Seismic waves from the Great 
Tohoku Earthquake on March 11, 
2011,
were detected all over the
globe.
Even ten days after the 
devastating event, ground motion
was still measurable.
On the illustration, you can see
the vertical ground motion 
plotted over time in hours
as it was recorded at the 
seismic Black Forest Observatory
(BFO).
The stronger seismic waves last 
for up to ten hours after the 
quake.
If we zoom into the seismogram 
ten days after the quake,
the first thing we see are the 
major earth tides with a 12-hour
period
- these are always there.
However, riding on top of them
are these small oscillations 
with a period of approx. 20 
minutes.
They are the reverberations of 
the earth,
which has been caused to 
oscillate like a bell by the 
earthquake.
Hello and welcome.
In this video, you will learn 
how to recognize the earth's 
free oscillations:
How they are observed, 
described,
and what their signals are used 
for in seismology.
In this video, I will show you 
various forms of these 
oscillations
and how they are classified.
The Tohoku earthquake on March 
11, 2011, will be used as an 
example.
With a magnitude of 9.1, it is 
the fourth-strongest earthquake 
over the past 100 years.
The stimulation of such free 
oscillations requires strong 
quakes,
which need to be measured with 
highly sensitive seismometers in
quiet, shielded locations.
For the first time, such signals
were observed 
during the even stronger quakes 
in Chile and Alaska in 1960 and 
1964.
To explain this phenomenon, I 
will utilize an oscillating 
string for simplification.
The string is under tension and 
starts oscillating when plucked.
Here, the fixed ends ensure that
a standing wave is formed.
It has two wave nodes: left and 
right.
At these points, a destructive 
superposition or interference of
the oscillation takes place.
In the middle, you can see 
constructive interference 
creating a wave trough.
We call this free oscillation 
the fundamental mode.
In addition, there are also 
additional modes, which can be 
seen here.
They are characterized by a 
higher number of wave nodes and 
troughs.
Due to this, they have a higher 
frequency and generate a higher 
tone.
They are called overtones.
Common to all modes is the fact 
that the position of the nodes 
and troughs
is not dependent on the 
stimulus,
but instead exclusively on the 
boundary conditions.
These include e.g. the length 
and thickness of the string.
Each free oscillation or normal 
mode
corresponds to an eigenfrequency
at which the oscillation takes 
place.
For the earth, the very same 
principle holds true - but in 
three dimensions.
It, too, has its characteristic 
normal modes and 
eigenfrequencies,
which are dependent exclusively 
on its form and interior 
structure.
They are stimulated by the 
constructive and destructive 
interference of seismic waves.
The frequency characteristics of
the normal modes of a string
are described with the help of a
Fourier analysis.
The corresponding mathematical 
function
- the eigenfunction -
is composed of sine and cosine 
functions.
In the spherical case of the 
earth, spherical harmonics are 
used instead.
Using them, the normal modes of 
the earth can be composed.
They are subdivided into 
spheroidal and toroidal modes.
Here is an example of a 
spheroidal normal mode:
the “breathing mode”, in which 
the earth behaves like a balloon
that is periodically inflated 
and deflated again.
Expansions and contractions 
deform the earth's surface.
Spheroidal normal modes are 
standing waves
which are composed of 
interfering P- and Rayleigh 
waves
as well as vertically polarized 
S-waves.
Just like them, they oscillate 
radially and tangentially to the
earth's body.
In contrast, there are also 
toroidal normal modes.
We can imagine them as a torsion
of the earth:
Here in the simplest case, a 
rotation of the northern 
hemisphere
relative to the southern 
hemisphere.
Toroidal normal modes originate 
from interfering Love- and 
horizontally polarized S-waves.
Hence, in the case of these 
standing waves,
the earth oscillates exclusively
tangentially.
Toroidal modes affect only the 
earth's mantle and the earth's 
crust,
since in the liquid outer core 
shear waves are not able to 
propagate.
The normal modes of the earth 
are described via a 
nomenclature:
Spheroidal modes are identified 
with S and toroidal modes with 
T.
Three integer indexes describe 
each mode.
Firstly: Small “l”.
This is the harmonic degree of 
the mode and for spheroidal 
modes,
it indicates the number of node 
lines on the sphere's surface.
The node lines correspond to the
node points of the oscillating 
string in 1D.
