WEBVTT
00:01.970 --> 00:05.790
The oscillation of a string on a musical instrument depends on its
00:05.790 --> 00:06.710
length and tension.
00:07.250 --> 00:09.230
It produces high and low tones.
00:09.870 --> 00:13.730
The mathematical description of this oscillation resembles that of
00:13.730 --> 00:14.430
seismic waves.
00:14.950 --> 00:18.770
But how exactly do seismic waves oscillate in the earth, and how do
00:18.770 --> 00:19.330
they propagate?
00:20.570 --> 00:23.830
The wave equation provides an answer to this question.
00:24.470 --> 00:28.070
It is important in order to understand the propagation of the various
00:28.070 --> 00:31.070
types of seismic waves in the earth and to evaluate them
00:31.070 --> 00:31.970
quantitatively.
00:39.710 --> 00:41.030
Hello and welcome.
00:41.970 --> 00:45.590
In this video, I will deduce how the movement of an oscillating string
00:45.590 --> 00:47.130
can be described mathematically.
00:47.910 --> 00:51.850
Furthermore, I will also show you how this simple description can be
00:51.850 --> 00:55.290
applied to the significantly more complex propagation of seismic
00:55.290 --> 00:55.590
waves.
00:55.750 --> 00:59.130
The assumptions and formulas necessary for this fit together like
00:59.130 --> 01:00.130
pieces of a puzzle.
01:00.490 --> 01:03.830
Let me now qualitatively introduce you to the puzzle pieces involved.
01:04.550 --> 01:08.930
Here, the string should be oriented parallel to the x-axis and is made
01:08.930 --> 01:10.870
to oscillate in the direction of y.
01:11.750 --> 01:15.870
Upon being plucked, it leaves its equilibrium state and deformation
01:15.870 --> 01:17.190
moves along the string.
01:17.830 --> 01:22.050
I will now describe the displacement u as a function of the location
01:22.050 --> 01:24.110
along the x-axis and the time t.
01:24.910 --> 01:28.230
For this purpose, we will now look at a small section of the
01:28.230 --> 01:29.190
oscillating string.
01:29.810 --> 01:32.570
This section is stretched by the tension tau.
01:33.390 --> 01:36.430
This tension can be described by the force F.
01:38.290 --> 01:40.890
To do so, we use Newton's first law.
01:41.270 --> 01:44.990
The force F equals the mass m times the acceleration a.
01:45.570 --> 01:50.190
Hence, the force on the upper end equals tau times sine theta 2 and
01:50.190 --> 01:53.490
the force on the lower end is tau times sine theta 1.
01:53.990 --> 01:56.510
The difference is the force acting on the section.
01:57.250 --> 02:01.430
This is the one-dimensional equation of motion for our string, the
02:01.430 --> 02:02.890
first piece of the puzzle.
02:03.590 --> 02:07.450
On the other side of the equation, we substitute the mass m with the
02:07.450 --> 02:11.550
density rho times the length dx of the section of the string reduced
02:11.550 --> 02:12.310
to 1d.
02:12.690 --> 02:16.130
The acceleration is the second-order partial derivative of the
02:16.130 --> 02:17.630
displacement u over time.
02:18.050 --> 02:21.650
If the displacement values are small, the angle theta is also small
02:21.650 --> 02:24.190
and the sine can be substituted by the gradient.
02:24.790 --> 02:29.990
F equals tau times du of x plus dx and t differentiated with respect
02:29.990 --> 02:32.050
to dx minus du over dx.
02:32.290 --> 02:35.930
In the next step, we approximate the first term in the brackets, that
02:35.930 --> 02:37.690
means the force on the upper end.
02:38.390 --> 02:42.850
This is the sum of the force on the lower end du of x and t over dx
02:42.850 --> 02:48.030
plus the change along the path dx, second-order derivative of u of x
02:48.030 --> 02:50.650
and t over dx squared times dx.
02:51.350 --> 02:53.970
These are the first two terms of a Taylor series.
02:54.690 --> 02:59.010
You can see that these two terms cancel each other out and the result
02:59.010 --> 03:06.070
is tau d2u over dx squared times dx equals rho dx d2u over dt squared.
03:06.910 --> 03:10.970
This can be rearranged to give the second-order spatial derivative of
03:10.970 --> 03:15.670
u equals 1 divided by v squared times the second-order time derivative
03:15.670 --> 03:16.390
of u.
