SUPERVISOR DECLARATION

“I hereby declare that I have read this thesis and in my opinion this report is sufficient in

terms of scope and quality for the award of the degree of

Bachelor of Mechanical Engineering (structure and material)”

Signature : ...................................................

Supervisor : …………………………………

Date : …………………………………

Signature : ...................................................

2nd Supervisor : …………………………………

Date : …………………………………

Projek Sarjana Muda ii

DECLARATION

“I hereby declare that the work in this report is my own except for summaries and

quotations which have been duly acknowledged.”

Signature : ..............................................

Author : ………………………………..

Date : ………………………………..

Projek Sarjana Muda iii

FOR MY BELOVED PARENTS,

SIBLINGS AND MY FREINDS

Projek Sarjana Muda iv

ACKNOWLEDGEMENTS

Assalamualaikum w.b.t

On this occasion, I wish to express, and thanks to my supervisor, Dr. Ruztamreen

b. Jenal, He has served me for help in completing the first undergraduate. Not forgetting

my parents and friends who helped me a lot in the completion of a Bachelor Degree

Project report. Without them I might not be fully complete these papers. Their service is

really huge to me.

Not forgetting the other parties involved, either directly or indirectly. Only God

alone can give back to them. With this, once again I extend my thank-you very much.

Wassalam.

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ABSTRACT

Non linear acoustics is one of the analytical methods used for the purpose of

reviewing whether a specimen has a disability or not. Apart from studying the specimen,

it also aims to analyze the defects that occur in the specimen. These methods include the

use or transfer of a function called "transfer function". Transfer function is used to

generate a graph and the graph, we can make comparisons when there is a defect in

material or not.

This relationship involves the material elasticity and nonlinear acoustic effects.

Elasticity of materials depends on the properties of young modulus of plate such as

aluminum which has elasticity 72400 MPa. Aluminum is easily established, cheap and

readily available. In addition, aluminum is a medium elasticity causes aluminum are

choose as a specimens selected experiments.

However, in this investigation, we used ABAQUS software is a software technology

without involving the destruction of the plate and analysis Finite Element Method

(FEM). These methods include the investigation of the properties of materials

(Aluminum). As such we can see the properties of aluminum that has a density of 2780

kg / m³ and the Poisson ratio of 0.33. From these characteristics, we can see the changes

that occur in the specimens, especially in the frequency. This is because the main focus

is the frequency.

Plates that do not have a disability will produce the same output frequency to input

frequency. This will happen because of reflection back to the plate when no valid target

out. Therefore, this method is very good because it can be reviewed in detail in the event

of a defect in the plate. However, this method requires high expertise because it requires

analysts with knowledge to make the analysis of published data output. Of the results

obtained showed that contrary to graph theory. Graph obtained is a not straight lined.

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ABSTRAK

Non linear akustik adalah salah satu kaedah analisis yang digunapakai untuk tujuan

mengkaji spesimen samada mempunyai kecacatan ataupun tidak. Selain daripada mengkaji

spesimen, ia juga bertujuan menganalisis tahap kecacatan yang berlaku pada specimen. Kaedah

ini meliputi penggunaan persamaan fungsi pemindahan atau dikenali “transfer function”. Fungsi

pemindahan digunakan untuk menghasilkan graf dan daripada graf tersebut , kita akan dapat

membuat perbandingan apabila wujudnya kecacatan pada bahan ataupun tidak.

Hubungan ini melibatkan kekenyalan bahan dengan kesan non linear akustik. Kekenyalan

bahan bergantung kepada sifat modulus young bahan tersebut contohnya aluminium yang

mempunyai kekenyalan 72400 Mpa. Aluminium bersifat mudah dibentuk, murah dan mudah

didapati. Selain itu kerana sifat kekenyalannya yang sederhana menyebabkan alluminium dipilih

sabagai spesimen ujikaji.

Walaubagaimanapun, dalam penyiasatan ini, kita menggunakan perisian ABAQUS iaitu satu

perisian menggunakan teknologi tanpa melibatkan kemusnahan plat dan melibatkan analisis

unsur tidak terhingga. Kaedah ini meliputi penyiasatan melalui sifat-sifat bahan (alluminium)

tersebut. Sepertimana kita dapat melihat sifat aluminum yang mempunyai ketumpatan 2780

kg/m³ dan nisbah poisson 0.33. Daripada ciri-ciri tersebut, maka kita boleh melihat perubahan

yang berlaku pada spesimen terutama melalui frekuensinya. Ini kerana fokus utama adalah

kepada frekuensi.

Plat yang tidak mempunyai kecacatan akan menghasilkan frekuensi output yang sama

dengan frekuensi input. Ini kerana akan berlaku pantulan balik pada plat apabila tiada sasaran

keluar berlaku. Oleh itu, kaedah ini adalah sangat bagus kerana ia dapat mengkaji secara

terperinci apabila berlaku kecacatan pada plat. Walaubagaimanapun, kaedah ini memerlukan

kepakaran yang tinggi kerana ia memerlukan penganalisis yang berilmu untuk membuat analisis

semula terhadap data output yang diterbitkan. Daripada keputusan yang diperolehi menunjukkan

graph yang diperolehi bercanggah dengan teori. Graph yang diperolehi tidak bergaris lurus.

