Mechanical Waves - Introduction

 

Let us take a simple example where we create a mechanical wave.

 

We take a rope or string in our hand and we move our hand up and down. We start a wave that will propagate to the right as shown in the figure of this link.

 

http://demo.webassign.net/ebooks/cj6demo/xlinks/c16-sec1-0005.htm

The wave arrives at point 1 and moves it up, then at point 2 etc

 

Can you figure out what the movement of point 1 will look like?

 

Let us try to mark a point with a pen.

 

It will actually move up and down in a vertical axis as seen at this animation

http://en.wikipedia.org/wiki/Harmonic_oscillator

 

Can you predict what will be the movement of the point in time? Can you calculate what will be the position of the point in 30 seconds for instance?

In other words can you model its movement? Or fit its behavior in a statistical model?

How could you try to do that?

 

 

You can place a ruler in a vertical position and use your camera to take photographs via its special function of taking pictures automatically in regular time intervals. You can process these photographs to measure the distance of the point in different time intervals.

 

You could use the example below with the values mentioned.

http://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html

 

 

 

 

You will get a series of points like this:

t x
0 0
0.2 0.631103
0.4 0.681973
0.6 0.10584
0.8 -0.5676
1 -0.71919
1.2 -0.20956
1.4 0.49274
1.6 0.742019
1.8 0.309089
2 -0.40802
2.2 -0.74999
2.4 -0.40243
2.6 0.315125
2.8 0.742956
3 0.487716

 

You could also use the example at the link that follows http://www.seilias.gr/index.php?option=com_content&task=view&id=85&Itemid=32

(Just press play. If you also want to follow the graph you can only tick “Θέση” which stands for “thesis” meaning “position”.)  Note: X=0.4*sin(5t)

 

 

You then ask a mathematician to fit these points in a statistical model.

 

You will get this formula:

X=0.75*sin(5t)

which in a more general form could be written as follows:

X=A * sin(ωt)

 

As mentioned in Wikipedia, Sine, in mathematics, is a trigonometric function of an angle.

 

Interestingly, we see that an angle is associated to our data.

 

How is that possible? How can you have something that moves up and down or back and forth and get something that resembles a revolving angle or circular motion?

 

This is where general culture becomes relevant. You may have heard that Galileo who was studying the movement of the moons of Jupiter using his newly discovered telescope, was seeing them going back and forth (Physics Textbook “Fundamentals of Physics”, by D.Halliday, R. Resnick, J. Walker, published by Wiley). Imagine that you know that all celestial bodies revolve around others and that someone tells you that there is a planet that goes back and forth.

 

This is also where a comment I heard from a physics high school teacher becomes relevant. She was complaining that stereometry (solid geometry or maybe one could say “3D-geometry”) is no longer taught in high school. She believed that this is a resource that could substantially develop imagination via visualization in space and could also enhance higher thinking functions.

 

Example: what does a circle give you when it revolves in space around its diameter? Answer: a sphere

 

Is is possible that a planet revolves around another and that you see it going back and forth?

 

How could you examine this motion?

 

Imagine placing a basket ball on the table to simulate Jupiter, and a tennis ball to simulate one of its moons, Callisto. You also use a hula hoop where you fix the tennis ball in order to simulate the orbit of the planet around Jupiter.

 

Let us say that the balls have little battery lamps inside and that the room is dark so that you can only see the two balls.

 

Is there any optical plane that you could consider that would allow you to apprehend the orbital motion as a back and forth motion?

 

While the planet is in orbit, imagine having your eyes at the level of the table; and also being very far behind.

 

Would you be seeing the planet moving back and forth?

 

It is actually as seeing the projection of the tennis ball moving back and forth on an axis.

 

Although the planet is in orbit, its projection is moving back and forth.

 

In the same way a back and forth motion can be projected as a circular motion. And this is where the angle is coming from.

 

To understand this better you can draw a circle with a diameter and one point A like in the drawing that follows. Draw a line vertically to the diameter. Point A' will be the projection of point A. Pick a point further down clock-wise and do the same. 

 

 

Mechanical Waves - Characteristics

 

Let us consider an example using the animation from this link, as described in the picture that follows.

 

 

 

Using our hand or the setting shown in the image, we do ups and down. The energy that we provide is transferred to the string.

