MCAT Physics Waves & Sound: Comprehensive Chapter 7 Summary

hi everyone today we're going over chapter 7 of physics for the mcat which covers waves and sound

this video will use slides made by my friend victoria instead of the usual notes that i hand write

so the slides may look a little bit different um but i promise you the content is the same and thank you so

much victoria for letting me use yours chapter 7.1 is about the general characteristics of a wave

to begin the mcat only covers sinusoidal waves which are smooth periodic waves that

kind of look like a sine or a cosine curve there are two kinds of sinusoidal waves

and the first is called the transverse wave so the transverse wave is the version of a wave that you are probably

most familiar with it is where the movement of the particle is perpendicular to

the movement of the wave so what this means is that if we draw our classic picture of a wave and let's

pretend this wave is moving to the right so what it means that the particle movement is perpendicular to the

movement of the wave is that particles here are moving up and down so they're moving

up and down which is perpendicular to the rightward travel of our wave on the other hand

longitudinal waves are waves that have parallel movement to the movement of the particles

this is much harder to draw out um because in effect both the wave and the particle would be moving

to the right and also to the left but what this means in practice is that imagine that you're standing in

a very long line of people and you're at the very end of the line and you shove the person in front of you

forward so that person would then shove the person in front of them for it and this would go

all the way down and then the person at the very front of the line would turn around and shove everyone

back so then you would have this movement that is parallel to the direction

of the line so everyone is shoving everyone forward and backwards and this is how sound waves work so

transverse waves are um forms of electromagnetic waves so you will often see some see light

depicted as a transverse wave however it's really hard to depict a sound wave

because how sound waves work is that each um individual air molecule is pushing the

one in front of it forward to propagate forward this is also a sinusoidal wave because

there are always people who are being shoved forward and people who are being shoved backwards

and people who are not being shoved at all and it turns out that this pattern forms a

sinusoidal pattern you can also graph a longitudinal wave using a sine curve but it wouldn't represent

the same perpendicular motion that the transverse wave did so what a peak would represent is that the

particles are currently being shoved forward and at the bottom this would represent

that the particles are being shoved backwards there and here at this node are particles that are not

shoved in any direction there are some relevant equations when dealing with waves the first is

propagation speed denoted by v and this is just how fast a wave is moving in the forward direction

this is equal to the frequency which is how often a wave hits a certain point so this is

in units of per second so one over a second or inverse seconds or

in units of hertz which is the same thing as an inverse second or one over seconds

um and this frequency is multiplied by the wavelength which is just the length of the wave

this is most common in meters so this gives the propagation speed in units of

meters per second the other equation is angular frequency this is almost the same thing as frequency except

you multiply it by 2 pi this is in units of radians per second this is most useful when

calculating harmonic motions such as in springs and pendulums this is not particularly

related to this chapter however you might be asked to calculate the angular frequency of something on

the mcat so just remember it's the regular frequency multiplied by 2 pi

there's some terminology that we use when we talk about waves so first the amplitude is the distance from the zero

point until the very highest point and since this wave is sinusoidal this is equivalent to the distance from

the zero point to the very lowest point the wavelength is how long it takes the wave to go an

entire cycle so we start at a particular point and then we go

an entire cycle and when the wave is back at where it started this is your wavelength and so this

wavelength is the same no matter what point you use for this calculation for example

if we use this point we have to go one entire cycle for this to be a wavelength so the very

peaks of the waves are called the crests and the very lowest parts are called the troughs

and the direction of travel is usually to the right here we can consider the phase of a wave

so we can say that waves are in phase when their crests and their troughs are at the same spot

so if i were to draw another wave here you can see that the crests and the troughs of this wave

line up with each other so they're in phase even though their amplitudes are not the same

if we were to draw a different wave so for this hypothetical wave these are clearly not in phase and so these are

out of phase by this amount and this amount is pi since the troughs of this wave line up

perfectly with the crest of this other wave so they are perfectly out of phase

we can also have different factors of in phase and out of phaseness we can have them be out of phase by 80

degrees 1 degree this is out of phase by pi or otherwise known as 180 degrees

because the period of every single wave is in units of two pi and so because these are exactly half a

wavelength off from each other they're off by pi or 180 degrees and when waves are off by

