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From YouTube: EOSC 350 Lecture 10: Seismic 2. Doug Oldenburg
Description
Second lecture on seismic. Refraction, waves & rays, reflection coefficients. Slides are available at https://github.com/ubcgif/eosc350website/raw/master/assets/3_Seismology/Seismology.pdf. The app demonstrated is available at https://www.3ptscience.com/app/SeismicRefraction
A
A
On
Friday,
just
to
kind
of
refresh
memories
about
what's
going
on
the
gun,
so
here
we
are
we're
at
the
top
of
an
earth
that
we
now
are
thinking
about
as
being
an
elastic
earth
and
the
whole
concept
of
having
particles
inside
the
earth
that
are
connected
by
Springs.
That's,
like
that's
a
good
thought
process
to
have
so
we
have
some
kind
of
energy
source
puts
energy
into
the
ground.
That
comes
back
up
here.
It's
recorded,
that's
a
control
source.
It
could
also
be
an
earthquake.
Here's
the
typical
wave
recording
from
an
earthquake.
A
The
first
thing
that
you
see
is
this
P
way-hey
and
that's
a
compression
away
where
the
particles
are
moving
back
and
forth
and
the
direction
of
propagation,
and
we
did
that
experiment
up
top
and
then
this
is
the
s
weight.
A
velocities
are
smaller
than
P
wave
velocities.
So
that's
coming
in
here
and
there's
some
other
stuff
in
here
that
we're
not
really
sure
what
it
is.
We'll
probably
talk
a
little
bit
about
that
and
then
there's
these
big
high-altitude.
A
A
A
A
We
didn't
talk
about
it.
I
make
a
mention
of
it.
Now
waves
decrease
in
amplitude
as
they
they
travel.
One
is
because
of
something
called
geometrical
spreading.
So
all
the
energies
that
goes
through
this
circle
must
also
go
through
a
larger
circle,
which
means,
if
energy
is
some
certainty,
amplitude
has
to
increase.
A
The
other
is
that
just
we,
as
we
saw
when
everybody
was
standing
up,
remember
like
the
first
person
had
a
big
amplitude
by
the
time
the
energy
got
to
the
back
part.
The
amplitude
was
actually
really
small.
That
must
mean
that
there's
some
attenuation
that
that's
going
on
in
there
and
so
energy
was
being
bossed
we're
dead.
If
anybody
that
was
putting
up
the
frat
Stephanie
was
moving
back
and
forth,
like
that,
a
lot
of
kinetic
energy
later
on
other.
A
Forms
of
energy,
yeah
and
basically
converted
the
heat,
so
friction
loss
sound.
It's
probably
not
so
much
in
that,
not
in
this
case
but
yeah.
So
then
we
had
the
four
different
waves,
the
body
waves,
as
well
as
the
that
the
surface
waves
and
all
had
different
names
and
different
particle
motions.
Then
we
saw
this
diagram.
This
was
a
movie,
that's
play
the
so
the
energy
is
going
off
of
here,
and
these
are
the
wave
fronts
that
are
propagating
through.
A
We
can
see
that
there's
different
things
happening
and
here
versus
here,
so
there's
energy,
that's
being
transmitted
into
here,
there's
actually
energy,
that's
being
reflected
from
the
bottom,
and
there
will
be
energy
and
maybe
see
that
too,
even
a
bit
better.
You
can
see
this
little
blue
line.
Yeah
I
can
see
it
better
in
this
one
than
I
could.
A
A
So
here,
what's
that
app,
how
many
people
actually
downloaded
it
and
you
need
two
people:
whoops
ooh,
better,
nothing!
Okay,
so
you
have
here's,
here's
our
app
and
it's
got
a
layer,
one
layer
up
top
another
layer
and
layer
underneath
so
there's
three
layers
and
each
of
those
layers
has
got
a
particular
velocity.
So
there's
a
velocity
one,
which
is
400
meters,
lots
to
2,000
velocity
three,
a
width
to
1500,
so
velocities
are
continuing
to
to
increase
the
simplest
arrival
that
we're
going
to
get
so
suppose,
you're
sitting
at
some
place.
