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Lecture Comments (5)

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Post by Peter Ke on April 5 at 07:00:14 PM

At 32:09, why is it 10^-1 and not -1?

1 answer

Last reply by: Professor Selhorst-Jones
Mon Feb 25, 2013 9:31 PM

Post by Valentina Gomez on February 25, 2013

I found this lecture quite interesting! Thank you for facilitating physics! It's not an easy task :)

1 answer

Last reply by: Professor Selhorst-Jones
Mon Dec 10, 2012 12:52 PM

Post by Meda N on November 30, 2012

How do I calculate the intensity of a sound wave measured in dBSPL?


  • Sound waves are transmitted as a series of longitudinal pressure differentials.
  • Depending on the medium and its specific conditions, the speed of sound can vary greatly.
  • "High" pitches correlate to high frequency sound waves, "low" pitches correlate to low frequency sound waves.
  • The intensity ("loudness") of a sound is determined by the power (P) of the source and the area (A) it is spread over:
    I = P

  • If the source of the sound spreads perfectly evenly in all direction (which turns out to be difficult in the real world), we can model it with a sphere. The surface area of a sphere is A=4πr2, which we can use with the above equation.
  • The human ear can hear a huge variety of intensities. We manage this issue by introducing a logarithmic scale for measuring intensity: the decibel (dB).
  • If you feel uncomfortable with logarithms, you might want to do a quick review of how they work. Check out the pre-calculus math section on to get refreshed with logs.
  • We define the idea of sound level (β) using decibels:
    β = (10 dB) ·log10 I

  • Since waves can interfere with one another, if they have different frequencies, the waves will come in an out of phase with each other. This is the beat frequency:
    fbeat = |f1 − f2|.
  • For objects traveling faster than the speed of sound, we can describe its speed with a Mach number: [v/(vs)]. Mach 1 is the speed of sound, Mach 2 is twice the speed of sound, etc.


Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Speed of Sound 1:26
    • Speed of Sound
  • Pitch 2:44
    • High Pitch & Low Pitch
    • Normal Hearing
    • Infrasonic and Ultrasonic
  • Intensity 4:54
    • Intensity: I = P/A
    • Intensity of Sound as an Outwardly Radiating Sphere
  • Decibels 9:09
    • Human Threshold for Hearing
    • Decibel (dB)
    • Sound Level β
  • Loudness Examples 13:44
    • Loudness Examples
  • Beats 15:41
    • Beats & Frequency
    • Audio Examples of Beats
  • Sonic Boom 20:21
    • Sonic Boom
  • Example 1: Firework 23:14
  • Example 2: Intensity and Decibels 24:48
  • Example 3: Decibels 28:24
  • Example 4: Frequency of a Violin 34:48

Transcription: Sound

Hello and welcome back to Today we’re going to be talking about sound.0000

Sound is one of the waves you encounter the most often in life.0005

Such as right now unless you’ve got me on mute, but probably you don’t.0007

So sound waves are longitudinal pressure differentials.0013

If you had just a normal column of air, all of the air molecules would be equally spaced.0017

If you were to push on it suddenly, you’d get this effect where you’d have a bunch of them get bunched up.0025

Then that bunching would translate into them being really far spaced out and then they’d get bunched up again and then they’d be far spaced out, and then bunched up again and far spaced out.0030

That bunching and spacing is the equivalent of a wave going up and down.0040

Suddenly we’re seeing the pressure deferential either positive pressure deferential or negative pressure deferential.0046

That’s how the wave information is being transmitted. That’s how energy is being transmitted.0052

Is through these compressions and rarefactions moving through it, the compression being where it’s bunched and rarefaction being where there is very few air molecules at that moment.0056

We’re able to have longitudinal pressure differentials and that’s what’s happening as I’m speaking right now.0065

My vocal folds, my vocal cords are vibrating the air and those vibrations are being transmitted to the mike0070

Which is then being moved, turned into data, which is then transmitted to your speaker, and then the speakers vibrate that same amount of vibration so that you hear my voice reproduced.0076

The speed of sound varies greatly from medium to medium, even within the same types of mediums; the speed of sound can vary based on other factors.0088

