Intro to wave superposition
Summary
TLDRThis video explores wave superposition, also known as interference, where overlapping waves create combined effects. It explains how waves add algebraically when they meet, resulting in constructive or destructive interference based on their phase relationship. The video uses interactive simulations to demonstrate these concepts, showing how different waves interact and the mathematical relationships governing their interference patterns.
Takeaways
- 🌊 The concept of wave superposition refers to the overlapping of two waves, which can also be called interference.
- 📐 When two waves meet, their effects add up algebraically, resulting in the combined height of the waves.
- 🔄 If two pulses are identical, their heights add up when they meet; if they differ, their combined height is the sum of their individual heights.
- 🔄 Negative pulses can result in a net effect of zero when they align perfectly, demonstrating destructive interference.
- ⏱ The timing of the wave's movement is crucial as it determines when interference occurs.
- 🌐 Different shaped pulses can interfere, resulting in a pattern that is the sum of the overlapping patterns at any given moment.
- 🌉 Standing waves are a special case of interference where waves travel in opposite directions and create a stationary pattern.
- 🔗 The resultant wave from two interfering waves is the algebraic sum of the two individual waves.
- 📉 The phase difference between two waves is critical in determining whether the interference is constructive or destructive.
- 🔄 Constructive interference occurs when the phase difference is a multiple of 2π, while destructive interference happens when the phase difference is an odd multiple of π.
- 📚 The relationship between phase difference and path difference is given by the equation Δφ = 2π/λ * Δr, where Δφ is the phase difference, Δr is the path difference, and λ is the wavelength.
Q & A
What is meant by 'superposition' in the context of waves?
-Superposition refers to the overlapping of two or more waves in position and time. When this happens, the resulting wave is the algebraic addition of the individual effects of the waves.
What are the two types of interference that occur during superposition?
-The two types of interference are constructive interference, where the waves combine to create a larger amplitude, and destructive interference, where the waves cancel each other out.
What happens when two identical waves meet during constructive interference?
-When two identical waves meet during constructive interference, their amplitudes add together, resulting in a wave with a combined height equal to the sum of their individual heights.
How does destructive interference occur between two waves?
-Destructive interference occurs when two waves meet, and one wave has a positive amplitude while the other has an equal negative amplitude. This causes their effects to cancel each other out, resulting in a net amplitude of zero at that instant.
What is a standing wave, and how is it formed?
-A standing wave is a type of wave that appears to be stationary, formed by the interference of two waves traveling in opposite directions. In a standing wave, certain points (nodes) remain still while others (antinodes) oscillate with maximum amplitude.
What is the significance of phase difference in wave superposition?
-Phase difference refers to the offset between the phases of two waves. It plays a crucial role in determining the type of interference. A phase difference of zero results in constructive interference, while a phase difference of 180° (or π radians) results in destructive interference.
What is the general expression for the superposition of two waves with a phase difference?
-The general expression for the superposition of two waves with a phase difference is given by the equation: 2A * cos(ϕ/2) * sin(Kx - ωt + ϕ/2), where ϕ is the phase difference, K is the wave number, and ω is the angular frequency.
How does the phase difference affect the resultant wave in a simulation of superposition?
-In the simulation, when two waves have a phase difference of zero, they interfere constructively, doubling the amplitude. When they have a phase difference of 180°, they interfere destructively, canceling each other out.
What are the conditions for constructive interference in terms of phase and path difference?
-Constructive interference occurs when the phase difference between two waves is zero or when the path difference is an integer multiple of the wavelength (mλ, where m is an integer).
What are the conditions for destructive interference in terms of phase and path difference?
-Destructive interference occurs when the phase difference is 180° (or π radians) or when the path difference is an odd multiple of half the wavelength ((m + 1/2)λ).
Outlines
🌊 Wave Superposition and Interference Basics
This paragraph introduces the concept of wave superposition, which is the overlapping of two or more waves resulting in a new wave pattern. The term 'superposition' is used when waves overlap in position and time, and this phenomenon is also known as interference. The video demonstrates what happens when two pulses meet and overlap, resulting in the algebraic addition of their individual heights. The concept of constructive and destructive interference is introduced, where constructive interference occurs when the combined height of overlapping waves is the sum of their individual heights, and destructive interference occurs when the waves cancel each other out, resulting in a net effect of zero. The video also touches on the idea that energy is not destroyed in destructive interference, but rather the effects cancel out. The simulation of wave interference is suggested as a way to explore these concepts further.