A
So
it's
this
guy
here
travels
along
here
right
along
the
surface
on
the
plot
beneath
we
have
what's
called
the
travel
time
curve.
So
this
is
the
same
distance
along
this
direction,
and
this
is
the
time
down
here
and
at
a
particular
location
which
was
67
meters.
The
arrival
time
which
is
given
by
this
point
here
is
about
point
1,
6
seconds.
A
So
these
spots
are
kind
of
companion
plots
and
we'll
be
using
them
a
lot.
The
next
ray
is
going
to
be
the
reflection,
so
this
is
energy
that
leads
here
goes
down
to
this
interface
and
bounces
back,
and
we
have
that
the
angle
of
incidence
is
equal
to
the
angular
deflection.
So
this
reflection
point
is
halfway
in
between
and
the
travel
time
curve
for.
That
is
something
looks
like
this,
and
we
talked
about
this
at
the
end
of
last
day.
A
So
we're
actually
going
to
think
about
a
way
that
comes
down
and
gets
reflected
or
meets
this
interface.
So
this
is
a
v1
and
this
is
v2,
and
this
way
this
is
going
to
come
down
and
now
it
meets
a
medium
with
different
properties
and
two
things
happen,
part
of
it
gets
reflected
and
part
of
it
gets
transmitted,
but
actually
it's
not
just
the
velocity.
A
That's
important
here
is
something
called
the
acoustic
impedance,
so
we're
going
to
represent
that
by
the
letter,
Z
or
Z
any
regular
from
and
that's
the
product
of,
the
density
and
the
boss.
That's
that's
the
acoustic
impedance.
So
as
energy
comes
down
here,
the
amount
that
gets
reflected
and
the
amount
that
gets
transmitted
that
depends
upon
that
change
of
the
acoustic
impedance
and
it's
given
by
this
formula
here.
The
reflection
coefficient
is
the
impedance
of
the
second
layer
depends
the
first
divided
by
the
sum
of
them.
A
So
the
reflection
coefficient
and
the
wave
that
goes
down
is
got
an
amplitude
T,
so
I've
got
a
reflection
and
I've
got
transmission
and
they're
relating
to
the
amplitudes
of
the
waves
that
just
get
reflected,
or
that's
me
now.
The
interesting
thing
about
a
reflection
coefficient
is
that
if
I
looked
at
this
formula
here
so
that
our
Z
2
minus
Z
1
over
Z,
2
plus
Z
1,
okay,
let's
suppose
that
I
had
a
medium
here
that
you
know
the
velocity
is
equal
to.
You
know.
A
A
In
which
case,
if
I
was
going
to
evaluate
the
reflection
coefficient,
what's
going
to
be
an
essential
number
here,
it's
going
to
be
close
to
one.
So
it's
going
to
be
in
you
know
like
10
to
the
6
minus
10
and
6
plus
10
squared,
which
is
approximately
1.
So
if
I
go
from
a
medium
with
low
impedance
to
high
impedance
I
get
a
reflection
coefficient
for
something
possible
and
the
bigger
that
difference
is
the
larger
the
amplitude.
What
happens?
I
go
the
other
way
suppose
that
the
medium
below
a
velocity
of
0.
A
A
A
Technology
is
feeding
me,
I
used
to
be
able
just
click
on
my
powerpoint,
and
it
would
come
up
and
here's
here's
a
point
here.
So
I've
got
a
way.
That's
coming
you,
okay
and
there's
actually
a
negative
reflection
coefficient,
so
the
wave
that
comes
back
is
coming
back
with
the
opposite,
polarity
and
wait.
You
can
see
also
that
there's
still
a
way
that
is
so
that's
the
whole
concept.
A
I've
got
energy,
that's
coming
in
hit
a
boundary,
some
of
it
comes
back
and
some
of
it
gets
transmitted
and
the
stuff
that
comes
back
is
going
to
depend
upon
whether
I'm
going
from
a
bigger
impedance
from
a
ball
feeds
to
a
lie
or
a
lie
to
a
bow.