Here’s some example mediums, so when see air at 0 degrees centigrade and one atmosphere of pressure, the speed of sound is 331 meters per second.0095

On a hot day like 30 degrees centigrade suddenly it’s moved up 349 meters per second.0104

This is because now the air has more energy in it, it moves faster so it’s going to bounce around so those pressure differentials are going to get moved faster.0110

Water at 0 degrees centigrade, one atmosphere is 1482 meters per second, whoops that shouldn’t be 0 degrees centigrade, that should be 20 degrees centigrade right there.0117

Water at 20 degrees centigrade, once again more energy, it moves around faster so it’s 1482 meters per second.0128

We increase the density by adding salt, turning it into sea water. We get an even higher one, 1520 meters per second.0134

Steel. Very uncompressible, so those pressure differentials get moved very quickly as its very hard for it to move.0140

That stiffness causes the information, the energy to be transmitted very quickly from one location to the next location along our steel column, bar, what have you.0146

So we have a much faster speed of sound. Depending on the material, depending on what we’re dealing with, speed of sound is going to change.0157

That’s something to keep in mind while we’re working with this.0163

Pitch. The human ear is capable of hearing a huge array of frequencies.0166

We’re able to hear the buzzing of a TV set, that sort of background hum of electronics.0171

We’re also able to hear the low base note of a kick drum.0178

That huge variety of frequencies that we’re able to hear is what we consider pitch.0183

A high pitch sound is one that is a large high frequency, like a violin or a piccolo or the right end of a piano.0189

A low pitch on the other hand is one that has a small frequency, like a double bass, a tuba, or the left end of a piano.0196

High frequency gives us that high sound like ‘Ah.’ That’s a higher frequency than when I talk like this in a low frequency voice’.0204

We experience this…experientially, we experience it qualitatively, as a feeling but it is also connected to a qualitative change in the way the thing is measured.0211

A young person with good hearing can normally hear sounds anywhere between 20 hertz and 20,000 hertz.0226

This is a really wide range of possible frequencies we can pick up.0232

As we get older, we start to lose some of that range, especially the higher end.0236

Certain diseases and injuries can also damage your ability to hear.0239

Pitches below human hearing a called infrasonic and pitches above human hearing are called ultrasonic.0243

Infra being Latin for below. Ultra being beyond and sonic being sound.0250

Certain animals can actually hear sounds that we can’t hear.0255

A dog whistle puts out a frequency that above human hearing in the ultrasonic range.0258

The dog’s ears are better calibrated to be able to hear those higher sound notes and so they’re able to pick it up.0263

Bats use ultrasonic sound to be able to locate their prey, locate the area around them, to be able to see, quote unquote ‘see’ the area around them.0268

Ultrasonic things have certain uses and so they’re able to send out an ultrasonic chirp and hear it back.0278

The way they hear it back, they’re able to pick up knowledge about the area around them.0286

We’re not able to hear those chirps because they’re ultrasonic, they’re above our hearing.0289

Intensity. Clearly the frequency or frequencies that we hear is part of the experience of sound but that’s only part of it.0296

The other part is how loud it is, the intensity of the sound.0302

A really loud sound is experienced differently than a really quiet sound.0306

That dynamic change is part of our experience of sound.0310

This is actually a qualitatively measure as well. It is how much power is being put out.0314

Remember waves are the transfer of energy through a medium.0320

An emitter must be putting out some constant supply of energy or a varying supply of energy but it makes it a little bit easier to talk about as if it’s constant first.0323

It has some power output, p, right?0330

If our emitter is putting out energy, putting out information to the world around it, it’s putting it out by putting out power.0333

Of course, if we’re far from an emitter, the power is more spread out and thus less loud.0340

If we walk really far away from an emitter it’s going to seem quieter to us. That’s because its power may remain constant but our experience of that power is now spread out over a wider distance.0344

It has to travel farther and so the sound waves might start off like this and they spread out a little bit but if we’re over here, they’re going to spread out and they’re going to be way wider out.0356

We’re going to get less information for the same area, so we’re going to have the intensity of the sound is going to be determined by how much area the power has to spread over.0366

