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# Decibel Scale

## Video transcript

all right I'm going to tell you about the decibel scale this is a scale that we use to figure out the loudness of a sound the equation that goes along with it looks like this beta equals 10 log logarithm base 10 of I divided by 10 to the negative 12 watts per square meter this looks intimidating let's talk about it and break it down beta is the number of decibels so this side gives you the number of decibels and we abbreviate decibel with a little D capital B so this is the number of DB decibel you've probably seen this number on your stereo somewhere we are adjusting volume because we're going to measure volumes in decibels 10 this 10 just denotes the fact that this is the deci bel scale and not just the bel scale if you didn't multiply by 10 you'd have the bel scale but this is multiplied by 10 we like the 10 we're going to call it decibel log will talk about log in a minute logarithm here I is the intensity of the sound wave so this is the intensity and in physics intensity is defined to be the power divided by the area and what this means think about it this way you've got your ear and a sound wave say is coming toward your ear if you imagine one so power is in what power is measured in watts area is measured in square meters so think about intensity this way if you had one square meter imagine one square meter of area here this doesn't have to be an actual physical object just imagine a square meter of area the power that passes through that area would be how many joules if you figure out how many joules pass through this one square meter if you ask how many joules per second pass through that one square meter how many joules of sound energy per second pass through the one square meter that would be the number of watts per meter squared which would be the intensity so watts is tools per second and so this gives you an idea of how much energy per second passes through a certain amount of area and this part of the equation is my favorite this is my all-time favorite right here this number this 10 to the negative 12th watts per square meter represents the threshold of human hearing and what that means is this is the softest possible sound you can hear any sound with an intensity less than that you won't even notice but if it's anything bigger than that a human ear that's healthy should be able to detect it here's why I like this number this is unbelievably small this is one trillionth of a watt per meter squared a trillionth what this says is that even if only one trillionth of a Joule per second passes through the square meter your ear would still be able to detect a sound that's that soft and if that doesn't impress you let me put it to you this way imagine we did have one watt let me put it to you this way if you had one watt how big of an area and a watt isn't really that much what is not a lot of power if you had one watt how big could you make this area how spread out how diluted could one watt be spread over how large of an area could this one watt be spread over and still be intense enough for the human ear to hear it what do you think football field you know I don't know a city no it turns out if you do the calculation I suggest you do it's interesting you would get that you can spread 1 watt over the entire land area of Germany about 3 times over and still it's intense enough for the human ear to hear that's how unbelievably sensitive our ears are it's it's actually I told you is unbelievable I can hardly believe it myself let's come back over to here so here's our equation this is the decibel scale Y log what you're thinking why in god's name did the physicists have to put logarithm in here these scare me they used to scare me too well I'll show you why here's the problem the fact that we can hear such a soft sound 10 to the negative 12 watts per meter squared means there's a huge range of human hearing this means we can hear from 10 to the negative 12 watts per square meter this is this is zero point zero zero see three four five six seven eight nine ten eleven with a 1 watts per square meter all the way where there's no upper limit you'll just blow out your ears but once you get to about one watt per square meter this is when it starts hurting this is painful you're not going to be happy over here your ears are going to start hurting you'll get start getting hearing losses not good so there's a huge range 12 orders of magnitude this 1 watt per square meter is a trillion times bigger than the side the scale is just way too big this is awkward we want to scale this smaller maybe like 1 to 100 to measure loudness we don't want to measure from 1 to a trillion or a trillionth to 1 and that's what log is going to do logs are great this is a trick physicists use physicists love this trick logarithms take really big or really small numbers and turn them into nice numbers that's why we're going to use the logarithm so let me show you what I mean logarithm if you don't remember here's what logarithm does log base 10 of a number equals here's what it does I'm going to stick a number in here let's stick a hundred-thousand what log does log is a curious guy logs always asking a question log always wants to know okay if I'm log base 10 log wants to know what number would I raise 10 to in order to get this number in here so log looks at this number in the parentheses this entire number here and ask what number should I raise 10 to in order to get a hundred thousand well we know the answer to that you should raise 10 to the fifth and if I raise 10 to the fifth I'll get a hundred thousands so oh if v is number I raise 10 to to get 100,000 and that's the answer to this that log base 10 of a hundred thousand is five and look what happened log took a huge number a hundred thousand and turned it into five well that's outstanding log can take huge numbers turn them into nice numbers the logarithm base 10 of 1 billion would be 1 billion is a big number that's hard to deal with but log takes 10 and ask what number could I raise 10 to in order to get a billion and I should raise 10 to the 9th because I've got 1 2 3 4 5 6 7 8 9 zeros here I raise 10 to the 9th to get this number so the answer to this question for the logarithm is 9 oops that's not 9 9 and that's why logarithms are good logarithm took this enormous number of billion and turned it into 9 so logarithms take enormous scales turn them into nice scales that's why we like this formula which is our decibel scale because it takes enormous intensities and small intensities turns them into nice intensities so let me show you an example with this equation really quick let's say you're talking to your friend maybe you're yelling at your friend you guys are having a heated exchange and so you're yelling he's next to you these are the sound waves coming at him you're yelling with an intensity of say 10 to the negative 5th that doesn't sound like a lot but that's actually you're pretty upset here that's pretty loud so I want to know how many decibels is this how do we figure out the depths decibels well here's what we do we use our formula for decibels beta number of decibels equals 10 log base 10 of the intensity over always 10 to the negative 12th watts per square meter because that's the softest sound we can hear what do I get 10 to the negative 5th is my intensity so I plug this into here and I'm going to get beta equals 10 times the log base 10 of 10 to the negative fifth because that's my intensity divided by 10 to the negative 12 now these are both watts per square meter so those cancel well what's 10 to the negative 5th divided by 10 to the negative 12 turns out that 10 to the seventh so I end up with 10 log of 10 to the seventh now I don't like log I'll be honest they freak me out but I could even do this 1 log of 10 to the seventh remember what log does it asks what number do I raise 10 to in order to get the thing in the parentheses well the number I raise 10 to to get the thing in this parenthesis it's already 10 to the 7 starting this form so I just raised 10 to the seventh to get 10 to the seventh so the answer to log base 10 of 10 to the seventh is just seven so my final answer beta the loudness the number of decibels is going to be 10 times log of 10 to the seventh was just seven because I had to raise 10 to the seventh to get 10 to the seventh so 10 times seven equals 70 I'm yelling at 70 decibels I need to calm down my friends going to start getting mad at me that's how you figure out how loud the sound wave is