180 degrees that's when they're perfectly not in phase if we were to take our wave that's

shifted forward by pi and we were to shift it another 180 degrees forward

this is now a shift of 360 degrees which is two pi and as you know 360 degrees is the same thing as zero

degrees and so now these waves are perfectly in phase we can apply these ideas of phase to the

principle of superposition which states that the displacement of a wave at any given

point is the sum of all of the interacting waves at that point so one type of interference is

called destructive interference which is two waves that cancel each other out and

so on our first wave i can draw one that is perfectly out of phase with our original wave so if we were to

add these two together we can note that the highest points of one correspond with the lowest points of the other

and so forth with here being our zero point and so if these were to perfectly cancel

each other out then our resulting wave would be completely flat we can also have

destructive interference that doesn't um correspond to a completely flat wave so if our perfectly out of phase wave

were to have a lower amplitude then our sorry drew that backwards if it were to have a lower amplitude than the

original wave then our resulting wave you see how these don't perfectly

cancel out however this has a destructive sort of effect where the resulting wave

will have a lower amplitude than the original wave another case is constructive

interference so constructive interference is when the second wave that's added has

the same phase as the original wave so in this case if the waves are perfectly

in phase then a resulting wave holds the same shape

but it has a greater amplitude so what's important to note about this wave that we drew here

is that although it doesn't cancel out completely this is still known as destructive

interference and not partially destructive interference because the waves are perfectly out of

phase with each other so therefore these peaks and these troughs

align with the original peaks and troughs of the original wave however when we look at partial

interference this is when the waves are out of phase by not a perfect multiple of pi

so if we look at the original wave a and the original wave b we can see that they are out of phase by

something that is not exactly pi and so this resulting wave which is partially constructive or partially

destructive will have um will have peaks and troughs that are not

exactly the peaks and troughs of a or b we can also talk about traveling waves and standing waves

so here we're talking about the movement of the entire wave as a whole so traveling waves are waves that move

in a particular direction this is like light because light moves in a direction or like sound because

sound also has to move forward for anyone to hear it on the other hand a standing wave doesn't propagate

so standing wave doesn't really move anywhere not the entire wave at least my favorite example of this is a jump

rope because you're swinging a jump rope up and down but your jump rope isn't really going

anywhere in waves that have more than one peak like this are usually caused by two

waves interfering with each other so they have what are called nodes and antinodes

nodes like here are places where the wave doesn't move anywhere and antinodes are places like

here that have maximum amplitude what this picture is trying to show is that your wave may start off like this

but as the two waves interfere with each other you will eventually get a wave with

lower amplitude and then even lower amplitude and then eventually you will have a point where

nothing is moving and then you will begin to go in the opposite direction

until eventually you reach the exact opposite sign you can think of this as like a long

extreme jump rope where each of these is like a jump rope that starts at the top

and then moves down and then moves lower and then when it reaches the bottom it swings

back up again chapter 7.2 is about sound and sound production most formally

is the mechanical disturbance of particles in a material along the waves direction of movement in

practice this is what i said before this is that what creates sound is that molecules in the air are running

into each other and this causes the molecules to propagate forward at a certain frequency

until it hits your ears and this frequency is then translated into what you know is

the perception of sound so the frequency of sound is v equals the square root of

b over rho this capital b is known as the bulk modulus which is just a description of the

medium's resistance to compression this bulk modulus term changes a lot depending on whether you're dealing with

a liquid a solid or a gas so gases have a very low resistance to compression gases compress

very easily so gases have a very low um bulk modulus however a solid or a liquid are much

more resistant to compression so it has a much higher bulk modulus this is why you might have heard that

sound travels faster in water and this is why because water has a much higher bulk modulus than air does

the bottom term here row is just the density of the medium and although the density of water is

much higher than the density of gas this bulk modulus term in the end is going to have a much higher

effect on the overall speed of the wave just because bulk modulus is a much more variable

variable than density i don't think you're going to have to calculate the speed of sound based on the bulk modulus

and the density on the mcat at least i've never seen it but you do need to know that in general

the speed of sound is fastest in solids and slowest in gases and in particular the speed of sound in

air at 20 degrees is going to be equal to 343 meters per second this is really

important to know the range of human hearing is between 20 and 20

000 hertz and remember that hertz is a measure of frequency so this is in units of

inverse seconds one over seconds and it's a measure of how often something passes so the doppler effect

is an effect where the actual frequency of the sound is different from the perceived

frequency of the sound when both the objects are moving so if we think back to our original

equation that the velocity of a wave is equal to the frequency times the wavelength

then the doppler effect will begin to make more sense so when things are moving toward each other

let's say we have a car oh okay let's say we have a turtle so we have a turtle that's going toward you

at 10 meters per second however you're also going toward the turtle at five meters per second so

relative to you this turtle is moving toward you at 15 meters per second

so you and the turtle are getting closer by 15 meters per second and so when you're going toward each

other this means that your effective velocity is faster than it was before

and therefore your effective frequency is higher than it was before so your perceived frequency will be

greater than the actual frequency if however this turtle was chasing you at the same 10 meters per second

but you're running away from the turtle now at five meters per second then any sound this turtle makes will

seem to be moving at five meters per second and this is because relative to you the

turtle is only getting closer to you by five meters per second and so the perceived frequency will be lower than