A high intensity is either going to be a small area or a very large power.0378

As we either make the power smaller or the area wider we’re going to experience less loudness, less intensity in the sound.0385

What area does our power spread over? This is actually a really difficult question because sound often behaves in very complex ways in real life.0394

For example if I was sitting in the shower, probably standing, but whatever.0402

If I was standing in a shower, say it’s just a square around me.0407

My ear is over here and I shoot out sound waves.0413

It is not just the sounds waves that manage to shoot backwards towards my ear and get picked up.0417

Some of these sounds waves are going to bounce off the container and bounce back to my ear.0422

They’re going to get spread out and some of the energy will wind up being absorbed by the container, but some of it’s going to get bounced back towards me.0427

This is actually, can be a really complicated question, that’s why your voice can sound and have such different dynamics depending on the room you’re in.0434

If you’re in a very large cathedral chamber and you’ve got all this wood around you, you’ve got this big echoey open space.0440

The wood picks it up, changes the vibrations slightly. If on the other hand you’re in a tiny, tiled bathroom, you’re going to get a different sound.0447

Once again, those very hard tiles vibrate; reflect almost all the energy, almost all that information.0455

It’s a small space so it all bounces back and gets picked up by your ear again.0462

Different spaces, different perception of the sound because it becomes reflected differently.0465

Really complicated question, so we’re going to do a reasonable approximation.0471

We’re going to treat sound as if it spreads in all directions equally and simultaneously.0475

An outwardly radiating sphere, so that makes sense.0479

If we’re on a clear plain, we’re out somewhere in a grass land and we shout.0482

Its going to expand everywhere at once. We’re going to have some point source and it’s going to burst outwards just in a continuing radiating sphere.0487

Remember, this is actually a sphere that it’s bursting out in.0494

The point source and it blows out in all positions. Some of it may wind up being reflected but it’s a reasonable approximation.0500

Not perfect, but it gives us something to hold on to and have an understanding of how sound works.0505

If that’s the case, what’s the surface area of a sphere?0510

Well the area of a sphere is equal 4 times pi times the radius of the sphere squared. 4 pi r squared.0513

If the center of an imaginary sphere is put on our emitter, so we have some imaginary sphere that we’re considering.0519

We put it so that the center of that sphere is around the emitter. We are standing r away.0526

The intensity of the sound is going to be that area of the sphere dividing the power of our emitter.0531

The intensity will equal the power divided by 4 pi r squared.0536

The area divides the power. If we’ve got that simplified version of the expanding sphere then we’re able to use this nice little formula.0541

Intensity is a great way to measure sound as a quantitative thing, but it’s not a very good way to talk about it.0551

The range of possibilities for human hearing is really, really large. 1 x 10^-12 watts per meters squared.0558

Sorry I should have mentioned that before. If intensity is equal to power over area.0567

Remember we talk about power as watts. Area, well we use for our unit of length, meters.0571

Area is going to be meter time meter or meters squared.0578

Intensity is equal to the watts divided by meters squared.0582

Human threshold for hearing is around 1 x 10^-12 watts per meters squared.0585

That’s a really, really small amount that we’re able to pick up.0590

The pain threshold is massively larger, 10^13 times more.0594

Pain threshold when we start experience real physical pain from the sounds going off around us is 10 watts per meters squared.0598

That’s a really huge array of different possible values that we’re able to pick up.0606

We might be able to talk about it intensity like this thing but our experience of sound isn’t with each of these little tiny amounts different or these very large possible amounts different.0610

Our experience is actually based on a different scale. It’s based on a log rhythmic one.0620

We experience sounds multiplicatively, so that’s where the idea of decibels is going to come in.0624

To deal with this, we’re going to introduce the idea of the decibel.0630

DB which uses a log rhythm to help us manage this vast range of possibilities.0633

If you don’t remember what a log is. You’re going to want to go back and you’re going to want to check either pre-calculus or calculus and get a real quick understanding of it.0638

The basic idea is log base a of x equals y. Means the same thing as x to the y equals, sorry, not x to the y. Screwed that up.0645

Log base A of x to equal to y, gives us a to the y equals the quantity x.0662

For example, log base 10, which is what we’ll be using for the log rhythm for decibels.0671