Secondly: The index “n”,
which indicates the overtone 
number, and which equals zero 
for fundamental modes.
The third index is the harmonic 
order “m”.
It represents the number of node
lines which pass through the 
poles.
For toroidal modes, “l” and “m” 
describe the number of curls
that make up the horizontal 
torsions.
For an idealized earth, i.e. a 
symmetrical sphere,
the value of “m” does not change
the eigenfrequency of the mode, 
which is why this index is often
not specified.
These labels allow for the 
systematic categorization of the
normal modes,
whereby each normal mode is 
characterized by its 
eigenfrequency.
In the frequency spectrum, they 
occur via their own maxima.
I will present a number of 
special modes with low indexes
- that means few node levels -
with reference to their 
eigenfrequencies in the 
spectrum,
which was obtained from a time 
series lasting several days
after the Tohoku earthquake at 
the BFO observatory.
For this purpose, I will use the
vertical ground movements.
They indicate the spheroidal 
normal modes.
First of all, note the X-axis in
this frequency spectrum.
It indicates values in the sub-
millihertz range.
The 0S2 mode has the smallest 
eigenfrequency, 0.3 mHz,
which corresponds to an 
oscillation period of approx. 54
minutes
- even surface waves with 
extremely long periods
only reach periods of a few 
hundred seconds.
0S2 is a fundamental mode.
This is how you can tell: “n” 
equals 0.
“l” equals 2.
This means that what we have 
here are two node levels
which in this case separate the 
earth at the equator and in a 
plane vertical to it.
In the animation, you can see 
this mode at an exaggerated 
scale.
We call this mode the “rugby” 
mode,
because the maximum deformation 
has an egg-like shape.
If we examine it more closely, 
we realize that 0S2 is split 
into four parts.
These constitute the orders “m”.
They are due to the influence of
external forces
such as Coriolis and centrifugal
forces, as well as effects of 
the heterogeneous earth.
An additional example can be 
seen here in the spectrum: 0S3.
It looks like this in the 
animation.
You can see how the number of 
node planes increases to three.
Now, you might ask where 0S1 is.
0S1 is not a seismic mode, but 
instead describes the 
translation of the entire earth,
which can only be brought about 
by external forces
and hence does not constitute a 
seismic mode.
But which mode did we see at the
start of this video?
In the illustration, we see 
slightly more than 14 periods 
within five hours,
that is approx. three per hour.
This corresponds to a period 
length of a good 20 min.
Hence, what we have here is the 
spheroidal “breathing” mode 0S0,
which has the largest amplitude 
of all of the earth's normal 
modes.
In this case, Rayleigh waves 
traveling around the earth in 
opposite directions
interfere constructively to 
create purely radial expansion 
and contraction.
The earth's free oscillations 
are important
for investigating the inner 
structure of the earth.
Due to the very low-frequency 
oscillations,
the wavelengths of the earth's 
free oscillations are extremely 
large,
which means that they indicate 
the average elastic properties 
of the earth
as a function of the earth's 
radius in a highly stable 
manner.
One other important use of free 
oscillations is the modeling of 
synthetic
- that means artificial 
waveforms for seismic tomography
or the analysis of seismic focal
mechanisms.
At the beginning, I introduced 
the earth's free oscillations as
interfering seismic waves,
but conversely, waves can also 
be described and calculated as 
the sum of various normal modes.
Hence, the same physical 
phenomenon
can be expressed using different
mathematical formulas.
This shows the equivalence of 
standing waves and traveling 
waves.
In this video, you learned about
the types of free oscillations 
in the earth
and how they are described.
They are subdivided into two 
major types.
Spheroidal modes describe 
combinations of radial and 
tangential oscillations.
Toroidal modes are exclusively 
tangential torsions of the 
earth.
Normal modes are described using
the harmonic degree “l”, the 
harmonic order “m”,
and “n”, which represents the 
overtone number.
The equivalence of standing and 
traveling waves
can be used in seismology to 
calculate synthetic seismic 
waves.
Free oscillations in the earth 
are also useful 
for accurately describing the 
earth's radial structure.
It is only at particularly well-
shielded locations
and with highly sensitive 
seismometers that it is possible
to evaluate these weak signals 
from the earth
- the BFO seismic observatory is
one such place.