So
the
reflection
coefficient
is
under
greater
or
less
than
one.
A
That
was
the
reflection
coefficient.
There's
also
a
transmission
coefficient,
that's
given
by
this
formula
here.
So
it
just
depends.
It's
just
to
said
whatever
the
impedance
of
the
first
layer
was
over
that.
So
that
tells
us
now
what
the
amplitude
of
that
that
transmitted
wave
was
okay,
so
that
gives
us
two
waves.
We've
got
a
direct
wave
and
we've
got
a
reflection.
We
know
something
about
that
reflections.
Some
of
the
energy
is
going
to
go
in
and
some
of
its
going
to
to
come
back,
and
we
also
know
about
the
law
of
reflection.
A
The
energy
in
a
weighted,
if
you're,
if
you're,
sitting
someplace
there's
some
kind
of
a
source
and
you're
receiving
some
energy,
then
you
can
think
about
okay,
how
that
energy
got
to
you
by
thinking
about
okay,
what
is
sort
of
the
minimum
anarchy
requirement
can
get
to
you
or
what
is
the
minimum?
Let's
say
in
this
case:
we're
talking
about
travel
time
path
to
get
you.
A
How
does
it
get
to
you
the
fastest
and
if
you
can
figure
out
what
the
trajectory
was,
the
docq
and
fastest
way
and
I
could
tell
you
how
that
energy
came
about
so
there's
an
interesting
thing,
so
we
saw
something
we
call
the
refraction
arrivals
and
there's
kind
of
interesting
analogy
to
that.
That
happens.
A
A
So
now
you
and
your
friend
are
out
swimming
dick
and
you're
a
certain
length
you're,
a
certain
distance
from
the
shore,
so
here's
person,
1
and
person
2,
is
setting
up
over
here,
get
to
your
boat
up
swimming
he's
over
there
you're
over
here
and
then
he's
in
distress
or
she
okay.
So
let
us
suppose
that
you
are
purchasing
to
and
you
want
to
try
to
save
person
1.
What's
your
captain.
A
Gonna
get
as
possible
good,
ok!
Now.
What
would
your
options
me?
You
got
to
go
across
the
water,
we'll
go
up
to
the
sand,
which
would
probably
be
quicker
to
move
across,
identify
and
perfect.
Ok.
So
if
you
went
this
way,
don't
be
like
a
direct
arrival
right
here,
you're
just
going
through
there,
it's
the
shortest
distance,
but
then
what
you're
really
interested
in
is
your
travel
time,
which
is
your
distance
divided
by
whatever
speed
you've
got
and
if
you're
swimming
your
speed
might
actually
be
fairly
small.
So
this
time
gets
to
be
very
block.
A
Let's
go
up
here
and
now
you
can
just
haul
it
across
here
right
because
you're,
you
know
here
you
have
a
speed
v1,
but
here
you
have
a
speed
v2
and
you
can
you
can
truck
that
and
go
over
here,
someplace
and
then
down,
and
you
might
actually
do
that
right.
So
you
you've
got
travel
times
to
go
from
here
to
here
yeah
and
you
got
two
of
those
right.
So
whatever
that
distance
is,
let's
suppose
that's
L,
so
it's
2
L,
divided
by
D
right
and
then
you've
got
whatever
X
distance.
A
A
A
You
know
so
I
could
go
straight
that
minimizes
the
distance
that
I
got
in
in
here.
So
that's,
that's!
Okay
and
I
could
come
straight
back
here,
but
that
actually
makes
this
distance
here
a
little
bit
longer.
So
now
you
start
wondering
well
wait
a
minute.
Maybe
this
problem
is
more
interesting
than
I
thought
and
the
question
is
okay.
What,
if
I'm?
If
that's
not
optimum
and
I
I'm,
going
to
you,
know
sort
of
slip
across
here?
