458 |
Week 11: Sound |
• Sound Intensity: The intensity of sound waves can be written:
I = 12 vaρω2s20
or
I = |
1 1 |
P02 = |
1 |
P02 |
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2 vaρ |
2Z |
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•Spherical Waves: The intensity of spherical sound waves drops o like 1/r2 (as can be seen from the previous two points). It is usually convenient to express it in terms of the total power emitted by the source Ptot as:
I= Ptot
4πr2
This is “the total power emitted divided by the area of the sphere of radius r through which all the power must symmetrically pass” and hence it makes sense! One can, with some e ort, take the intensity at some reference radius r and relate it to P0 and to s0, and one can easily relate it to the intensity at other radii.
•Decibels: Audible sound waves span some 20 orders of magnitude in intensity. Indeed, the ear is barely sensitive to a doubling of intensity – this is the smallest change that registers as a change in audible intensity. This motivate the use of sound intensity level measured in decibels:
µI ¶
β= 10 log10 I0 (dB)
where the reference intensity I0 = 10−12 watts/meter2 is the threshold of hearing, the weakest sound that is audible to a “normal” human ear.
Important reference intensities to keep in mind are:
–60 dB: Normal conversation at 1 m.
–85 dB: Intensity where long term continuous exposure may cause gradual hearing loss.
–120 dB: Hearing loss is likely for anything more than brief and highly intermittent exposures at this level.
–130 dB: Threshold of pain. Pain is bad.
–140 dB: Hearing loss is immediate and certain – you are actively losing your hearing during any sort of prolonged exposure at this level and above.
–194.094 dB: The upper limit of undistorted sound (overpressure equal to one atmosphere). This loud a sound will instantly rupture human eardrums 50% of the time.
•Doppler Shift: Moving Source
f ′ = |
f0 |
(956) |
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(1 |
vs |
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va |
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where f0 is the unshifted frequency of the sound wave for receding (+) and approaching (-) source, where vs is the speed of the source and va is the speed of sound in the medium (air).
• Doppler Shift: Moving Receiver
vr |
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f ′ = f0(1 ± va ) |
(957) |
where f0 is the unshifted frequency of the sound wave for receding (-) and approaching (+) receiver, where vr is the speed of the source and va is the speed of sound in the medium (air).
Week 11: Sound |
459 |
• Stationary Harmonic Waves |
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y(x, t) = y0 sin(kx) cos(ωt) |
(958) |
for displacement waves in a pipe of length L closed at one or both ends. This solution has a node at x = 0 (the closed end). The permitted resonant frequencies are determined by:
kL = nπ |
(959) |
for n = 1, 2... (both ends closed, nodes at both ends) or:
kL = |
2n − 1 |
π |
(960) |
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2 |
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for n = 1, 2, ... (one end closed, nodes at the closed end).
• Beats If two sound waves of equal amplitude and slightly di erent frequency are added:
s(x, t) = |
s0 sin(k0x − ω0t) + s0 sin(k1x − ω1t) |
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(961) |
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= |
2s |
sin( |
k0 + k1 |
x |
− |
ω0 + ω1 |
t) cos( |
k0 − k1 |
x |
− |
ω0 − ω1 |
t) |
(962) |
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2 |
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which describes a wave with the average frequency and twice the amplitude modulated so that it “beats” (goes to zero) at the di erence of the frequencies δf = |f1 − f0|.
11.1: Sound Waves in a Fluid
Waves propagate in a fluid much in the same way that a disturbance propagates down a closed hall crowded with people. If one shoves a person so that they knock into their neighbor, the neighbor falls against their neighbor (and shoves back), and their neighbor shoves against their still further neighbor and so on.
Such a wave di ers from the transverse waves we studied on a string in that the displacement of the medium (the air molecules) is in the same direction as the direction of propagation of the wave. This kind of wave is called a longitudinal wave.
Although di erent, sound waves can be related to waves on a string in many ways. Most of the similarities and di erences can be traced to one thing: a string is a one dimensional medium and is characterized only by length; a fluid is typically a three dimensional medium and is characterized by a volume.
Air (a typical fluid that supports sound waves) does not support “tension”, it is under pressure. When air is compressed its molecules are shoved closer together, altering its density and occupied volume. For small changes in volume the pressure alters approximately linearly with a coe cient called the “bulk modulus” B describing the way the pressure increases as the fractional volume decreases. Air does not have a mass per unit length µ, rather it has a mass per unit volume, ρ.
The velocity of waves in air is given by |
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va = s |
B |
≈ 343m/sec |
(963) |
ρ |
The “approximately” here is fairly serious. The actual speed varies according to things like the air pressure (which varies significantly with altitude and with the weather at any given altitude as low and high pressure areas move around on the earth’s surface) and the temperature (hotter molecules push each other apart more strongly at any given density). The speed of sound can vary by a few percent from the approximate value given above.