03:16.690 --> 03:20.970
Here, the wave velocity v equals the square root of tau divided by
03:20.970 --> 03:21.130
rho.
03:21.130 --> 03:25.230
This means that the velocity v depends on the physical properties of
03:25.230 --> 03:28.770
the string, the tensile stress tau and the density rho.
03:29.490 --> 03:33.370
A higher tension results in a greater restoring force and hence in a
03:33.370 --> 03:34.050
higher velocity.
03:35.550 --> 03:40.430
A higher density describes a larger inertial mass, thereby reducing
03:40.430 --> 03:41.470
the wave velocity.
03:42.410 --> 03:46.570
That is why the thinner, more highly tensioned strings of a ukulele
03:46.570 --> 03:50.410
produce higher-pitched sounds than those of a contrabass.
03:50.410 --> 03:52.850
This is the one-dimensional wave equation.
03:53.370 --> 03:57.010
I will now show you what this means for the oscillation of the string.
03:57.310 --> 04:00.850
For this purpose, we need to find a solution for the displacement u,
04:01.190 --> 04:02.890
which fulfills the wave equation.
04:04.350 --> 04:08.050
The solution of the wave equation can generally be expressed as a
04:08.050 --> 04:11.310
function of the spatial variable x plus-minus the velocity v
04:11.310 --> 04:12.830
multiplied by the time t.
04:13.490 --> 04:16.810
This can be shown directly by forming the partial derivatives over
04:16.810 --> 04:17.650
space and time.
04:17.650 --> 04:21.690
One special form of this is the movement of a harmonic wave.
04:22.490 --> 04:27.390
u of x and t equals A cosine of omega t plus-minus kx.
04:28.650 --> 04:33.390
Hence, both the location x as well as the time t occur in the argument
04:33.390 --> 04:34.890
of the trigonometric function.
04:35.390 --> 04:39.170
This means that the wave oscillates over time, but also in space.
04:39.970 --> 04:44.130
At a fixed point in time, let's look at a photo of the string, there
04:44.130 --> 04:46.990
are wave peaks and troughs at various locations.
04:47.890 --> 04:52.050
Or at a fixed location, let's hide the right and left parts of the
04:52.050 --> 04:54.150
string and concentrate on a small section.
04:55.190 --> 04:58.630
In this case, the wave peaks and troughs change over time.
04:59.690 --> 05:03.250
The spatial distance between two wave peaks is called the wavelength.
05:04.070 --> 05:06.270
The time interval is called the period.
05:07.450 --> 05:11.230
To further describe the oscillation, we now also need to observe the
05:11.230 --> 05:14.770
boundary conditions and initial conditions of the tensioned string.
05:14.910 --> 05:17.830
How is it held in place and where and how is it plucked?
05:19.110 --> 05:21.570
But that would exceed the scope of this lesson.
05:21.910 --> 05:25.590
I have now derived the one-dimensional wave equation and introduced
05:25.590 --> 05:26.670
solutions for it.
05:28.210 --> 05:31.730
This is the basis for further developing it for three dimensions.
05:32.370 --> 05:37.470
As is the case in 1D, density determines the propagation velocity of
05:37.470 --> 05:38.810
seismic waves in the earth.
05:38.810 --> 05:40.290
But not only that.
05:40.690 --> 05:44.390
The elastic properties of what is beneath the surface also influence
05:44.390 --> 05:45.850
the velocity of the waves.
05:46.810 --> 05:50.630
In order to put together the wave equation for 3D cases, we require
05:50.630 --> 05:54.190
another set of puzzle pieces which were not necessary for 1D.
05:54.510 --> 05:57.390
I will now introduce them to you in a simplified manner.
05:57.930 --> 06:01.550
As was the case with the string, stresses also exist within the earth,
06:01.630 --> 06:04.510
and these stresses are also transformed into their formations.
06:04.850 --> 06:07.430
The how is described by a state equation.
06:08.180 --> 06:11.430
It relates the elastic properties of the earth to each other.
06:12.870 --> 06:14.710
For this, we use Hooke's law.
06:15.790 --> 06:19.410
It describes in general how stress deforms a physical body.
06:20.550 --> 06:26.670
Stress sigma ij equals C ijkl times deformation epsilon kl.
06:27.610 --> 06:29.350
Sigma and epsilon are tensors.