Log base 10 of 10 equals 1 because 10^1 equals 10.0676

But log base 10 of 100 is equal to only 2 because 10 squared equals 100.0684

Similarly, 1,000, log 10 of 1,000 would be 3.0690

For simplicity we’re just going to start saying log even though we’re going to be talking about log base 10.0694

You could have the base of anything but we’re going to be talking about log base 10.0699

For ease we’ll just say log from here on out.0703

This is the important thing, if you don’t know, we’re going to wind up needing to use some of the properties that log rhythms have later on in the examples.0705

You might want to go back and refresh some of your memory on this.0711

On with decibels. We’re going to define the sound level beta, and this guy right here is called beta.0714

B E T A. He’s another one of our Greek letter friends.0721

The sound level beta, by referring to the ‘I knot’, the intensity, the basic intensity, the lower threshold of our hearing.0725

We’re going to utilize log. So beta, the sound level is equal to 10 decibels and that gives us a nice chunky number that we can hang on it.0732

10 decibels times log of the intensity of the sound that we’re listening to divided by that base threshold I knot.0740

Log of I over I knot times 10 decibels gives us our sound level.0750

Notice, a rise in 10 decibels means 10 times the intensity because we’re dealing with log 10.0755

If we go from 10 decibels to 20 decibels, that means the intensity has gone up by a power, not by a power of 10, has gone up multiplicatively by 10.0761

Because of log has gone from 1 to 2, it has gone up one number.0770

Going up one number we multiply by 10 decibels, so we’re going to have a change in the power of 10 times that we started off with.0775

If we go from 100 decibels to a 110 decibels, the exact same thing going on, because it’s a difference of 10 decibels.0782

That means our log creates a difference of 1, so it’s times 10.0788

If we had a difference of 20 decibels, it’d be a difference of a 100.0792

Sorry not a difference, a multiplicative 100.0796

100 times the power. If we had a difference of 30 decibels, it’d be 1,000 times the power, etc., etc.0799

Remember, we’re dealing with a log so it’s going to be connected to how exponents work.0806

You’re going to have to remember how logs work. If you remember that, sorry to harp on it so long, but if you don’t remember it right now, go back, relearn it really quickly.0810

It will make this understanding how decibels business work really, really easy if you compared to what you’re standing with right now if you don’t remember how logs work.0816

Here’s some loudness examples. A bunch of examples.0825

The example sound and the sound level that it gives you.0829

The auditory threshold for human hearing is at 0 decibels, because remember I over I knots.0833

If that’s equal to 1, log of 1 equals 0.0838

Light rustling leaves, very light, vague slight sound. 10 decibels.0841

A whisper, 20 decibels. Normal conversation gives us anywhere between 40 and 60 decibels depending on the loudness of the conversation.0846

A washing machine might come in around 50 decibels although it’s going to depend on the washing machine.0853

Car at 10 meters, once again, depends on the type of car.0858

Hearing damage from long term exposure, so this just means that you’re being exposed to something constantly and you’re not using any sort of hearing protection.0862

Notice how many things are going to be above that. A busy highway at 10 meters. A chainsaw at 1 meter.0868

A rock concert. So if you’re going to rock concerts regularly. If you’re constantly using loud motor driven things.0874

Either working around large powerful engines, going to like a construction yard or doing any sort of serious construction work.0881

You’re going to definitely want wear hearing protection because you’ll start to experience hearing damage.0890

If you’re shooting at a range, you’re definitely going to want to have at least one form of hearing protection on if not two forms of hearing protection.0894

Look at how incredibly high that is. 150-160 decibels, that means that we’re a 1,000 times potentially or even more depending on the kind of gun being shot.0900

A 1,000 times the power of a pain threshold. Guns really, really powerful amount of noise coming out of them.0910

Any sort of explosion, really powerful amount of noise coming out it, you’re defiantly want to wear some sort of hearing protection if you’re around that sort of noise.0918

Keep in mind; anything that’s particularly loud can cause hearing damage over long term exposure or even short term exposure if it’s very loud.0926

You’re going to want to keep that in mind because when you get older it’s going to be a real disappointment when you start to lose your hearing.0934