A
So
you
can
appreciate
that
might
actually
require
a
little
bit
of
geometrical
thinking.
Grades.
Actually
yeah
get
a
final
derivation,
but
that
is
the
case
right,
because
if
you,
if
you
just
swim
straight,
that's
not
going
to
be
the
shortest
time.
If
you
go
too
far
over
here,
then
that's
not
the
shortest
time,
so
there's
got
to
be
someplace
in
the
middle
and
there's
actually.
Some
kind
of
special
angle
called
the
critical
angle
that
if
you
put
yourself
in
that
direction,
you
hit
the
ocean
the
beach
run
across
and
get
to
them
the
fastest.
A
So
let
us
look
so,
let's
suppose
we're
at
yeah,
so
we're
at
this.
This
particular
point
and
let's
first
of
all
look
at
the
the
direct
arrival
and
then
let
us
look
at
an
arrival.
Then
that
actually
comes
down
here
and
it
travels
along
on
that
interface
and
then
comes
back
up
here.
So
that's
exactly
what
we
were
just
doing
before,
and
so
that
would
be
the
first
refraction.
A
By
using
this
guy
here,
so
what
is
this
so
then
on
this
side
here?
So
this
is
time
right.
So
now,
I
can
look
at
any
particular
time
here.
So
this
is
point
zero,
three
seconds
and
I.
Look
at
these
wait
pass
there.
You
see
they're,
partially
in
solid
and
they're
partially
in
dashed,
and
what
that
means
is
that
at
0.3
seconds,
okay,
I'm
sitting
up
here,
so
let's
even
go
earlier.
A
So
you
can
see
what
happens
when
you
start
off
the
direct
goes
this
way
while
it's
going
down,
it's
been
impacted.
Goes
this
way,
so
those
two
our
equal
lengths
here
at
point,
O
two
seconds:
you've
gone
the
same
distance
but
he's
kind
of
seems
like
he's
going
in
the
wrong
direction.
But
if
you
now
make
this
a
little
bit
more
so
at
point
O
three
seconds,
you
can
see
this
so
now
you
can
see.
A
A
So
that's
your
that's
sort
of
the
manifestation
of
what's
happening
and
why,
as
you
have
these
different
velocity
structures,
you
can
get
waves
that
travel
in
a
perhaps
a
non-intuitive
path
and
actually
arrive
back
where
there's
where
you'd
want
them
to
have
again
in
an
earlier
time.
The
the
travel
time
curve
looks
like
this,
so
this
is
the
arrival
times
for
the
direct
way,
and
this
is
the
arrival
times
for
that
refracted
wave
there's
a
couple
of
things.
First
of
all,
we
don't
get
a
refracted
wave,
so
we're
certain
distance
away.
A
So
that's
called
the
critical
distance
and
then
once
that
refracted
arrival
starts
to
come
and
he's
propagating
at
a
particular
velocities.
So
he's
got
a
slope
that
looks
like
this,
and
the
thing
that
we're
going
to
be
most
interested
in
is
what's
called
the
first
arrival.
So
here's
time
T
is
equal
to
equal
to
zero
and
as
we
go
to
these
various
stations
as
we
go
down
progressively
in
time,
we're
first
of
all.
Yet
the
first
way
to
ride
is
the
director
arrival
here.
A
But
if
we
go
farther
off
than
the
first
way
to
arrive
is
actually
back
to
the
rival.
So
what
we're
going
to
do
is
I'm
going
to
look
a
little
bit
more
at
this
and
then
we're
going
to
be
able
to
take
these
curves
here,
dissect
them
and
we're
going
to
do
two
things
they're
going
to
get
the
slope
of
this
one
to
give
us
the
velocity
of
the
upper
layer.
A
We
use
the
slope
of
this
one
to
give
us
the
velocity
of
the
lower
layer
and
we're
going
to
use
this
thing
here
called
the
intercept
time
to
get
the
layer
thickness.
