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About this Lesson
- Type: Video Tutorial
- Length: 13:24
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- Posted: 07/01/2009
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This lesson was selected from a broader, comprehensive course, Physics I. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/physics. The full course covers kinematics, dynamics, energy, momentum, the physics of extended objects, gravity, fluids, relativity, oscillatory motion, waves, and more. The course features two renowned professors: Steven Pollock, an associate professor of Physics at he University of Colorado at Boulder and Ephraim Fischbach, a professor of physics at Purdue University.
Steven Pollock earned a Bachelor of Science in physics from the Massachusetts Institute of Technology and a Ph.D. from Stanford University. Prof. Pollock wears two research hats: he studies theoretical nuclear physics, and does physics education research. Currently, his research activities focus on questions of replication and sustainability of reformed teaching techniques in (very) large introductory courses. He received an Alfred P. Sloan Research Fellowship in 1994 and a Boulder Faculty Assembly (CU campus-wide) Teaching Excellence Award in 1998. He is the author of two Teaching Company video courses: “Particle Physics for Non-Physicists: a Tour of the Microcosmos” and “The Great Ideas of Classical Physics”. Prof. Pollock regularly gives public presentations in which he brings physics alive at conferences, seminars, colloquia, and for community audiences.
Ephraim Fischbach earned a B.A. in physics from Columbia University and a Ph.D. from the University of Pennsylvania. In Thinkwell Physics I, he delivers the "Physics in Action" video lectures and demonstrates numerous laboratory techniques and real-world applications. As part of his mission to encourage an interest in physics wherever he goes, Prof. Fischbach coordinates Physics on the Road, an Outreach/Funfest program. He is the author or coauthor of more than 180 publications including a recent book, “The Search for Non-Newtonian Gravity”, and was made a Fellow of the American Physical Society in 2001. He also serves as a referee for a number of journals including “Physical Review” and “Physical Review Letters”.
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If you have a wave traveling down a string, it's useful to talk about the power transmitted by that wave. That's generally true of any kind of wave. Power is an interesting and important physical quantity. When you have waves in three dimensions rather than one dimension--like, for instance, sound emanating and spreading out through a room--it's useful to define a new physics quantity called "intensity." Intensity is not just the power; it's the power per area. Because if I'm standing here and I've got a little one meter squared window in front of me, what I might care about is not the total power in the whole wave but the power per square meter that's coming towards me. If you have a wave heading towards something, intensity, which is defined to be power per area, is often a physically useful and interesting quantity.
It's got units of power per area. That's watts per square meter. And if you think about it, power, remember, is energy per second. So you could also think of this as energy each second passing through one square meter. It's joules per second per square meter. And certainly when you're listening to sound waves and you've got a little area on your eardrum, what you're interested in is the energy per second per square meter that's entering you. So it's a very commonly used physical quantity for any kind of wave any time you have power passing through area in any situation.
If it's sound waves, I can even write down a formula. I won't derive this, but it's the formula for the intensity of a sound wave traveling through air with density , with speed of sound v--typically 330 meters per second or so. If the frequency of the sound is increased, the intensity increases like the square of the frequency. This is the omega of the wave. And fmax--remember when you're talking about sound waves, one way of describing it is the displacement of the air molecules from their equilibrium position, and fmax tells you the amplitude of displacement. So that's a very handy formula sometimes when you're working with sound waves.
And an equivalent formula can be written down, because sometimes instead of talking about the displacement of the molecules people prefer to think about over-pressures. Sound can also be though of as simply a longitudinal sinusoidal wave of over-pressure. And remember that Tmax--I've never derived this but you can--is related to fmax. It's fmax times the quantity times v times . So if you know fmax and these quantities, you can figure out Tmax, or vice versa. You could substitute this into the formula and figure out intensity as a function of the maximum over-pressure of the sound, and it's going to be proportional to the square of that. So that's a useful formula sometimes.
You know, intensity is energy per second per square meter. If it's entering into your ear, you would think that that's a measure of loudness. The more intense the wave, you would think that it's going to be louder. So what happens if you double the intensity of the sound wave entering your ear? You barely notice it. If you increase the intensity by a factor of 10, you're starting to notice it. So your ear does this funny thing where it can really detect a very large range of intensities. Your ear can detect an intensity as low as, if you have a normal ear, 10^-12 watts per square meter. It's a fantastically tiny energy per second per square meter passing into you, and you can probably detect that. It's just a little quiet rustle. You can also go extremely far in intensity and still hear it and not go deaf. You can hear one watt per square meter. You can detect that. It's loud; it's painful, but you can certainly distinguish sounds going from this intensity to that.
So it's this funny thing. As the intensity goes up by a lot, your ear seems to just think it's getting steadily louder and louder. So we often distinguish between intensity and loudness or sound level, and in fact we do it quantitatively rather than qualitatively. And here's the formula. Here's the intensity; that's watts per square meter. And is, by definition, the sound level or the loudness. So this is the mathematical connection between intensity and loudness, and your ear does a pretty darn good job of doing this mathematical operation and your brain interprets as loudness.
This is a weird formula. Let's just stare at it for a second. You take the intensity and you divide by I. I is just some defined reference intensity. And for sounds, I is usually 10^-12 watts per square meter, because that's typically the smallest or lowest intensity that's detectable. And then this formula will tell you what's the loudness of any other louder intensity. Actually, it works for intensities less than intense but it's just a mathematical function. You can plug in any number you want; I is a constant.