Beats. So from earlier work we know that waves interfere with own another.0942

If the waves have the same frequency, the amount of constructive or destructive interference comes from how in or out of phase they are.0947

As they sync up, they’ll manage to make themselves louder but as they come out of sync they’re going to wind up becoming quieter as they become more and more destructive on one another.0953

What if they’re different frequencies? From before it was just a question of how out of phase that they are and that determined how much destructive or constructive interference we had.0964

What if they’re different frequencies? That means we’re going to have the constructive interference, the amount that they’re syncing up is going to change as they cycle through their different frequencies.0972

They have a 1 hertz difference in their frequencies, every second they’re going to cycle from fully in phase to fully out of phase.0981

We’re going to experience the sense of them working together and then canceling one another out.0990

Working together and then canceling one another out. That’s going to be called the beat frequency.0994

The beat frequency is equal to the difference between the two numbers. It doesn’t matter which one is the larger one and which one is the smaller once because it’s the absolute value.0998

That’s how we’re going to experience it. It’s going to just be the difference between the two numbers.1007

They waves will come in and out of phase based on the difference in their frequencies.1011

It doesn’t matter if its 10 hertz and 11 hertz or a 1,001 hertz and a 1,000 hertz.1015

That’s still going to be the same beat frequency from our experience.1021

Visualizing this and visualizing beats, drawing them out, really difficult potentially.1025

They’re really easy to hear, so that’s how we’ll demonstrate this one here.1030

In a few moments we’re going to play an audio clip. The order of the clip will go like this; a 440 hertz pure tone just by itself.1033

Then a 445 hertz pure tune by itself. Now notice, these sound pretty darn close, these sound really alike.1040

If you’re not good at hearing differences in small frequencies you might not even be able to notice it.1049

It’s just barely noticeable that these two things are different tones, but when we play them simultaneously right after that, 440 hertz and 445 hertz.1054

We’ll be able to really notice the fact that they’re not the same thing anymore. We’re going to notice a beat frequency of 5 hertz as they pull in and out of phase.1066

We’re going to hear that sound. The next one we’ll play after that is 440 hertz compared to 441 hertz.1073

You’re going to hear a beat frequency of 1 hertz as they come in and out of phase.1079

You’ll hear the sound louder and then quiet down to nothing and then louder and then quiet down to nothing.1082

Then finally, we’ll have the first frequency stay at 440 hertz and just remain there while the second one will slide from 441 hertz to 450 hertz over ten seconds.1087

We’re going to hear those beat frequencies go from 0 to 10 hertz.1098

We’ll hear that beat frequency of getting louder, getting quieter. It’s going to speed up over those 10 seconds as the beat frequencies become larger and larger and larger.1101

Okay, so we’re going to listen to it and there we go.1109

Okay, now we have some idea of how those beat frequencies work.1132

The thing that’s really interesting to notice is how those two tones that don’t really seem that different suddenly produce this really noticeable phenomenon in this beat frequency.1136

At 440 hertz and 441 hertz, heard on their own we wouldn’t be able to notice the difference, but put right next to one another, put right on top of one another, not next to each.1143

Put right on top of another, suddenly that beat frequency becomes really, really noticeable.1153

We’re definitely able to hear those things. As the slide happens we start to hear those sounds change, we start to hear those beat frequencies change more and more.1158

In fact, you might have noticed at the very end, as the tail end of it. It starts to sound like the beat frequency is in some way its own sound.1165

That’s kind of true. The beat frequency becomes perceived as its own sound as the two sort of fight one another and we get this extra sound that is the two working against one another or together.1170

They’re creating this extra sound. Sound is a very complex phenomenon because once again we’re hearing many, many different waves.1182

It’s easy to talk about as a single wave, but in real life we’re hearing many, many different waves working together, working against one another all simultaneously.1188

Because we’ve had a lifetime experience listening to sound, we’re able to have an idea of how to turn all that auditory information into something we can actually process and operate on.1195

Once again, we’ve had a lifetime experience; we know what these sounds mean.1205

But really, when we you get right down to it, it’s a lot of very complicated information.1209