the actual frequency because your perceived velocity is lower and therefore your perceived frequency

is lower and this is the basis of echolocation as well

here we have the equation for the doppler effect you might see a lot of different equations for the doppler

effect but this one equation sums them all up so here we have the frequency so this is

the frequency that you're going to end up observing this is the original frequency denoted

by f naught this is a zero and not an o for observer and so this um f naught which is the

original frequency or the emitted frequency is multiplied by this term

this is the speed of sound so remember that i said the speed of sound in air is 343 meters per second

and so on top we have the speed of sound plus minus the speed of the observer

and so this velocity of the observer is either added or subtracted from the speed of

sound depending on whether the observer is going away from the source or toward the

source so if you're going toward the source you would add and if you're going away from

the source you would subtract on the bottom half you would have the velocity

of sound plus or minus the velocity of the source and so this would depend on whether the source is going toward or

away from the observer and so if the source is going to go toward the observer this

would be a minus sign and if the source is going to go away from the observer this would be a plus

sign to remember which one goes on the top and which one goes on the bottom

i heard this mnemonic once and i've been unable to forget it ever since so we consider the top the observer or

the detector and we consider the bottom the source or s so here we have dom over

sub i've never been able to forget this ever since then so for a bit of a logic check on why

these signs make sense so why for the source if it's moving toward you it's a minus sign and for the observer

if it's moving away from you it's a minus sign we can think of what would happen if an

observer were to move away at one meter per second and the source were to move toward you at one

meter per second so if we use this equation we can see that the proportion of v

minus 1 and v minus 1 are the same so this cancel out to 1

so we have the frequency equals the original frequency which makes perfect sense because if the

source is moving at one meters per second to the right and you are also moving at

one meter per second to the right then you guys aren't moving at all with respect to each other

so if on the other hand the source was moving toward you at one meter per second but you were

moving toward the source at one meter per second we would have the um observed frequency

equals the original frequency times v plus one over v minus one

and so we can see that this is going to be a value that's greater than one and so we're going to end up with an

observed frequency that's going to be greater than the original um

frequency it makes sense that this observed frequency is now going to be greater than the original frequency

because if you're moving together at in this scenario two meters per second this is going to end up with a higher

velocity than your original velocity and as we know things that move faster have a higher frequency

the intensity of a wave is going to be measured by power over area which is going to be in units

of watts per meter squared so what this means is that if you have a flashlight bulb with a certain amount of

watts and you hold this really close to a sheet of paper this is going to be very

intense because this amount of power is spread out over a small area

but if you back up a lot then this light will be much less intense now because it's spread over a very large area

note that this has nothing to do with the frequency the frequency doesn't have anything to do with the intensity

because in terms of sound the frequency is going to be what pitch your sound is and in terms of light the frequency is

going to determine what kind of light it is like is it visible is it going to be microwaves

it's also important to note that intensity is proportional to the square of the amplitude so most

accurately this should be written as i is proportional to the square root of

amplitude and intensity is also proportional to one over the distance squared um

so this is what i was talking about before where if you had a flashlight and you backed up from a piece of paper

then the intensity which would be much lower and what's important is that there is a square factor in each of these

things so you might be asked questions like if the amplitude of the wave were to double

and the power was to also double what would happen to the intensity of your wave or yeah of your wave

and what you would calculate is 2 squared because this is the result of the doubling of the amplitude

multiplied by 2 because this is a result of the doubling of the power and this would equal 8. so you would say

that your light got 8 times more intense if on the other hand you were told that

the distance doubled and also the amplitude doubled then this would be

2 squared divided by 2 squared equals 1. so nothing would happen to your intensity

so i don't think these thresholds are really important at all one that is important

is the softest intensity so this is the threshold of human hearing the threshold of human hearing is 1 times 10 to the

negative 12 watts per meter squared you might want to memorize this along with the speed of

sound and so what i do think is important is this equation here so what this equation tells you is how

to convert basically from an intensity of sound to decibels here we have our final

decibel level and you can see that this is denoted by final sound level you also have an

initial sound level um and so our threshold of hearing here is set arbitrarily to zero decibels

this is just because this is the lowest we can hear and so if you wanted to convert

any type of sound you would use this equation so here we would plug in our initial

intensity which is going to be r0 decibel 1 times 10 to the negative 12 watts per

meter squared and here we're gonna have zero decibels and here we're going to plug in what we

want our intensity to be so if you wanted to know about a sound that's at 1 times 10

to the negative 9 watts per meter squared you would plug that in here and you

would take this ratio and then you would take a log of it it's okay if you don't remember

how to take logs in your brain i don't either but what i do know is that this ratio here

is going to be 10 to the third and so the log of 10 to the third

is just going to be 3. so whatever the factor of difference is that's the number that you're going to