06:29.850 --> 06:33.070
They describe stress and deformations in 3D.
06:33.770 --> 06:37.750
C is a fourth-order tensor and describes how stresses are transformed
06:37.750 --> 06:38.470
into deformation.
06:39.930 --> 06:43.830
It is assumed that the material found in the interior of the earth has
06:43.830 --> 06:47.450
approximately the same physical properties in all spatial directions.
06:48.110 --> 06:49.490
We call this isotropic.
06:50.150 --> 06:52.590
This simplifies the application of the law.
06:53.230 --> 06:57.330
Stress sigma then appears as a function of the deformation epsilon and
06:57.330 --> 06:59.370
the Lame parameters lambda and mu.
07:00.050 --> 07:02.670
They describe the elastic properties of the earth.
07:03.410 --> 07:05.710
In this case, mu is the shear modulus.
07:06.170 --> 07:08.470
It indicates how easily a medium can be sheared.
07:09.070 --> 07:11.970
The greater this value, the more readily it shears.
07:12.450 --> 07:16.790
This state equation will be inserted into the homogeneous 3D equation
07:16.790 --> 07:18.330
of motion in the next step.
07:18.930 --> 07:22.710
It is similar to the equation of motion in 1D and describes a
07:22.710 --> 07:26.690
propagating wave without taking into consideration the stimulating
07:26.690 --> 07:29.710
force that means far away from the seismic source.
07:30.670 --> 07:35.090
This combination allows the equation of motion to be formulated using
07:35.090 --> 07:36.110
displacement terms.
07:37.470 --> 07:40.930
The many Nabla operators, put simply, stand for the spatial
07:40.930 --> 07:42.330
derivations in 3D.
07:43.230 --> 07:47.590
We solve this equation by separating the displacement field U into a
07:47.590 --> 07:50.590
scalar potential phi and a vector potential psi.
07:53.190 --> 07:57.750
Even if this appears complex at first glance, this separation leads to
07:57.750 --> 08:00.250
two simple solutions for the equation of motion.
08:01.390 --> 08:05.430
The seismic wave equations for p- and s-waves, in 3D of course.
08:07.350 --> 08:10.950
These solutions have a form similar to that of the one-dimensional
08:10.950 --> 08:15.210
wave equation, but describe seismic longitudinal waves and shear
08:15.210 --> 08:15.550
waves.
08:16.470 --> 08:21.290
The wave velocities are therefore vp equals the square root of lambda
08:21.290 --> 08:27.090
plus 2 mu divided by rho and vs equals the square root of mu divided
08:27.090 --> 08:27.570
by rho.
08:28.550 --> 08:31.990
That brings us to the end of this qualitative derivation.
08:32.630 --> 08:35.750
I will show you the solution of both wave equations here.
08:36.890 --> 08:40.670
Both equations describe plane waves which propagate in an arbitrary
08:40.670 --> 08:41.290
direction.
08:42.690 --> 08:47.210
In complex exponential notation of the trigonometric function, we can
08:47.210 --> 08:50.350
recognize the similarity to the harmonic wave in 1D.
08:57.670 --> 09:01.910
You have seen how the derivation of the wave equation consists of four
09:01.910 --> 09:02.250
steps.
09:03.090 --> 09:06.110
First, writing out the equation of motion.
09:06.790 --> 09:10.430
Second, applying the state equation of the material properties.
09:11.290 --> 09:15.570
Third, transformation to the wave equation and fourth, solving the
09:15.570 --> 09:17.750
equation that describes the wave propagation.
09:18.490 --> 09:21.990
The solution of the wave equation then indicates how the waves
09:21.990 --> 09:23.970
oscillate in time and space.
09:24.950 --> 09:28.590
Furthermore, it also indicates what the propagation velocity of the
09:28.590 --> 09:29.510
waves depends on.
09:29.990 --> 09:32.850
In the case of the string, it is density and tension.
09:33.550 --> 09:37.670
And in the earth, it depends on density and elastic properties, but to
09:37.670 --> 09:39.730
different extents for p- and s-waves.
09:40.810 --> 09:45.350
You have learned that the derivation of the wave equation in 1D and 3D
09:45.350 --> 09:49.710
is composed of four similar parts, allowing an oscillating string or a
09:49.710 --> 09:52.510
seismic wave to be described in its entirety.