It’s because we’re so good at understanding and analyzing. We’ve got brains and they’re developed to do this.1213

They’re able to make sense, to get some meaningful information.1218

Finally, sonic boom. What happens when an object exceeds the speed of sound?1222

It moves faster than sound, producing a conical shockwave in its wake.1227

When you go past the speed of sound suddenly a lot of things are going to change.1232

Normally if you’re moving slower than sound, the sound wave, the fact that you’re pushing on the air in front of you tells the sound in front of it because it’s basically translating that information through the speed of sound.1236

Tells it to get out the way, there’s this really fast guy coming. So it’ll be able to create a moving pressure wave and you’ll be able to sort of push the sound out of your way before you have to actually run into it.1248

Suddenly once you’re moving faster than sound, you’re movement is faster than the sound can propagate the information of your moment coming.1260

So you’re actually slamming, you have to…your aircraft or spaceship has to be able to slam the air out of the way.1267

You’re actually slamming all of those molecules, so suddenly the fact that you’re slamming all of these molecules, slamming all of these atoms out of the way.1275

There is going to be way more friction, way more drag, way more heat produced on your thing.1283

Whole bunch of complicated ideas to talk about here, but we’re going to just talk about real quick, real simple idea.1287

We’re going to see the fact that every time you emit a sound or just the fact that you’re going faster than sound is able to handle.1293

That information is emitted spherically out in a sphere from you, or a circle if we’re looking two dimensionally.1300

The next time you emit it, you actually manage to already pass the edge of that expanding circle.1306

See over here, you emit here, but by the next time you emit, you’ve already passed the edge of that expanding thing.1312

It’s a little harder to see over here because we’re looking so far behind in time.1319

Over here where we’re looking closer to the instant that this picture is taken, we’re able to see the fact that the emitting of information happens after you passed the front of the information that you’re coming.1323

We’re going to get this depending on the speed that we’re moving, we’re going to see this conical wave front out of you.1337

This sonic boom appearing and that’s what people on the ground or other people in the air would wind up hearing is that slamming of air getting transmitted to them.1343

That slamming of air can only move at the speed of sound, so you’re able to actually get passed the sound of your own coming because you’re going faster than the speed of sound.1352

Also if the speed of sound is vs and the object is traveling at v, we give it a Mach number.1365

V over vs. So Mach 1 is just the speed of sound, you’re traveling at sound.1370

Mach 2 is two times the speed of sound because you have to be traveling double the speed of sound etc., etc.1375

Mach 5 would mean that you’re managing to move at five times the speed of sound.1381

Really interesting ideas here but we really have quite enough understanding or time at this moment to tackle all of them.1384

But we’re getting the chance to dip our toes in what’s going on here.1391

We’re ready for some examples. On a warm summer evening, the speed of sound in the air is 340 meters per second. If you see a firework explode in the air, so boom, firework explodes, and we’re standing over here.1395

We see the firework explode almost instantly because light travels so incredibly fast as we’ll talk about in the next section.1408

We see…we can effectively pretend that it’s instantaneous.1416

We see it explode the instance it explodes, but it takes some time for the sound to reach us.1420

It explodes and there is some distance between us and it. It’s going to take 4 seconds.1424

The speed of sound is 340 meters per second and the time that it takes is 4 seconds, we just figure out the distance is equal to the velocity times the time.1433

So 340 times 4 seconds and we see the fact that it must be 1,360 meters away from us because it takes time for that much distance to be crossed by those pressure waves coming from the explosion.1445

However light 1,360 meters is practically nothing to light. Light moves super, super fast as we’re going to talk about in the next section.1461

We’re able to effectively treat it as moving over that distance as instantaneously.1472

If we’re really, really far distance, light wouldn’t necessarily at something that we can treat as instantaneous but the distance for the sound would be so incredibly far at that point that we pretty much wouldn’t be able to hear it at all.1478

Example 2. If you double the intensity of a sound, what increase does that cause in decibels?1489

We start off with some sound at intensity I, what increase are we going to see in the decibels?1495

Say our old sound came in at b, so old intensity is as same as beta.1500

Then our new intensity is going to make beta new.1509

What is beta new? Beta new, remember, I new, intensity new is equal to 2 times I old, which for e, we’ll just say I old equals i.1515

