Mechanical Waves Explained: Amplitude, Frequency, Wavelength, and Harmonics

in this video we're going to work on some basic problems associated with mechanical waves

so here we have a graphical representation of a wave what is the amplitude and wavelength of

the wave shown below the amplitude is the distance between the midpoint and the highest point of

the wave so it's from zero to three so in this example the amplitude is three meters

now to find a wavelength you could find the distance between the crests of the wave

or the distance between the troughs or you could find

the distance that it takes to complete one cycle so from here to here

so how many cycles do we have in this wave so this is one complete cycle

that's two cycles and that's three cycles so in three cycles the wave traveled a

distance of 10 meters so we got to find the distance it travels in one cycle

so that's going to be 10 meters divided by 3. so the wavelength is going to be 3.33 meters

so this position is about 3.33 and then here it's 6.67 and then it's going to be 10 meters

number two determine the amplitude period and frequency

of the mechanical wave shown below so feel free to try this if you want to so the amplitude is the distance between

the midpoint or the center line and the crest of the wave so in this example we can clearly see that it's 10

meters so it's basically equal to that number now what about the period and the

frequency of the wave the period is the time it takes to make one complete cycle

so notice that the five seconds corresponds to the first quarter of the cycle

so at the second quarter which is here it's going to be 10 seconds at the third quarter it's 15 seconds

and at the full cycle at the end of it it's going to be 20 seconds so the time it takes to make one

complete cycle is 20 seconds the frequency is 1 divided by the period

so it's going to be 1 over 20. so 1 over 20 will give us a frequency of 0.05 hertz

and so that's the answer for this problem so here's another example

go ahead and calculate the amplitude of this particular mechanical wave to calculate the amplitude

it's going to be the difference between the highest value and the lowest value divided by 2. so

let's say h is the highest value l is the lowest value and so it's going to be 12 minus 4

divided by 2. 12 minus 4 is 8 8 divided by 2 is 4. so the amplitude in this example is 4

meters now the midpoint between 4 and 12 is 8. if you add 12 plus 4 you get 16 divided

by 2 that's eight and as you mentioned before the amplitude

is the distance between the center line and the highest point so 12 minus eight is four and that will

give us the same answer now what is the period

of this particular wave so notice that the six seconds corresponds to three-fourths

of a cycle which would basically stop here at the bottom

so if we divide it into three six divided by three is two so the quarter cycle would be at two

seconds the half cycle would be at four and so a full cycle would be 8 seconds

therefore the period is 8 seconds and now we can calculate the frequency so the frequency is 1 over the period so

that's 1 over 8 seconds and 1 over 8 will give us a frequency of

0.125 hertz and so that's how you can calculate the amplitude the frequency the wavelength

and the period of a mechanical wave represented by a graph

number four a mechanical wave moves in a string with a speed of 125 meters per second

if the frequency of the wave is 250 hertz what is the wavelength so how can we find the answer for this

problem the speed of a wave is equal to the wavelength times the frequency

and so the wave speed is 125 meters per second we're looking for the wavelength and the

frequency is 250 hertz so to calculate the wavelength it's simply the wave speed

divided by the frequency so 125 divided by 250 is 0.5

so it's 0.5 meters number five a two meter long string with a mass of

0.10 kilograms is plucked with a tension force of 500 newtons

what is the linear density of the string the linear density is simply the mass per unit length

the mass of the string is 0.10 kilograms and the string is 2 meters long so 0.10 divided by 2

will give us a linear density of kilograms per meter now let's draw a picture so let's say if

we have two boundaries and a string is in between them if we apply a force

or tension force on the string we can create a wave pattern on a string and so you might get a wave

and what we need to do in part b is calculate how fast that wave is moving

if we pluck it with a given tension force the velocity of the wave

is equal to the square root of the tension force divided by the linear density

the mass per unit length so in this example the tension force is 500 newtons

and the linear density is 0.05 so 500 divided by 0.05 that's 10 000. if we take the square root of that that

will give us a speed of 100 meters per second so that's the answer to part b

part c determine the frequency of the wave if it has a wavelength of 0.25 meters

we know that the wave speed is the wavelength times the frequency so the frequency has to be the wave

speed divided by the wavelength so it's going to be 100 meters per second divided by a wavelength of 0.25 meters

and so that's going to give us a frequency of 400 hertz

and that's the answer to part c number six a standard wave is moving at 130 meters

per second on a 2.5 meter long string what is the fundamental frequency of this wave feel free to try if you want

to to calculate the fundamental frequency you can use this equation

it's nv over 2l now n is one when dealing with the

fundamental frequency so it's going to be 1 times the wave speed which is 130 meters per second

and the length of the string is 2.5 meters so 2 times 2.5 is 5

so it's 130 over 5. so the fundamental frequency is 26 hertz

so that's the answer to part a now part b what is the frequency of the third

harmonic that's going to be f3 the frequency of the third harmonic is

simply three times the fundamental frequency the fundamental frequency being f1 the second harmonic is f2 the

third harmonic f3 fourth harmonic f4 and so forth so the third harmonic is simply three

times the fundamental frequency or three times 26 which is 78 hertz

now let's move on to part c how many nodes and antinodes are present in the third

in the wave of the third harmonic so the first harmonic looks like this so that's f1

the second harmonic looks like that that's f2 the third harmonic

let me draw that better that's the third harmonic so

these are the nodes the third harmonic has four nodes the antinodes

are where you have the maximum displacement or that's where constructive

interference occurs so there's three anti-nodes the nodes occur with destructive

interference that's when the amplitude is zero so the number of nodes

is equal to n plus one and the number of antinodes is equal to n

so for the third harmonic n is three notice that the antinodes in n are the same

and the number of nodes is four it's n plus one so if we're dealing with the seventh harmonic

the number of nodes is going to be seven plus one it's gonna be eight the number of antinodes which

i'm going to know a n that's gonna be seven now let's move on to part d

what is the frequency of the fifth overtone you need to know that the first overtone

i'm just going to put ot for overtone the first overtone corresponds to the second harmonic

the second overtone corresponds to the third harmonic so the fifth overtone

corresponds to the sixth harmonic the frequency of the six harmonic is going to be six times the fundamental

frequency or six times 26 hertz and so it's going to be 156 hertz so

that's the frequency of the fifth overtone which is the sixth harmonic

now what is the wavelength of the third overtone so the third overtone

represents the fourth harmonic so the formula that we can use to calculate

the wavelength of the fourth harmonic is this equation it's going to be 2l over n

so it's twice the length of the string two times two point five divided by four

so it's five over four so the wavelength of the fourth harmonic or the third overtone

is 1.25 meters now there's another way in which we can calculate it

so we could find the frequency of the four harmonic which is going to be four times 26

and that's 104 hertz and then we could use this equation the speed is the wavelength times the

frequency so the wavelength that is the the wavelength for the fourth harmonic

is going to be the speed divided by the frequency of the four harmonic

so the speed is 130 and the frequency of the fourth harmonic is 104 hertz

so if you take 130 and divided by 104 it will give you the same answer of 1.25 meters

and so that's it for this video you