🌊 Exploring Wave Superposition with Phase Differences
The second paragraph delves deeper into wave superposition, focusing on the role of phase differences between waves. It explains that when two waves with the same amplitude and wavelength have a phase difference, their superposition results in a new wave pattern that is the algebraic sum of the two original waves. The video uses a mathematical identity to simplify the expression for the resulting wave when there is a phase difference. The identity used is 2 * cos((a - b)/2) * sin((a + b)/2), where 'a' and 'b' represent the phase angles of the two waves. The video also discusses how to use an interactive physics simulation to explore wave interference and how changing the phase difference affects the resulting wave pattern. The concept of standing waves is briefly mentioned as a special case of wave interference.
🌊 Constructive and Destructive Interference Explained
In this paragraph, the video script explains the conditions for constructive and destructive interference in more detail. Constructive interference occurs when the phase difference between two waves is an integer multiple of 2π (or 360 degrees), resulting in the peaks and valleys of the waves aligning perfectly, thus doubling the amplitude. Conversely, destructive interference happens when the phase difference is an odd multiple of π (or 180 degrees), leading to the peaks of one wave aligning with the valleys of the other, effectively canceling each other out. The video uses a simulation to visually demonstrate these concepts, showing how changing the initial conditions of the waves can result in either constructive or destructive interference. The relationship between phase difference and path difference is also discussed, highlighting the importance of understanding these concepts for studying wave behavior.
🌊 General Conditions for Constructive and Destructive Interference
The final paragraph summarizes the general conditions for constructive and destructive interference. It reiterates that constructive interference occurs when the path difference between two waves is an integer multiple of the wavelength (Mλ, where M is an integer), and destructive interference occurs when the path difference is an odd multiple of half a wavelength ((m + 1/2)λ). The video script emphasizes the importance of these equations for understanding wave interference and suggests that they will be practiced in class. The paragraph concludes by reinforcing the significance of these concepts and their application in solving wave-related problems.
Mindmap
Keywords
💡Superposition
💡Interference
💡Destructive Interference
💡Constructive Interference
💡Phase Difference
💡Wavelength
💡Standing Wave
💡Path Difference
💡Wave Speed
💡Trigonometric Identity
Highlights
Introduction to wave superposition and interference.
Definition of superposition as overlapping waves.
Explanation of constructive and destructive interference.
Demonstration of wave interaction through a simulation.
How waves of different heights combine during superposition.
Impact of negative waves on superposition and interference.
Importance of time in wave motion and interference.
Different shapes of waves and their interference patterns.
Interactive physics simulations for wave interference.
Standing wave formation and its characteristics.
General formula for wave superposition involving phase differences.
Use of trigonometric identity in wave superposition calculations.
Derivation of the resultant wave equation from phase differences.
Visual representation of constructive interference in simulations.
Visual representation of destructive interference in simulations.
Explanation of phase difference and its impact on wave interference.
Conversion between phase difference and path difference using wave number.
Conditions for constructive interference in terms of phase and path differences.
Conditions for destructive interference and their mathematical representation.