Because that’s what we had it as before. If we double the intensity, I, of a sound, what increases does that cause in decibels?1530

So I starts off being the same as beta and I new is going to be equal to beta new and I new equals 2 times i.1537

Beta new is equal to our formula for decibels is 10 decibels times the log base 10 of 2 I, because that’s the intensity of our sound, divided by I knot.1544

10 decibels times log base 10 of 2 times I over I knot, you can’t stop me from separating that.1562

Still the same thing; either multiply it on the numerator, just multiplying the whole fraction, same thing.1575

Now, because the nature of log rhythms, we can separate those two. Log of xy is equal to log x plus log y.1580

At this point we’ve now got log 10 of 2 plus, and we should multiply because remember that 10db has to go over the entire thing.1590

Plus log 10 of I over I knot. We spread that out, we distribute, we get 10db times log 10 of 2 plus 10db log 10 of I over I knot.1602

Or remember, log 10 of I over I knot times 10db, that’s just…that’s the general expression for what something has as a sound level, what something is in decibels.1628

If that’s what we’ve got and we started off originally with I, then that means this whole thing on the right is just equal to our original beta. Beta old, right?1639

Our original beta is over here on the right and now we just have to figure out 10db times, well log base 10 of 2 is approximately equal to 0.301, plus that old beta.1648

That means 3.01 decibels plus the old number of decibels. The increase that we have is how much we’ve added to the old number of decibels.1666

So our increase in sound is that 3.01 decibels. If you double the intensity, you don’t experience a doubling in the number of decibels.1679

Not by a long shot. You’re just going to wind up adding on 3.01 decibels to the amount that you had originally.1687

Doubling the intensity does not mean doubling the decibels. Anything between intensity and decibels has to go through a log first.1694

We have to calculate this stuff out otherwise we’re going to wind up tripping over stuff.1701

Now we’re going to see something again about the importance of how much logs are going to play with this stuff by a slightly more complicated example.1705

Assume we can treat sound as an expanding evenly in all directions. So that same spherical idea we talked about before.1712

It’s not perfect but it’s a pretty good approximation and just like when before when we got rid of air resistance.1719

Not perfect but often a good approximation for a basic physics idea.1724

If we’re currently 1 meter from an emitter and we hear a sound at x decibels, what distance away from the emitter will lower the sound by 10 decibels, i.e. bring it to x-10 decibels?1728

To begin with, let’s note the fact that the emitter, no matter what distance, the emitter is still going to be putting out p equals the power.1739

The intensity is equal to the power over the area and since we’re dealing with a sphere, we’ve got power over 4 pi r squared.1752

Now we can start using our decibels connection, so we know that for the new one that we want to create, we want to see the distance r that we have to be at.1764

What distance r will create x minus 10 decibels. Our formula, 10 decibels and for ease, I’m just going to drop the base but we still know I’m talking about log base 10.1774

Of the power, divided by 4 pi r squared. Now power doesn’t change because power is constant for this emitter, is going to be equal to x minus 10 decibels.1785

Whatever we started with, lowered by 10.1796

If we want to know what this is, we know what x is, right?1798

X was the original amount. So let’s write everything in again so we can keep our equation proper.1802

X, x was the original number of decibels. So the original number of decibels was what the original r was.1813

Our old r was 1, right? We used to be 1 meter away from the sound source, so it’s going to be pi times 1 square minus, then we have to keep going up from what was above, 10 decibels.1818

Now at this point, that means we’ve got 10 decibels showing up everywhere.1835

This becomes…we divide by 10 decibels, these cancel out here and here and this becomes 1 right here.1840

At this point we can now do something else. Once again, if you don’t remember too much about logs, you’re going to want to double check, I mean you’re going to want go back, relearn this really quickly.1848

We can cancel out a log by raising it to whatever the power of the base is.1858

In this case, since we know those two sides are equal, 10 to each of those sides would still be equal.1863

So we raise both sides with a power underneath of 10. So 10 to the log p over 4 pi r squared is equal to 10 to the quantity log power over 4 pi, because 1 squared is just 1, minus 1.1867