 

If we keep our eyes on the first green bead and we watch it for 30 seconds (or 50/100 of a minute as the timer would indicate), we will notice that it has moved up to a maximal displacement then down to a maximum and that it has returned to its original position.

 

At the same time a disturbance or wave form has been created on the string as shown in the image. 

 

Let us see what will happen when 1 minute is completed.

 

 

 

 

The first green bead has repeated the course that we mentioned. Also, we see two waveforms of the same shape. The disturbance has moved horizontally on the string.

 

The energy of our hand is transferred to the string. It is transferred from one molecule to the other, from one bead to the other, causing their vertical displacement. By watching, for example, one green bead we see it moving up and down (oscillating) on a vertical axis.

 

We also see wave forms (shapes of waves) moving horizontally to the right.

 

In conclusion, as the wave form is propagated horizontally along the string, the beads are moving vertically up and down (oscillating).

 

What are the characteristics of this motion?

 

How “big” it is and how “quick” it is, and also how much energy it transfers.

 

We saw that there is a vertical displacement of the beads and a horizontal displacement of a shape or form, the waveform.

 

Let us consider the vertical displacement of the beads.

 

The vertical displacement refers to how big or how “ample” is the vertical motion of the beads. This would mean how far up can a point go from the reference line.

 

The answer is x=0.75cm

Therefore, the amplitude of the motion is 0.75cm.

 

How quick or how frequent is the up and down?

 

How frequent is the up and down?

Two are happening every second.

Two ups and downs per second=2 cycles per second=2 Hz

Therefore, the frequency of the motion is 2Hz.

 

How quick is the up and down? How long does it take for the motion to complete itself? It is a motion that repeats itself with time. It is a periodic motion. How long does it take for the fundamental motion, one up and down, to complete itself?

 

Since two ups and downs are happening every second, one is happening every 0.5 seconds.

Therefore, the period of the motion is 0.5 seconds.

 

We saw in a previous section that this periodic motion is associated with an angle.

How quickly does this angle “run”?

We can measure an angle in degrees, but we use radians in the International System. One radian is approximatively 57.3 degrees. Running a circle or doing 360 degrees would correspond to 6.28 radians or 2π radians.

One up and down would correspond to doing 360 degrees or 2π radians.

One up and down is one period.

Therefore, our angle runs 2π radians per period.

We say that the angular frequency is:  ω=2π/Τ=2π/0.5=4π=12.56 (rads/s)

 

The equation of the motion will be:

x= 0.75* sin(12.56t)

 

 

Let us consider the horizontal displacement of the wave.

 

The horizontal displacement refers to the distance between repetitions of the shape of the wave (wave shape). The wave shape is considered to have a length and this is called wavelength λ.

 

How do we calculate how quickly the wave will run on the string?

By measuring how far the disturbance goes in a specific time.

We know that the wavelength is the distance between repetitions of the shape wave. When the wave shape will be repeated this means that the wave front will have run one wavelength.

The shape of the wave is repeated every one period.

So, since a wave runs distance equal to a wavelength in one period, its velocity will be:

v=λ/Τ=λ*f

 

 

Which factors influence the velocity by which a wave travels through the string?

Our hand provides energy to the string and this energy is transferred to adjacent sections in the string. Let us say that the “first microscopic segment or first molecule will pull the second” etc. It will be easier to pull the next molecule segment if the string is light and not heavy and if it is elastic and not rigid.

Compare a string to a rope and an elastic band to a rigid one.

 

Easy: The more elastic the medium, which means the less tension a medium has, the bigger the velocity.

Difficult: The heavier the medium, or the more mass a medium has for a specific volume (density) the  more resistance to motion or more inertia it will demonstrate.

 

The following formula links the velocity with the elastic property, τ,  and the inertial property, μ, of the medium

v= SQR (τ/μ)

 

 

What will happen if we put more force in the pulse of our hand? iIs it going to change the amplitude or the frequency?

It will only change the amplitude.

Here is an interesting discussion : http://www.physicsclassroom.com/class/waves/Lesson-2/Energy-Transport-and-the-Amplitude-of-a-Wave

 

As mentioned:

A wave is an energy transport phenomenon that transports energy along a medium.

Our hand provides energy that is introduced in the medium and that causes the vertical displacement of its molecules and the propagation of a wave.

Here is the energy transported by a wave:

E=1/2μvω2y2