If you take the logarithm of--supposing the intensity was 10^-12 watts per square meter, so this would be the log of 1, which is zero--that's a zero decibel sound. So you take the log, you multiply by 10, and this is the symbol for decibels. Capital B is "bels." "Deci" is 1/10^th. Nobody ever, as far as I know--I never use "bels"; I always use "decibels" when I'm talking about loudness.
Let me remind you about how the logarithm works. Just a mathematical function you can always just plug in on your calculator if you need to solve problems or understand the physics of situations with sound. The logarithm is defined in the following way. Supposing you take the logarithm of x, some number, and you call that y. What does this really mean? The meaning of the logarithm is that 10 to the y power is x. So what is the logarithm of 100? What's the log of 100? It's the logarithm of 100, the logarithm of 10^ squared. So 10 to the 2 power; the logarithm is just 2. So it's easy to think about logarithms when you have a power of 10 in there. The log of 1,000 is 3. When the number inside goes up by a factor of 10, this number is increasing by one. This is not the natural log. Careful on your calculator. It's the log base 10. So what does a logarithm function do? It brings big numbers down and brings them to scale. As this number is getting huger and huger and by factors of 10, this is just increasing by one each time. And then there's this annoying but defined "multiply by 10" there and that gives you .
Let's think of an example, a specific example. You're at a rock concert and you've got front row tickets, so you're four meters away from this huge speaker and it's blasting out sound. And a typical number in that situation might be an intensity of 1 watt per square meter passing into your ears. What's the decibel level? Well, you take 1 watt per square meter and you divide by 10^-12. That's in the denominator so this fraction becomes 10^12 and the log of 10^12 is 12. Ten decibels, so is 12 times 10 or 120 decibels. For most people that's the threshold of pain. It's really not pleasant to be sitting there and you'll cause some ear damage if you sit there for a long time. That's why the musicians probably have some kind of earplugs in because they can't be exposed to this for a long period of time. A hundred and twenty decibels is an incredibly loud sound. And 1 watt per square meter--I don't know; it doesn't look like all that big of a number, but it's a lot of energy entering into your poor little ear.
Suppose you want to make it softer, what do you have to do? You walk away. And supposing that you wanted to decrease the intensity on your ear, the intensity of sound, by a factor of 100. You want to go from 1 watt per square meter to 0.01 watts per square meter. So if you've got 0.01 watts per square meter as your new intensity, then 0.01 divided by 10^-12 is now 10^10, and you take the log of that and you multiply by 10, and you get a of 100 decibels. So the sound is 100 times less intense, and the decibel or loudness level has gone just from 120 to 100. It's still very loud but it's no longer painful.
Interesting question: how far away do you have to go to bring the intensity level down by a factor of 100? You might think you've got to go 100 times further away. You were starting at four meters in my example. But that's not right. You don't have to go 100 meters away because sound is traveling out in spherical waves. It's spreading itself out over a progressively larger and larger area. And think about it. Intensity is defined to be power per area. Power divided by the area of a sphere is 4r^2. This is the energy per second being dumped out by the speakers, and this tells you that intensity doesn't go like 1 over r. It goes like 1 over distance squared. So if you want to decrease intensity by a factor of 100, you only need to increase the distance by a factor of 10, because then you're going to square that. So going from 4 meters to 40 meters takes the decibel level, the loudness level, from 120 down to 100.
Sometimes, I should just mention, people use this formula for other phenomena of nature besides just sound waves. So you can relate any intensity of some wave to some reference level. It doesn't have to always be 10^-12 watts per square meter. You can talk about intensity A and intensity B as your reference. You can say, "How much more loud is intensity A than intensity B?" and you just plug into this formula, and will tell you the answer.
Let me show you a little picture that's just kind of fun. It's about human response to intensity and loudness. This is a graph of intensity in watts per square meter on the left axis. The lower axis is frequency, ranging from 20 to 20,000 hertz. That's the kind of normal human range of hearing. The right-hand scale is a linear scale. It's just the decibels corresponding to these intensities, going from zero to 120. And notice, this yellow region is what you can detect if you have normal hearing, average hearing. And the bottom line here is tipped up. What that means is that you are quite sensitive to reasonably high frequencies around 2,000 to 5,000 hertz. You can detect those at this 10^-12 level. But if it's a low frequency, low pitch sound, it needs to be louder. It needs to be a lot more intense, a lot louder, in order for you to detect it. And if you're whispering, what do you tend to do? You're trying to make a low intensity sound wave, so you raise the pitch of your voice a little bit. You try to move up to higher frequency so that the person who's listening will be more sensitive. And here's my rock band at 40 meters now, not four meters, at 100 decibels. So just kind of interesting to stare at this graph and think about what kind of threshold of pain do you have and what can you just barely hear.
Intensity is power per area. It's watts per square meters. It's a useful typical measure of energy passing per second per square meter that you use in a lot of situations. Perhaps because of our human ear, our psychology, or maybe just because it's mathematically convenient, people will often, instead of talking about intensity, talk about loudness in decibels. One is related to the other. If you know intensity, you can calculate , and vice-versa using the inverse logarithm, 10^n. So you can talk about either one. It just depends on the physics problem or situation that you're involved in which unit system you would rather use. Either one is telling you about intensity, a useful typical quantity.
Intensity and Loudness Page [3 of 3]
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