Transcripts
today in this video we're going to start
chapter 17 oops chapter
17 that talks about
superposition of
waves now the word superposition is just
fancy
for for uh in our case for
overlapping what if we overlap two waves
let me see is it this one no
there you go super position it's also
called interference when two waves or
two pulses overlap in position and time
we call it we call that superp position
or interference and let's run this thing
you see we have one pulse traveling to
the right one pulse traveling to the
left and we'll see what happens when
they meet when they meet well they
overlap and the effect is well just the
algebraic addition of their individual
effects right that's all that's superp
position if I let them run we'll see
them
interfere I can we don't we don't need
to show the individual this is just the
the Rope the string what it feels what
it does is well the it follows the
addition of the individual contributions
now
now in this case two pulses were
different let's if we make the two
pulses the same like three and
three in height look when they
meet their height combined height is six
right if one of them is one and the
other one is three then the combined
height is four that's all now what if
one of the pulses is
negative well let's in this case three
and minus three well 3 + -3 is zero and
sure enough for an instant when the two
pulses align perfectly the net effect is
zero now the energy is still there we
you didn't destroy the energy this is
still there all the work
right so remember this has to do with
time as well the wave moves has a speed
travel so at one point they will they
will interfere and nothing nothing will
happen nothing will appear to to be
shown right this this type of
interference when everything cancels we
call it destructive interference in the
other in the other section in the other
hand this is constructive
interference of course the interference
can happen uh between pulses of any shap
shape like this one for instance this is
like an M the other one is like a ramp
and when they
meet you always get the
resultant
of the patterns that are that are
overlapping at that instant of time yeah
this is this is um this is a nice
simulation you can play with this with
this
um with this
patterns if you want let me see if I can
move this yeah here it's called o
physics interactive physics simulations
go to waves go to wave pulse
interference super position number two
and then you can have some fun with
this you can also make not only pulses
but waves interfere and if and if we do
this let's play this you see well in
this case we have one wave traveling to
the right one wave traveling to the left
this is a peculiar um type of wave that
we will study in this chapter it's
called a standing wave but is not the
only possibility let's let's go in
opposite directions no they're in the
same direction but there's the same
wave where is the other
one
yeah let me see one is like
this um show red
I don't know where the red one is oh
there you go same direction so I'm
sending the red one and the blue one and
I get the purple one so you play with
different wavelengths and then you get
different results of
course
um you
see what you get is the net effect of
adding the two
waves right like this purple is the red
and the blue that's all in this case
they're traveling in the same direction
if I make them travel in opposite
directions well this is what you're
going to get at at all times what you
get it's at this point is is the red
plus the blue the red plus the blue the
red plus the blue is algebraic so it
could be positive or could be
negative all
right now I'm I'm interested in this
type of interference this is the first
special case of interference that we
will um we will um study okay in general
interference you have the red function f
of XT and then you have the green
function G of XT so the blue function is
just the red plus the green that's it
you add one wave plus the other one you
get the blue wave that that simple let's
do
it so
let's say we have function
one which is one wave just a normal
traveling
wave okay and then we have the other one
which is that is going to be the same
wave meaning same amplitude same
wavelength same X same frequency same
Omega but to this one I'm going to add a
different face so the only difference
between these two is that the face is
not this same okay this is the first
scenario of superposition right if the
face are not the same so the total wave
is just going to be one wave plus the
other so a
sin KX - Omega t + a sin KX - Omega t +
5 and that's it that's a superimposed
wave now I'm going to use a little
identity
s of a + S of B is 2 cosine of a minus B
/
2 s of a + b / 2 okay I'm going to use
that identity in which this is a and
this is
B and that's going to give
us so I'm going to I'm going to factor
the a of course then is going to be sign
so this all this is a so oh
actually actually so s of a plus s of b
i factored the a so we're going to have
2 cosine a minus B so
KX - Omega t - KX - - plus Omega t - 5
and all this over two and then
s of a + b KX - Omega t+ KX - Omega t +
5 and all that's over two and then what
happens here is the KX cancel the KX
Omega T cancels Omega T and then what we
get is a 2 cosine of - 5 / 2 and on the
other side we get S of KX KX that's
2K 2 KX - Omega t - Omega t - 2 Omega t
+ 5 and all of this over
two right that's so what we get is 2 a I
want to put the two outside
cosine - 5/ two and then
sign the two the two cancel this two and
this two cancel and this becomes so KX -
Omega t + 5 / 2 okay and this is the
expression we were looking
for oops that's the total that's the
wave right that's the wave you get when
the only difference is the phase
difference meaning phase difference in
this case we call it five and that's
what the book calls it I will call it
Delta
F but if you read the book it's going to
say just fine and that's the reason is