Now 10 to the log, that cancels out and we get what’s inside, p over 4 pi r squared equals p over 4 pi.1886

One thing to notice is that we’ve got 10 to the…we can separate it, 10 to the xy is equal to 10 to the x times 10 to the y.1901

It just gets…Sorry, 10 to the x plus y. Misspoke there.1909

10 to the x plus y is equal to 10 to the x times 10 to the y because we just add things together.1912

X squared times x becomes x to the 2 plus 1 x cubed.1918

We have to remember that and we’ve got still over here, 10^-1 because we’re separating those two different things.1923

At this point we can now cancel p over 4 pi from both sides and what we’ve got now is 1 over r squared equals 1 over 10.1929

At that point, we’ve got r squared equals 10, so r is equal to the square root of 10, which is approximately equal to 3.16 meters.1943

So notice that because the way we did this, this whole thing occurred on the fact that we were dealing with r squared over here, here right.1963

If instead of starting at 1, we’d start it at r old, that r old would still have shown up over here and we would have had that.1973

What it is; is it’s not just going 3.16 meters away. It’s not that at all, its 3.16 meters times whatever the original distance was.1981

It’s our original distance times 3.16. So in our case, we looked at a slightly easier one because we dealt with an initial distance of 1 meter, but we can also expand it to a slightly more complicated thing.1990

It’s actually route 10 times the original distance that we started at if we want to get a 10 decibel lower.2001

Lowering it by 10 decibels means multiplying our original distance by route 10, which is why we get going from 1 meter to 3.16 meters.2008

Say we started off at 5 meters and we wanted to lower it by 20 decibels, then we’d have to take that 5 meters originally, multiply it by route 10 twice.2017

Route 10 twice, becoming 10, so we’d have to go from 5 to 50 meters if we wanted to be able to get a 20 decibel lower in the sound.2026

It’s important to note, once again, dealing this stuff can be a little confusing at first because we’re dealing with logs.2036

You’re definitely going to want to work on calculating this thing out by hand because you might be surprised by how some of the results are going to work out.2042

Think about this, the fact that we’re going from 1 meter to 3.16 meters to get a lower of 10 decibels might not be inherently oblivious at first because we’re working with the way log rhythm works.2047

Log rhythm can be a little, a little new, a little odd at the first time you’re dealing with it because we’re normally can…we’re normally…the way we understand the world first is through additive understanding.2058

Adding things together. It is a little bit more difficult to think in terms of multiplication through exponents and that’s how logs look at the world.2070

Definitely the sort of thing, I definitely want to caution you if you have to do a problem with decibels, make sure you’re working through the formula of how we get decibels first.2078

Otherwise it’s really, really easy to wind up making mistakes.2085

Final example, a nice easy example to finish things off with.2088

Say we have a 256 hertz tuning fork and we put it to a violin and we hear a beat frequency of 3 hertz.2092

What are the possible frequencies that the violin is emitting?2101

Remember, f beat is equal to the difference in those two things.2103

If we’ve got a 250 hertz tuning fork and we’re comparing it to the frequency of the violin and we’ve got 3 hertz beats coming out, then the two possibilities for the violin are the two things that are 3 away from 256.2110

We’ve got the frequency of the violin, it must currently be emitting. The violin can emit many things but we’re going to treat it as if you’re just bowing on a single thing, at a single tension.2127

At this point, we’re going to see the two possibilities are 253 hertz, the lower one or 259 hertz.2136

The two things that wind up being 3 hertz away from what we’re comparing at.2146

That’s how beats frequencies work if you need to solve anything with beat frequencies.2152

Once again, if you’re not used to logs, go back, definitely going to behoove you to take 20 minutes just refreshing yourself on how logs work.2155

Otherwise some of this is going to be really, really complicated to understand how decibels work to solve any problems with it.2162

If you’re working with decibels, definitely want to work through it, otherwise you can easily trip yourself up.2166

You want to remember everything is based on multiplicative ideas.2172

That’s how you want to be approaching it through log rhythmic scale.2175

Alright, hope it made sense. Hope you have a better understanding of how sounds work and next time we’ll wind up talking about light waves. Thanks.2178