Gelombang dan Optik

Merupakan Kumpulan Materi Kuliah Gelombang dan Optik

Gelombang Optik

4 SKS

Dosen:

1. Dr. Andhy Setiawan, S.Pd., M.Si.

2. Lina Aviyanti, S.Pd., M.Si.

Sumber Buku:

Drs. Taufik Ramlan Ramalis, M.Si.

1. Kinematika Gelombang

A. Pendahuluan

B. Bentuk Umum Persamaan Differensial Gelombang

1. Solusi Persamaan Gelombang

2. Superposisi Dua Gelombang dan Layangan

3. Kecepatan Grup dan Dispersi

4. Efek Dopler

C. Energi dan Momentum Gelombang

1. Penjabaran Persamaan Gelombang Melalui Kekekalan Energi

2. Intensitas Gelombang

3. Rapat Momentum

4. Impedansi dan Daya Gelombang

D. Pemantulan dan Transmisi Gelombang

1. Koefisien Pantul dan Koefisien Transmisi

2. Hukum Snellius

E. Soal Latihan

Wave Kinematics

We bring our studies to a close in this chapter about waves. They are some of the most common experiences we share: both the sounds we hear and the light we see are waves. At the same time they are subtle and pervasive: we shall see that they lie at the heart of the fundamental physics we contemplated in Chapter 7.

For our purposes, a wave is a periodic disturbance or deformation of the medium it is in. For example, sound waves are pressure variations which deform the air (or other medium) in which they travel; even empty space serves as a medium for electromagnetic waves (ie., light or radio waves). The very existence of waves is a consequence of the medium's tendency to be at rest. Once energy is transmitted to the medium, conservation of energy requires that if the medium is to come to rest at one place, it must move the energy somewhere else. That movement is the wave; we say that the wave "propagates" through the medium.

We will distinguish between several types of waves. Waves may be "transverse" or "longitudinal". Transverse waves disturb the medium at right angles to their direction of travel, while longitudinal waves disturb the medium in the same direction as they are travelling. Electromagnetic waves are transverse, while sound waves are longitudinal. We will also distinguish between travelling and "standing" waves (below). Finally, we mention that waves occur as "wavetrains", which are arbitrarily long and continuous, and as pulses, which are more compact in nature.

The propagation speed of a wave (denoted c) is the rate at which a given peak of the wave travels. The wavelength (λ) is the distance between peaks, and the frequency (ν = ω / 2 π, where ω is the "angular" frequency) is the number of peaks per unit time. The period (T = 1 / ν) is the time between two peaks (recall periodic functions in math), and the wave number (k = 2π / λ) is proportional to the number of peaks per unit distance. (The factors of 2π in the angular frequency and wave number are necessary so that the trigonometric functions we will be using will be periodic in the period and wavelength, respectively.) Here we see a wave train on a string as a function both of time and of distance:

(The vertical scale for this example is in meters; for a sound wave, it would be in units of pressure, and for light waves it would be a field strength.) Its frequency ν is 4 "Hertz" (Hz), so its period is .25 s; its wavelength is 0.4 m. Dimensional analysis gives us the "dispersion" relations

c = λ ν = ω / k

(since c depends on the frequency, waves of different frequency tend to disperse). Therefore for this wave, c is 1.6 m / s. For sound waves in air, c is equal to the Sqrt ( 1.4 R T / (28.95 g / mol)); for electromagnetic waves, c is 1 / Sqrt (μ ε). The relationship between the speed of light and the permittivity and permeability was one of the first clues that light is an electromagnetic wave.

Finally, the amplitude (A) of a wave is the distance from the resting (equilibrium) state of the medium to the peak of the wave. The wave above has an amplitude of 2 m. The energy of a wave is proportional to A ², and its intensity (I) is the power it delivers per unit area (P / 4 π r ² for a spherical wave).

It should be noted that this discussion applies to linear waves. These waves are idealizations of the waves found in nature, and are a good approximation for small amplitudes. They are called linear because they arise as solutions to linear differential equations. Other waves are "nonlinear", and their dispersion relations depend on their amplitude as well as their frequency. These waves are much more difficult to analyze, and are corresponingly more interesting! The most common examples of nonlinear waves are the "breakers" which you see on the ocean shore. We will focus on linear waves in this chapter.

There are two main types of solutions to the wave equation, as we indicated above: travelling and standing waves. Travelling waves have the form

y = A cos ( k x - ω t + δ).

