This course introduces acoustics by using the concept of impedance. The course starts with vibrations and waves, demonstrating how vibration can be envisaged as a kind of wave, mathematically and physically. They are realized by one-dimensional examples, which provide mathematically simplest but clear enough physical insights. Then the part 1 ends with explaining waves on a flat surface of discontinuity, demonstrating how propagation characteristics of waves change in space where there is a distributed impedance mismatch.
Lecture 5-1 Part 3. Snell's Law 1
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來自 Korea Advanced Institute of Science and Technology(KAIST) 的課程
Intro to Acoustics (Part 1)
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從本節課中
Waves on a Flat Surface of Discontinuity 2
You will going to learn more about the waves on a discontinuity. With the Snell's Law, you will be able to find out the wave propagation after transmission and reflection on discontinuity.
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Yang-Hann KimProfessor
Mechanical Engineering
Okay, now let's move to another case.
The question is what is going to happen
if you have wave instant of [INAUDIBLE]?
To surface of this continuity.
What if there is wave of likely instant to for example,
partition, general partition.
And what if the wave instant to, for example, plate.
That is our next topic of discussion.
Okay, to see the fundamental physics associated
with oblique instant, let's see the following graph.
Okay, the first question is is instant to
and the r is reflected angle and
the t is transmitted angle.
What is the relation between i and R intersected t.
That is governed by very well known law and
I'm sure some of my students know this.
Surprisingly when you are in elementary school, that is called Snell's Law.
Okay?
Maybe some of the students feel very guilty that you do not remember
Snell's Law, but it doesn't really matter, okay?
Let's see some detail physics associated with the Snell's flow.
Okay instant wave this
flat surface of discontinuity with this angle, okay.
Then it propagated this way, therefore this is the wavelength
of instant wave I mean suppose it propagate this way
like that, and this is the point B and this is point A.
Okay this is propagate, [SOUND] at certain time,
the incident wave approach, I mean reach over there, A.
Right?
And then it travels.
What the distance?
This distance.
Okay.
And the reflected wave reflected and propagated that direction,
therefore this is the wavelengths due to the reflection.
And those are transmitted waves therefore
this is the wavelengths of transmitted wave.
And projected distance upon this discontinuity has to be same, right?
Projected distance of zeta t over there, and
projected distance of this wavelength over there, and
projected distance of this length over there has to be same, right?
Geometric curve identity.
So let's express
that one in terms of,
in terms of the geometry.
Okay, delta y, delta y, delta y has to be same.
Lambda i/sine theta or
back to here, I can say delta y, and this is the angle,
so what is the relation between lambda 2 and lambda t and delta y?
Lambda t, this is lambda t.
And this is a theta T.
So therefore
this is a theta T, so therefore delta Y,
psi theta is lambda T, right?
So, this is true.
So, I can say delta y is the lambda
i over sine theta i, and this is true and this is true.
And this is the Snell's Law.
And of course this is true too.
Move shows this and this is not interesting,
so let's se what it means, by that
physically okay.
So let's look at this one and that one.
This is interesting, because depending on c0 and
c1, you can have interesting phenomena.
Let's see over there.
If I convert this in wave number form,
and wave number is omega/C0.
So therefore this term is k0 sine
omega 0, and this term is k1 sine omega.
Okay, I sine omega I, sine theta I.
So this is simply the projected wave number on the surface.
Right?
So if K0 is very,
is larger than KI, okay.
This is the k0 and I draw the picture over there.
At a certain angle this is k0 sine 0.
So that is the k0.
And sign set to zero is this lens, as that lens
has to be equal to K1 sign set at zero.
So in this case, the K1 sign set to zero over there
has to be the same as K set to 1 has to 90 degrees.
So if the angle of incidence
is larger then this critical angle then this has to go over there.
And it is not possible for k1 sin theta 1 can be this
so geometrically there is a very interesting phenomena happen.
So in this case, how mathematically can we achieve this case?
This the case.
Going over there, this is the case we should
have imaginary angle that has to be greater than pi over 2, okay.
This is the case, because I mean K K zero,
sine zeta zero is larger than this.
Therefore, the angle has to be greater than pi over two.
And that angle has to be something that represent the physics that we do not know.
And this angle, we call critical angle.
Okay.
And this happened physically for a case when we have a medium.
K is omega over c.
So when c is small that means I am in error.
When c is large, that means there is water.
Now if some instant angle is greater than
critical angle, then there is no transmission.
To the water.
Actually what happened is the surface where there's a surface wave is generated.
So if there is a sound wave coming from water, in this case, K zero.
Is larger than k1 in this case, in any case her is a transmission.
Let me continue to this rather strange concept in the next electro,
and let me summarize what we've learnt today.
We've reviewed the Mass Law and we
studied this Mass Law is quite relevant with the Mass Law
we observe in single system in high frequency.
So we were sufficiently persuade
that the mass law is valued only for
the high frequency reason, and
the essential elements I attempted,
I have attempted to.
And to you into this lecture is transmission
is composed by two part one is fluid
and the other one is.
The fundamental physical basis to get that
formulation is observing that PI plus PR on
the left hand side of partition is equal to
PT minus J omega Y multiply by Z zero,
that means that the transmission is have transmission
can be considered as two times the instant pressure to that
is blocked pressure plus the radiation of partition.
And the radiation of partition introduced fluid loading impedance.
That is
written over
there.
That's the central part that I attempted to deliver to you guys.
And then we show that
everything we learned today can be seen as this picture.
This picture shows an absolute region,
and stiffness controlled region, and damping controlled region.
And then we introduce the Interesting physical behavior
happen we have oblique instant aware.
And next lecture where we talk a little bit more about
the physics associated with oblique instant and
I would like to persuade you the instant angle has to be same as,
I mean reflected angle has to be same as the eastern angle.
Of course transmitted angle be
different by seeing those snares low
in a wave number low Which is not, I mean, very well known.
So, I personally think that looking
in wave number provide more physical insight.
Okay.