end up with here and so if we have a factor of three difference we then multiply this by 10

here and our final answer is going to be zero decibels

plus 30 decibels equals 30 decibels oops 30. so the take home lesson here is that

decibels are measured in a log scale and also they're measured in factors of 10. so

if you were to have a sound that were to increase by 10 decibels this would equal an increase by 10 times

if you were to have a sound increase by 20 decibels this would mean an increase of a hundred

times and if you have something increased by 30 decibels

this is an increase of a thousand times and we can see this very clearly here with our 10 to the negative 12 watts per

meter squared and 10 to negative 9. 10 to negative 10 to the negative 9 is a thousand times bigger

than 10 to the negative 12. one thing the mcat really likes to ask about is what would happen

if the intensity were to double so we know that if the intensity were to double

the number of decibels would definitely not double because doubling would represent an increase of

many many factors a quick rule of thumb is that if your intensity were to double

then your number of decibels would increase by plus six decibels this really short trick saved me a lot

of time in actually having to calculate what would happen if it were to double

attenuation is also called damping and this is just what happens when a wave tries to travel a really long

distance and some non-conservative forces happen to act on it so if you're speaking and your voice

goes very far then other factors like friction or the wind would cause the amplitude

of these waves to go down i was taught a different convention for naming open and closed pipes and this book teaches

i'm gonna go with what this book teaches but i want you to know to not be confused because i can't be the only

person who is taught a different convention for this so a pipe with closed boundaries

means that the ends are tied down and so these form nodes because if the ends are tied down like

in a jump rope then there is never any displacement at the ends however if you have an open boundary

which means the ends are free so if this wave were to end right here and the ends were free to move

up and down these are called open boundaries and they're marked by antinodes because right here

you can move up or down freely a harmonic is a way of describing the frequency inside a string

or inside a pipe so if we look at the system here which is what the kaplan

book calls a string system this is when both ends of this string are tied down

and this is true for all of these pictures of the string system this is what my physics class would have

called a closed closed system because both of the boundaries are closed and their nodes

so if we think about the lowest possible frequency we can fit in the system the lowest possible frequency is this

half wavelength here we can't fit anything lower and so this would be called the fundamental

frequency if we then increase the frequency we can see that the second

um the second lowest possible frequency for this is going to be one full wavelength

and the third is going to be one and a half wavelengths so these are known as the second harmonic

and the third harmonic and we can see in the equations relating the wavelength and the length

of the pipe that we're starting to form an equation and this equation is that the wavelength

is going to be equal to 2 l over n where n is the number of harmonics

so in the fundamental harmonic or the first harmonic n is equal to one and l is going to be

the length of your pipe or string next we take a look at open pipes so open pipes are marked with

two open ends and we have an open end we know that we would have an anti-node here which means that we are at maximum

amplitude so we can see that we have here the exact opposite of the situation that we

had in the string which is that this first or fundamental harmonic would also

only fit half a wavelength but it would fit half a wavelength in the opposite way where the node is in

the middle and their antinodes on the side you can also see this for the second harmonic

where we also have antinodes on the side and this also forms a complete wavelength

and so the equation to relate wavelength to l is this exact same equation for the

open pipe lastly we have the closed pipe which is going to be a little bit different from

the open pipe and the string for a closed pipe you have one

closed end as well as one open end and therefore you have one node and one antinode at the end and so

you'll note here that what's drawn here is a quarter of a wavelength and you'll also note

that here in the harmonics we only have odd numbered harmonics i had to alter this

picture a little bit because it didn't fit with what the book is teaching

and so the reason why we only have our odd harmonics is because if you were to have n equals

2 and we were to have twice the number of wavelengths so then we would have

half a wavelength this would indicate that we would have this segment here which has two closed

ends and so we we can't have that we can only have odd numbers because we have to have

one closed and one open side so you can see that the lowest number of frequencies we can fit

with a close and an open side is a quarter of a wavelength the second lowest we can fit is one and

a half wavelengths and then we have two and a half wavelengths so

the equation to relate um the harmonic the length and the wavelength of a closed pipe

is going to be lambda equals four l over n where n can only be odd

this is very important you never want to plug an even number in here and you never want to get a multiple of

wavelength that is a half or a hole lastly to talk about how ultrasounds work

so very high frequency sound waves are emitted into your body and they'll bounce off your tissues and

bounce back to the receiver and depending on the density of the different tissues

they'll bounce back at different frequencies which gives us an image of what's inside your body

a doppler ultrasound uses a um a similar idea however a doppler ultrasound

can be used to determine whether blood is moving away or toward you um based on the frequency of

the emitted and the observed frequencies so this is the end of chapter seven thank you so much for watching

and i hope this video helped you out