So
out
of
this
basic
travel
time
analysis,
we
can
get
two
velocities
and
a
layer
thickness
and
that's
extremely
helpful
for
a
lot
of
problems,
especially
when
you're
looking
for
bedrock
or
something
like
that.
A
So
we've
done
that
doctor
friend,
oh,
the
one
thing
that
we
didn't
mentioned
explicitly
was
Snell's
law.
How
many
people
have
heard
of
Snell's
law?
Has
anybody
have
not
heard
of
Snell's
law?
And
what
context
did
you
have
Snell's
law
in
optics?
Yes,
so
usually
that's
where
it
is.
It
gives
you
a
refractive
index
of
something
and
then
the
light
comes
in
and
it
bends
away
and
the
relationship
between
the
angle
of
incidence
which
we'll
call
theta
1
and
the
angle
of
refraction,
which
we
call
theta
2,
is
given
by
this
relationship
here
that
sign.
A
So
if
we've
got
something
that's
coming
in
at
some
particular
angle,
theta
1-
and
it
leaves
at
some
particular
angle-
theta
2.
The
thing
that
we're
interested
in
here
is
like
okay,
what
is
this
angle
of
refraction?
So
we
can
just
manipulate
this
so
that
just
gives
us.
You
know
sine
theta
2
is
equal
to
v2
over
v1
times
sine
theta.
A
So
a
v2
over
v1
feature
is
graded
B
1,
then
this
number
is
greater
than
1.
So
that
means
that
sine
theta
2
is
going
to
green
side.
Theta
1
theta
2
3.
So
it's
going
to
so
he
2
is
greater
than
these
1.
This
angle
is
bigger
than
that
angle,
so
this
thing
gets
refracted
away
from
the
normal.
That's
how
we'll
refer
to
this
as
the
angle
of
refraction,
whether
it's
refracting
towards
the
the
normal,
which
is
this
way
or
away
from
the
normal.
A
A
A
So
if
theta
2
was
equal
to
90
degrees,
then
what's
happening
so
they
the
energy
comes
in
like
that
is
now
propagating
I
guess
so
the
energy
is
propagating
right
along
this,
this
interface
and
that
is
called
a
critical
refraction.
So
when
theta
2
gets
to
be
equal
to
90
degrees,
then
we've
got
energy.
That
comes
right
along
here
and
we
call
we
stay
off.
The
energy
has
been
undergone
a
critical
refraction.
A
The
other
name
for
that
is
set
feet.
So
this
energy
is
not
propagating
along
here,
okay
and
as
it
does
so
any
any
way.
That's
propagating
law
is
continuing
to
put
stuff
up.
So
imagine
it
sort
of
at
some
little
time.
Just
then
this
lady's
hit
here,
you
could
think
about
it's
kind
of
like
an
equivalent
source
that
you
know
the
weight.
A
Something
strikes
here
and
there
starts
to
be
a
wave
front
that
comes
off
here
and
then
at
a
time
later,
the
wave
it's
come
across
here-
and
it's
like
this
so
as
the
time
goes
on,
the
propagation
of
the
wave
has
reached
this
point.
But
the
energy
has
kind
of
gone
up
here,
so
you
get
a
wavefront
that
is
emanating
from
this
surface,
that's
blasting
right
up
to
the
light
right
up
to
the
overlying
surface.
A
So
you
can
see
this
is
a
straight
line
in
here
and
we'll
go
back
and
take
another
look
at
that
movie
and
see.
If
we
can,
we
can
actually
see
this
guy,
but
this
is
the
sometimes
it's
called
the
head
wave
coming
in
or
the
critically
refracted
wave
that
is
providing
energy
to
the
receivers
at
the
surface.
A
Does
this
help?
Did
you
see
this
guy?
So
this
was
the
snapshot
of
what
that
movie
was.
You
could
probably
see
it
better
on
your
screen.
So
when
you
do
this,
go
on
your
own
computer
screen
and
you
watch
those
wave
fronts
if
you
hit
it
just
at
the
right
point.
Okay,
you
see
that
the
the
way
that's
coming
in
here
is
not
I
mean
that
wavefront
is
is
way
over
here.
