Oscillations and Waves

What are waves? In fact, there are many different types of waves e.g.

• Mechanical waves require a material medium to travel (air, water, ropes). These waves are divided into three different types.
  • Transverse waves cause the medium to move perpendicular to the direction of the wave. (e.g. on a rope, where the rope moves up and down and the wave propagates along the rope)
  • Longitudinal waves cause the medium to move parallel to the direction of the wave. (e.g. acoustic waves, where the molecules oscillate parallel to the propagation direction)
  • Surface waves are both transverse waves and longitudinal waves mixed in one medium.
• Electromagnetic waves do not require a medium to travel (light, radio).
• Matter waves produced by electrons and particles. Well, these waves are a bit different from the others and you really don't want to know why. No, believe me you don't want to know.
• Gravitational waves Gravitational waves are a natural part of general relativity and astrophysics as they mainly stem from huge gravitational objects rapidly changing form, like supernovae explosions, binary stars revolving about each other, masses falling into black holes and so on. So if you are interested in this kind of waves delete this tutorial immediately and consult a psychologist.

In this treatment we will focus on electromagnetic waves, but since they have a lot in common with mechanical transverse waves, we will use the analogies in order to visualize some important relations. In general a wave (at least in the case of the types of waves discussed here) is a transfer of energy from one point to another with no transfer of mass between the points. It is important to realize that a wave is quite a different object than a particle. A stone thrown though a window transfers energy from one point to another, but this involves the movement of a material object between two points. A common example of a wave is a wave on the ocean which carries energy, as they cause erosion on the shore, but material (i.e., water) is not continuously being transferred onto the shore. For example if you put a piece of cork on top of a wave the cork will stay at approximately the same distance from the shore (just hopping up and down), while the wave transfers its energy to the shore. Another example of a wave is a sound wave, which is vibrations of air molecules which propagate from one place to another. These also carry energy, but do not involve the mass movement of air from one place to another. In the case of a Synthetic Aperture Radar (SAR) we deal with Transverse Electro Magnetic waves (TEM - waves or just EM - waves). Since they have a couple of properties in common with transverse mechanical waves, we will start with transverse mechanical waves in order to visualize some properties of waves.

For mechanical waves we need a medium for the wave to propagate. If we consider a rope fixed at one end a few feet above the ground and held by you at the other end as such a medium, a wave can be created by vertically shaking the end of the rope. By shaking the end of the rope only once a single vibration is produced. This is a pulse. By moving a rope regularly up and down, a traveling or periodic wave is produced.

If we move the rope regularly up and down, any point on a transverse wave (i.e. on the rope) moves up and down in a repeating pattern. The shortest time that a point takes to return to the initial position (one vibration) is called period, T. The number of vibrations per second is called frequency (f) and is measured in Hertz (Hz). The equation for frequency is therefore, given by:

$$f = \frac{1}{T}$$ (1.1)

The shortest distance between to two adjacent corresponding locations on the wave (e.g. peaks, the highest points, and troughs, the lowest points) is the wavelength, $\lambda$ as shown in Fig. 1.1

Figure 1.1: Geometrical representation of the wave. If the horizontal axis is in time units L is the Period T of the wave, if the horizontal axis is in space units L is the Wavelength $\lambda$

The frequency f of a wave and its wavelength $\lambda$, are related to each other via the propagation velocity v

$$\lambda = \frac{v}{f}$$ (1.2)

The propagation velocity of a wave is, in general, dependent upon the media through with the wave propagates. For example, the propagation speed of sound is much higher in water than in air. For the sake of completeness we should mention, that the media can have some quite nasty effects on the propagation. The velocity of a wave can be defined in many different ways, focusing on different aspects or components of any given wave. For example, in a wave packet (consisting of a collection of waves) each wave moves with a phase velocity, while whole wave packet moves with a group velocity, which can be different from the phase velocity. This effect is called dispersion and shall not be discussed here in more detail. In this treatment we will only consider the group velocity of the wave packet, which corresponds to the classical particle velocity.

An other important feature of a wave is the amplitude A. The amplitude A can have slightly different meanings depending upon the context of the situation. The most general definition is, that the amplitude is the displacement of the medium from its undisturbed position to the top of a crest. In some publications it is referred as the maximum positive displacement of the medium from its undisturbed position to the top of a crest. In other discussions, though, the term amplitude takes on a slightly more complicated meaning. In some contexts, amplitude means the displacement of the medium from its undisturbed position to its disturbed position at a certain point along the wave. The definition of amplitude of a wave as the distance from a crest to where the wave is at equilibrium, is used to measure the energy transferred by the wave. The bigger the distance (i.e. the amplitude), the greater the energy transferred. While in discussions about interference, amplitude usually means the displacement of the medium from its normal position at certain points, and this displacement can be positive or negative. In general, waves may have any shape, e.g. saw tooth like oscillations. However, in the case of SAR sine and cosine functions (called harmonic functions) are used to create the waveform. Oscillations which can be described by a single term of the sine or cosine function are called simple harmonic oscillations (SHO). Simple harmonic oscillation can be represented as the projection of uniform circular motion onto one axis Fig. 1.2

Figure 1.2: Projection of uniform circular motion onto one axis

The phase angle $\omega t$ t in Fig. 1.2 corresponds to the real angle $\omega t$ through which the point has moved in circular motion. In Fig. 1.2, the motion started with initial phase $\phi$. At time t, the rotating point is at the angle = $\omega t$ + $\phi$ This is also the total phase of the oscillating point. Its position is described by x(t) = A cos ($\omega t$ + $\phi$).

The process of energy transport is, in general similar for electromagnetic waves and mechanical transverse waves. A wave is generated by an oscillating source (in case of the rope mentioned above, the oscillating source is the hand moving up and down) Due to the coupling of the elements of the media the each element forces it's neighbor to start the same oscillation with a time delay. In this way the wave propagates through the medium. If we observe one element of the wave we find, that each element of the wave oscillates around its undisturbed position. During this oscillation process the for each element of the wave an exchange between two energy forms takes place. In case of the rope the each element changes between potential energy and kinetic energy. Each element of the rope changes continously from potential energy and kinetic energy. E.g when the element is at the maximum distance from the undisturbed position it has the maximum potential energy and no kinetic energy (at the turning point the speed of the element is zero) on the way down it accelerates and gains kinetic energy until it gets the maximum speed (i.e. kinetic energy) at crossing the undisturbed position where the potential energy is zero.