string was related to w2A2v.
  • For sound, however, we need the idea of
    power/area which we call Intensity.

  • This intensity is also related like power
  • I a w2A2v (here A is amplitude).
  • But as sound spreads out, the area for this power
    increases, and so the Intensity falls off. For a
    point source, the area for the power is that of a
    sphere (4pr2). For a point source of sound, this
    takes the form of an inverse square law for I I
    P / 4pr2 .

Sound in air
  • For an ideal gas, the bulk modulus, B, is simply
    equal to the pressure, P. Thus, the speed of
    sound in air is v B/r1/2 P/r1/2 . But
    from the ideal gas law,
  • PV nRT, P nRT/V by definition,
  • r m/V. Thus, v P/r1/2 nRT/m1/2 . We
    can replace the m/n (total mass per total moles)
    by M, the molar mass)
  • v gRT/M1/2 , where g CP/CV 1.4 for a
    diatomic gas like air has to be introduced due to
    thermodynamic considerations.

Sound in air
  • v gRT/M1/2
  • For air, g 1.4, R 8.3 Joules/mole-K, T is
    the temperature in Kelvin, and M (a mixture of N2
    and O2) is .029 kg/mole.
  • Thus at room temperature (75oF24oC 297 K), v
    1.4 8.3 J/mole K 297 K / .029 kg/mole1/2
    345 m/s 770 mph.
  • At higher altitudes we have lower temperatures
    and hence lower speeds.

Human Hearing Pitch
  • A standard human ear can hear frequencies from
    about 20 Hz to about 20,000 Hz. As you get
    older, however, both ends tend to shrink towards
    the middle. This will be demonstrated during
    class, and you can hear for yourself what the
    different frequencies sound like and what your
    limits are.

  • How do we understand what people say? Does it
    have to do with frequency or intensity?
  • Of course, we can talk loudly or softly, which
    means we can talk with high or low intensity.
  • We can also sing our words at different pitches
  • So what goes into talking?

  • Along the same lines both a piano and a guitar
    can play the same note, but we can tell whether a
    piano or a guitar did play that note. What is
    going on?
  • It turns out that both talking and musical
    instruments are based on resonance standing
    waves are set up in the mouths of people and in
    the instruments.

  • We can have the same fundamental frequency set up
    on a string l/2 L in both a guitar and a
    piano. But this indicates that there are several
    wavelengths that obey this. These several
    wavelengths are called the harmonics, with the
    longest wavelength (1) being the fundamental
    (longest wavelength, shortest frequency).

  • Although a guitar and a piano may have the same
    fundamental frequency, the higher harmonics may
    resonate differently on the different instruments
    based on their shape.
  • In the same way, we can form different words at
    the same fundamental frequency by changing the
    shape of our mouth.

Fourier Analysis
  • It turns out that the ear is a great Fourier
    Analyzer - that is, it can distinguish many
    different frequencies in a sound. (The eye is
    not like this at all!)
  • It is hard to make computers listen to and
    understand speech because the computer has to be
    taught how to Fourier Analyze the sounds and
    interpret that analysis.

Human Hearing Volume
  • The volume of sound is related to the intensity
    but it is also related to frequency because the
    efficiency of the ear is different for different
  • The ear hears frequencies of about 2,000 Hz most
    efficiently, so intensities at this frequency
    will sound louder than the same intensity at much
    lower or higher frequencies.

Intensity W/m2
  • The ear is a very sensitive energy receiving
    device. It can hear sounds down as low as 10-12
    W/m2. Considering that the ears area is on the
    order of 1 cm2 or 10-4 m2, that means the ear can
    detect sound energy down to about 10-16 Watts!
  • The ear starts to get damaged at sound levels
    that approach 1 W/m2 .
  • From the lowest to the highest, this is a range
    of a trillion (1012)!

Intensity need for a new unit
  • Even though we can hear sound down to about 10-12
    W/m2, we cannot really tell the difference
    between a sound of 10-11 /- 10-12 W/m2 .
  • The tremendous range we can hear combined with
    the above fact leads us to try to get a more
    reasonable intensity measure.
  • But how do we reduce a factor of 1012 down to
    manageable size?

Intensity decibel (dB)
  • One way to reduce an exponential is to take its
    log log10(1012) 12
  • But this gives just 12 units for the range.
    However, if we multiply this by 10, we get 120
    units which is a nice range to have.
  • However, we need to take a log of a dimensionless
    number. We solve this problem by introducing
    this definition of the decibel I(dB)
    10log10(I/Io) where Io is the softest sound we
    can hear (10-12 W/m2) .

  • The weakest sound intensity we can hear is what
    we define as Io. In decibels this becomes
  • I(dB) 10log10(10-12 / 10-12) 0 dB .
  • The loudest sound without damaging the ear is 1
    W/m2, so in decibels this becomes
  • I(dB) 10log10(1 / 10-12) 120 dB .

  • It turns out that human ears can tell if one
    sound is louder than another only if the
    intensity differs by about 1 dB. This does
    indeed turn out to be a useful intensity measure.
  • Another example suppose one sound is 1 x 10-6
    W/m2, and another sound is twice as intense at 2
    x 10-6 W/m2. What is the difference in decibels?