The
direct
way
that's
going
in
here
is
sort
of
sitting
up
here
and
there's
this
band
of
energy.
A
A
That's
right,
so
it's
going
to
come
down
and
it's
going
to
refract
down
this
way,
so
this
will
be
theta
2.
So
if
V,
if
V
2
is
less
than
V
1,
then
this
quantity
here
is
lasting.
So
that
means
that
theta
2
is
the
last
good
theta
1.
So
that
means
it's
Sara
this
refracting
down
this
way.
So
what
happened
to
the
refraction
weight
in
this
case.
A
A
A
Basically,
a
high
velocity
to
a
low-velocity
medium
energy
gets
refracted
down,
but
there
is
no
critical
way.
That's
that's
ever
generated,
and
so
it
it
now
just
looks
like
this.
So
the
movie
you
can
see,
there's
just
nothing.
I
can
happen.
You
still
there
reflection
to
get
a
direct
way.
You
get
a
reflection
and
you
get
energy
that
goes
down,
but
there's
no
evidence
for
what
that
layer
is.
That
actually
turns
out
to
be
really
important
point,
because
what
you're
trying
to
do
and
generally
this
is
this-
is
what
happens
in
certain
engineering
problems.
A
Well,
you
try
to
find
out
what
this
thickness
is
and
you're
going
to
do
that
on
the
basis
of
the
refraction,
so
that
can
only
occur
if
V
2
is
larger
than
me
now.
Fortunately,
that's
usually
the
case,
because
this
is
maybe
the
bedrock
which
is
but
a
lot
higher
velocities.
Then
you
know
unconsolidated
material
up
top,
and
so
it
usually
turns
out
that
V
2
is
greater
than
1
and
so
yeah
you're
kind
of
good
to
go.
But
if,
if
that's
not
the
case,
we're
going
to
be
hooked
with
this
refraction
analysis.
A
So
there's
a
few
visualizations
of
refracted
energy
house.
It
travels
through
the
earth,
I'm
not
going
to
show
them,
but
you
might
just
sort
of
play
around
with
them.
It's
kind
of
interesting,
as
you
put
in
different
types
of
velocity
structures,
just
how
the
waves
can
kind
of
move
around
and
current
and
for
low
velocity
zones,
and
things
like
that.
We're
going
to
come
a
little
bit
at
the
end
towards
low
velocity
zones,
but
you
know
firing
those
guys
up
play
around
with
them.
A
Okay,
so
what
I
want
to
do
is
kind
of
turn
our
attention
to
really
analyzing
what
we've,
what
we've
got,
because
now
we
want
to
turn
this
information
into
some
real
numbers
where
we've
got
the
velocities
and
we've
got
the
depends
of
the
layer
thicknesses.
So
here
was
our
ray
pass.
That
told
us
what
kind
of
waves
we
have-
and
here
was
our
travel
time
curves
and
now
we're
going
to
use
the
information
in
here
in
particular
the
first
arrivals
as
I
said
before,
to
tell
us
you
know:
what's
there
so.
A
We're
going
to
experience
that
so,
let's
suppose,
you've
got
a
direct
wave,
that's
traveling!
So
here's
its
pdp-1,
here's
Pierce
mark
shot,
is
and
we're
at
some
distance
X
shot.
So
at
some
time,
T
is
equal
to
ax,
upon
B
1
we're
going
to
get
an
arrival
and
for
what
we
had
been
doing
before
the
arrivals
were.
You
know
just
listed
as
like
kind
of
a
alarm
certain
impulse.
In
reality,
what
travels
is
actually
not
just
an
impulse,
but
is
something
called
a
wavelet.
A
A
What
that
does
is
that
every
time
you
see
a
pulse
here,
then
you
get
a
replication
of
the
wave
load
that
comes
in
at
that
particular
time,
so
that
the
seismic
signal
attea
is
not
really
an
impulse
like
that.
That
looks
like
a
replication
of
this
this
wave
look.