because they are assuming that the phase
initially is zero okay but and it's true
one of the fa here the phase of this one
was zero so phase or Delta phase means
phase
difference all
right and uh it's just it just I will I
will explain this more in class what the
phase difference means in terms of let's
say how two waves are generated but when
you have this is a very general
expression and and um the way it looks
like is in the simulation here if I play
it look I'm sending the red the Red Wave
and I'm sending the Green Wave at this
point they are in face you see Peak with
Peak valid with valid Peak with Peak
valid with valid so when they add at all
times they are in constructive
interference because what the peak Peak
with Peak get you get twice the peak
valy with Valley you get twice the
valley
right that the equation tells us that
here when the face difference is zero
cosine of 0 is 1 and you get twice the
original wave you see how this is the
original wave the phase difference is
zero you get just twice the original
wave here you see it's the same thing
and that's what we are seeing when the
when the when the face is is zero you
get twice the wave twice as loud or
twice as intense but if you have a
difference and I'm going to put Express
the difference with
this okay yeah you see how um look at
the r red and green what I'm saying is
at the
beginning the the red and the green let
me let me do this let me do this better
okay there you go here look where the
green where is the initial conditions of
the green
and the initial conditions of the red
here they're the same so the face of the
red at the beginning which is Rel it's
expressed by this gray block the face of
the Red Wave and the face of the Green
Wave are exactly the same so you have a
pH of zero if I change the
face you see now the faces are different
this red red one is here the green one
is somewhere
there so the the total wave at all times
is almost non-existent because you are
almost always in a in a position of
destructive interference look Peak with
Valley Valley with Peak Peak with Valley
so they destroy each other and at this
point you are completely out of phase
the phase difference here is 180° or
actually pi and if we look at the
equation what would happen if you put 5
equal to to 180 well 180 over 2 is 90
cosine of 90 is zero you don't get a
wave right you get destructive
interference now let's take a look when
you have a
wave that's ugly let's do it again when
you have a
wave right um
here with
this I'll grabe this one when they are
in Phase well they look like this they
have zero degrees or zero radians in
facee difference so they are in
constructive interference when you're
completely out of phase then you have
Peak with Valley so the the phase
difference Delta fi when the Delta when
the Delta F when the phase difference is
equal to 2 pi 360 de face
difference well the the the distance or
the the the difference in
path between this point and this point
which is called Delta R it's equal to
Lambda that's the path difference that
means one wavelength is ahead of the
other one by one wavelength or the one
wavelength is behind the other one by
one wavelength and you can divide these
two equations and what you get is that
the path difference the phase difference
over the path difference is 2 pi/ Lambda
and if you remember that's the wave
number so this is a nice equation to
have in your equation sheet because
allows you to convert from phase
difference to path difference and we're
going to practice this also in class
lastly let's consider these two waves
here we know they are completely in
Phase so constructive interference here
destructive interference let's consider
when they are in constructive
interference when they are in
constructive interference what we can
say
is conditions for
constructive well what about we need the
phase difference to be zero right or the
path difference to be zero MERS when
there is no path difference or not or no
phase difference we know we have
constructive interference Peak with Peak
value with value but what happens if one
wave is ahead head of the other one by a
full wavelength like this one you see
now what you get is you get this peak
interfering with this peak and this peak
interfering with with the with the next
one so in this case the path difference
the phas difference
is my God the phas difference is 2
pi and the path difference is one
wavelength and you still get
constructive interference
what if you are two wavelengths
ahead I want to put them down here what
if you are two wavelengths
ahead right this peak instead of
interfering with this peak is
interfering with that one so you're two
wavelengths ahead so now you're two
wavelengths ahead whatever 4 PI right
and then if you were three lambdas 6 Pi
or four lambdas you will always get
constructive interference in actually
the main the in general if the path
difference is M Lambda where M can be
any integer 0 1 2 Etc you get
constructive
interference for destructive
interference let's take a look here is
constructive but here is the structive
you see Peak with Valley valy with Peak
so the conditions for the structive
interference is
R you get
destructive if the
phas or the
path well when the phase is
pi right or the path is Lambda over 2
you get destructive interference but the
same will happen with three Lambda over
2 right there will be one wavelength and
a half ahead like this and you will get
the structure interference the same will
happen with 5 Lambda / 2 7 Lambda / 2 so
in general you get destructive
interference when the path difference is
m + 12 of Lambda again m is 0 1 2 3 Etc
so these two equations are very
important
um they are General General scenarios of
the equation we first derived with the
cosine 5 / 2 and these two equations are
important we're going to do problems
involving these two in class
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