  • Calculating for each
  • I(dB) 10log10(1 x 10-6 / 10-12) 60 dB
  • I(dB) 10log10(2 x 10-6 / 10-12) 63 dB .
  • Notice that a sound twice as intense in W/m2 is
    always 3 dB louder!
  • This is the result of a property of logs
    If I2 is twice as intense as I1, then in terms
    of dB I2(dB) 10log10(2I1)
    10log10(2)log10(I1) 10.3log10(I1)

Distance and loudness
  • For a point source, the intensity decreases as
    the inverse square of the distance. Thus if a
    source of sound is twice as far away, its
    intensity should decrease by a factor of 22 or 4.
    How much will its intensity measured in dB
  • I(dB) 10log(1 x 10-6 / 10-12) 60 dB , and
  • I(dB) 10log(1/4 x 10-6 / 10-12) 54 dB.
  • (Notice that 4 is two 2s, so the decrease is two
    3dBs for a total of 6 dB.)

The Doppler Effect
  • The Doppler Effect is explained nicely in the
    Computer Homework program (Vol 4, 5) entitled
    Waves and the Doppler Effect.
  • fR fS(v /- vR) / (v /- vS)
  • where speeds are relative to the air, not the
    ground, and the /- signs are determined by
    directions (use common sense!).

Electromagnetic Waves
  • For waves on a string and sound waves, we can get
    a wave equation from Newtons Second Law.
  • So far in Physics 251 weve talked about electric
    and magnetic fields. Can the fields wave ?
  • If so, where do we start to try to get a wave
    equation for the fields?

Electromagnetic Waves
  • The basic equations for electric and magnetic
    fields are the basic four equations weve dealt
    with in this course
  • Gausss Law for Electric Fields
  • Gausss Law for Magnetic Fields
  • Amperes Law
  • Faradays Law
  • Together these four laws are called
  • Maxwells Equations .

Electric Field Wave Equation
  • Weve written Maxwells Equations in integral
    form, but they can also be written in
    differential form using the curl and the
    divergence (Calculus III topics). By combining
    these equations we get the following wave
  • ?2Ey/?x2 moeo ?2Ey/?t2
  • Compare this to the wave equation for a string
    T ?2y/?x2 m ?2y/?t2 .

Electric Field Waves
  • ?2Ey/?x2 moeo ?2Ey/?t2
  • This has the solution Ey Eo sin(kx ? wt fo)
  • and the phase velocity of this wave depends on
    the parameters of the space that the wave is
    going through v ?1/(em) .
  • Recall that eo 1/(4pk) where k 9 x 109
    Nt-m2/Coul2 , and mo 4p x 10-7 T-m/A .
  • Thus electric waves should propagate through
    vacuum with a speed of (you do the calculation).

Electric and Magnetic Waves
  • Maxwells Equations also predict that whenever we
    have a changing Electric Field, we have a
    changing Magnetic Field. Thus, we really have an
    Electromagnetic Wave rather than just an isolated
    Electric Field wave.
  • Does this EM wave carry energy? If so, how does
    that energy relate to the amplitude of the field

Electric Field and Energy
  • Recall the energy stored in a capacitor
  • Energy (1/2)CV2 where V Ed and for a
    parallel plate capacitor where the field exists
    between the plates C KA / 4pkd.
  • Thus, Energy (1/2)KA / 4pkdEd2
  • (1/2)eVolE2 where Vol Ad, and
  • e K(1/4pk) . Note that E2 is proportional to
    the energy per volume!

Magnetic Field and Energy
  • Recall that the energy stored in an inductor is
  • Energy (1/2)LI2 where L for a solenoid is
  • L m N2 A/Length and B for a solenoid is
  • B m(N/Length)I .
  • Thus, Energy (1/2)mN2A/LengthBLength/mN 2
  • (1/2)VolB2/m where Vol ALength . Note
    that B2 is proportional to the energy per volume!

Energy and EM Waves
  • Energy/Vol (1/2)eE2 and
  • Energy/Vol (1/2)B2/m
  • Note that Energy density (Energy/Vol) when moving
    (m/s) becomes Power/Area, or Intensity.
  • Also from Maxwells equations, we have for EM
    waves that Eo/Bo c ?1/(eomo) .
  • Putting this together, we have I E B / mo .

Intensity and the Poynting Vector
  • Maxwells Equations also predict that
    electromagnetic waves will travel in a direction
    perpendicular to the directions of both the
    waving Electric and the waving Magnetic Fields,
    and that the direction of the waving Electric
    Field must be perpendicular to the direction of
    the waving Magnetic Field. This is stated in the
  • S (1/?o)E x B
  • where S is called the Poynting vector and gives
    the Intensity of the electromagnetic wave.

  • From Maxwells Equations we also predict that
    electromagnetic waves should carry momentum,
    where the amount of momentum depends on the
    energy per speed p Energy / c .
  • (In relativity, we will see that electromagnetic
    energy can be considered to be carried by
    photons, where photons have mass. From
    relativity, E mc2, and p mc, so p E/c.)
  • If the light is absorbed the material will
    receive this amount of momentum. If the light is
    reflected, the material will receive twice this
    amount of momentum.

Radiation Pressure
  • Pressure is Force/Area, and Force is change in
    momentum with respect to time. Hence,
    electromagnetic radiation should exert a pressure
    on objects when it hits them.
  • Radiation Pressure F/A
  • (dp/dt) / A d(Energy/speed)/dt / A
  • Power/speed / A Power/Area / speed
  • If the radiation reflects, then the momentum is
    twice and so the radiation pressure is also twice.