The (dimensionless) argument of the cosine function is called the "phase", and is of the form b (x - c t); we therefore see that it describes a wave which translates to the right (for positive c) in time (recall translation of functions from algebra). Since

cos (a + b) = cos (a) cos (b) - sin (a) sin (b),

we have

y = A [cos (k x - ω t) cos (δ) - sin (k x - ω t) sin (δ)].

δ is called the "phase angle", and effectively allows us to specify the relative "starting point" of the wave at time zero. By experimenting with various values of δ (ie., 0, π / 2, π, 3 π / 2, 2π), we see that we can produce waves which have any given initial value (between - A and A) at time zero (see the "Travelling Waves and Beats Mathematica Notebook).

The other main type of solution to the wave equation is the standing wave, with the form

y = f (x) g (t).

As you might guess, it does not travel (translate in time), but rather oscillates "in place". We will discuss standing waves further in Section C.

Problems

(©1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.)

Title	Wave Kinematics - Measurement, Modelling And Application
Authors	S. J. Buchan, Steedman Science & Engineenng
Conference	Wave Kinematics and Environmental Forces, March 24 - 25, 1993, London, UK
Series	Advances in Underwater Technology and Offshore Engineering. Wave Kinematics and Environmental Forces, Volume 29
ISBN	0-7923-2184-7
Copyright	1993. The Society for Underwater Technology
Language	English Abstract

Measurements of near bottom wave-induced pressures and orthogonal, horizontal velocity components, at several locations around Australia, have demonstrated that directional wave spectra and wave kinematics may be routinely monitored

This presentation briefly addresses the measurement system, discusses various analysis techniques which have been applied to the data, and illustrate subsequent application of the information in numerical wave modelling and determination of marine engineering criteria

Particular applications include studies on shallow water pipeline stability, near shore wave penetration, dredge spoil dispersion and scour

INTRODUCTION

During recent years, recognition of the importance of directional wave information in many facets of coastal and ocean engineering design has been increasing Goda et al (1978) have illustrated the importance of wave directionality for studies of wave diffraction and refraction in harbours, Fornstall et al (1978) have demonstrated the inadequacies of unidirectional wave theories in describing wave kinematics, and Sand et al (1981) have discussed the applications of directional wave information to the hydrodynamics and design of offshore structures and pipelines

Buchan et al (1984) describes the development of a shallow water directional wave recording _system, based on the Sea Data 635-12s Directional Wave Recorder The instrument measures fluctuations in absolute pressure and horizontal current, and records the information internally on magnetic cassette It is usually deployed in a sea bed frame

Following Fornstall et al (1978), data processing involves the modelling of wave kinematics using linear wave theory, allowing the application of spectral analysis techniques The directional distribution of the sea state is described by a "spreading function" If the sea state is composed of a few resolvable, predominant wave trains, the spreading function may be satisfactory approximated by the widely accepted cosine 2s model of Longuet-Higgins et a1 (1963) Such a description of the sea state is based on only three frequency dependent parameters These are E(f), the omni directional energy density, 8(f), the principal direction of propagation of waves of particular frequency f, and s(f), a measure of the sharpness of the spread of wave directions about the pnncipal direction

Because of the directionality or "short-crestedness" of most sea states, application of unidirectional wave theory is known to cause over-estimates of wave-induced currents from surface wave measurements Simultaneous surface wave height and near-bottom wave-induced current measurements have allowed calibration of unidirectional linear wave theory for specific oceanographic environments Consequently surface wave data may be used to compute characteristic near-bottom wave-induced velocities

Since then, the findings and recommendations of this study have been incorporated in a new generation of Wave Radar, and this paper will describe the rationale for the changes, and will describe how they have been implemented Finally, data obtained from the Esso Odin Field using the new version of the Wave Radar will be presented

The statistics of individual wave heights are known to approximate the Rayleigh distribution Analysis of near-bottom wave-induced currents shows that wave-induced currents follow a similar distribution As a result, individual current speeds may be statistically related to characteristic wave-induced velocities

Disusun Oleh:

Arip Nurahman

&

Kawan-kawan

Pendidikan Fisika, FPMIPA. Universitas Pendidikan Indonesia

&

Follower Open Course Ware at MIT-Harvard University, Cambridge. USA.

Semoga Bermanfaat dan Terima Kasih