So
that's
why
this
particular
diagram
up
here
show
us
something:
that's
not
just
an
impulse,
but
actually
has
a
little
bit
of
a
wait
for
the
the
proper
mathematical
term.
Is
that
we're
actually
going
to
convolve
this
wavelet
with
our
impulse
response
here
and
generate
our
seismogram?
A
A
So
that's
what's
happening
up
here.
The
wavelet
is
kind
of
looking
like
this
guy
over
here,
and
what
we
have
here
is
a
representation
of
our
seismic
signals.
Then,
so
here
is
going
to
be
our
direct
wave.
Here's
our
refracted
wave
and
there's
also
going
to
be
a
reflection,
so
we're
going
to
take
these
guys
and
break
them
apart
and
really
try
to
analyze
the
specific
characteristics
of
this
of
this
travel
type
curve.
A
A
All
that
means
is
we're
plotting
time
this
way,
instead
of
that
way
and
you're
gonna
see
both
both
of
these
plots
kind
of
regularly,
as
as
we
go
through
here,
so
you
just
have
to
get
used
to
the
fact
that
in
some
manifestations
there's
time
going
down
this
way
and
others
we
decide
to
plot
up
this
way,
but
here's
the
way
I
like
to
plot
it
from
the
point
of
view
of
thinking
about
travel,
time
and
analysis.
So
we've
got
first
arrivals,
then
that
looked
like
that
and.
A
These
first
arrivals,
so
if
we
measure
them
awesome
up
here,
so
we
here's
our
seismic
plot,
so
you
can
see
that
each
one
of
them
now
is
got
this
little
on
it
and
we
picked
the
first
arrival
so
when
the
energy
has
just
started
to
come
out
for
each
trace,
we
pick
this
guy
record.
The
time
pick,
this
guy
record
the
time
duck
Duck
Duck,
and
then
we
plot
them
up
so
at
each
offset.
We
plot
what
the
time
is.
Okay
for
that
simple,
we
get
this
plot.
We
look
at
these
guys
here.
A
So
that
means
that
our
first
arrivals
now
could
be
something
now
that
looks
like
this
and
then,
if
we
have
subsequent
ones
again,
we
can
have
refraction
from
here.
Well,
look
at
this
face
a
little
bit
later
so
now,
I
just
want
to
take
you
through
these
guys.
I'm
gonna
do
this
yeah,
so
in
the
GPG
I.
Don't
want
to
do
all
of
the
derivations
here
on
the
board.
A
How
much
time
did
it
take
and
that's
again
easy
is
this?
Is
the
swimmer
problem
right?
So
we
take
whatever
length
we
have
here
/
this
upper
velocity,
whatever
distance
we
have
in
here,
/
the
below
lower
velocity
and
then
another
one
for
the
up.
So
we've
got
a
time
down
time
across
time.
Up
treat
three
pieces
of
time.
We
put
them
together,
we
can
get
an
equation
for
what
the
travel
time
is
as
a
function
of
distance.
A
The
one
comment
that
I
made
earlier
was
that
there
was
something
called
a
critical
distance,
so
there
is
a
time
before
which
there
is
no
critical
refraction.
So
the
first
critical
refraction
is
coming
down
like
this.
So
here's
the
way
it
comes
down
and
then
it's
going
to
get
diffracted
across,
but
by
the
time
it
comes
down
here,
it's
already
out
at
this
distance,
and
then
it's
got
to
get
back
up.
A
A
A
For
the
refracted
I'll
go
through
this
in
more
detail
later
at
the
beginning
of
next
one
next
week,
but
I
just
want
to
show
it
just
in
case
it's
going
to
kind
of
come
up
in
the
lab
this
afternoon.
We
can
also
show
that
the
travel
time
for
this
way
here
as
this
particular
formula,
that
the
time
is
equal
to
X
/
b2,
which
is
the
distance
here,
divided
by
the
velocity
of
this
lower
medium
plus
some
intercept
time,
and
we
have
an
equation
for
that.