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TABLE OF CONTENTS

CHAPTER CONTENTS PAGES

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

CONTENTS vii

LIST OF FIGURE ix

LIST OF TABLE xii

NOMENCLATURE xii

APPENDIXS

1.1 Gantt Chart PSM 1

1.2 Gantt Chart PSM II

1.3 Result Frequency uncracked (1-6)

1.4 Result Frequency uncracked (932-939)

1.5 Result Frequency cracked (1-6)

1.6 Transfer Function Matrix

1.7 R Value Analysis

CHAPTER 1 INTRODUCTION

1.1 Background 1

1.2 Objective 3

1.3 Outline of the Thesis 4

1.4 Problem Statement 4

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CHAPTER 2 DAMAGE DETECTION BY USING NONLINEAR

ACCOUSTIC EFFECT

2.1 Introduction 5

2.2 Crack Detection Method 7

2.2.1 Analytical Modeling Method 7

2.2.2 Finite Element Method (Fem) 9

2.3 Non-Linear Acoustics 11

2.3.1 Surface Acoustic Waves 11

2.3.2 Nonlinear Acoustic Mechanisms 13

2.3.3 Non Linear Elasticity 13

2.4 Harmonics Generation 15

2.5 Sidebands Generation And Amplitude

Modulation

15

2.6 Summary 16

CHAPTER 3 FINITE ELEMENT ANALYSIS

3.1 Flow Analysis 17

3.2 Modal Analysis 19

3.3 Modal Analysis Using Finiti Element

Method (FEM) 19

3.4 Uncracked Plate 20

3.5 Cracked Plate 23

3.6 Frequency Response Matrix *H(ώ)+ 26

3.7 Model Diagram 27

3.8 Modulus Intensity ( R Value) 28

CHAPTER 4 MODAL ANALYSIS OF ALUMINUM

4.1 Finite Element Analysis (FEA) Result 29

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4.2 Mode Shape 29

4.2.1 Mode Shape Uncracked Results 30

4.2.2 Mode Shape Cracked Stiffness 34

4.3 Natural Frequency Response Function

For Plates With Several Of Stiffness Value

At Center Area

37

4.4 Transfer Function Result 39

CHAPTER 5 RESULT ANALYSIS

5.1 Analysis Results 41

5.2 Optimizing Of Frequency Range 41

5.3 Natural Frequency Shifting 43

5.4 Modulation Intensity (R Value) Against

Stiffness

45

5.5 Conclusion 47

CHAPTER 6 CONCLUSION AND RECOMMENDATIONS

6.1 SUMMARY OF RESEARCH 48

6.2 CONCLUSIONS 49

6.3 RECOMMENDATIONS 50

CHAPTER 7 REFERENCES

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LIST OF FIGURE

NO TITLE PAGES

Figure 2.1 : specimen with impact damage area 6

Figure 2.2 : step simulation to analysis of specimen 7

Figure 2.3 receptance analysis 9

Figure2.4 : A cracked beam is decomposed into three plain beams

where the cracked section is represented by a short reduced

cross section beam

10

Figure 2.5 : (a) Finite Element Method 10

(b) Average of R value against crack size with various

damping factors

11

Figure 2.6 : Particles move in ellipses in a surface acoustic wave. The

amplitudes decrease exponentially with depth

12

Figure 2.7 : stress-strain curve for sandstone of nonlinear and linear

elasticity

14

Figure 2.8 : stress-strain curve for the glass characteristics of nonlinear

and linear elasticity

14

Figure 3.1 : procedure of analysis 18

Figure 3.2 : Method to create and analysis part using Finite Element

Analysis 0f ABAQUS software

20

Figure 3.3 : Model Plate for uncracked (150 mm x 400 mm x 2 mm ) 22

Figure 3.4 : Material properties of the aluminum plate model for FE

analysis

23

Figure 3.5 : model aluminum plate 24

Figure 3.6 : selected stiffness first and second plate 25

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Figure 3.7 : dimension of sensor 27

Figure 4.1 : picture uncracked with stiffness, E = 69 Gpa 33

Figure 4.3 : Plate response 38

Figure 5.2 : Graph optimizing frequency range and total quantity 1700 42

Figure 5.4 : result of natural frequency shifting versus stiffness for the

aluminum plate at various vibration modes excitation from

FE method

44

Figure 5.6 : Graph of material stiffness against the R value 46

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LIST OF TABLE

NO TITLE PAGES

Table 4.2 : natural frequency result 37

Table 4.4 : natural frequency from z-direction 39

Table 5.1 : Total number of Frequency range

42

Table 5.3 : Percentage of natural frequency shifting of uncracked and

cracked plate

43

Table 5.5 : Diagram shows the stiffness reduction with the average R

value

45

NOMENCLATURE

Bo Mean value of the modulation signal

Bı Peak-to-peak variation around the modulation

E Modulus of elasticity

fο Fundamental frequency

H i j (w) Transfer function value at frequency between points i and j

wn Natural frequency at mode n

U i/j Mode shape value at location i/j for mode n

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CHAPTER 1

INTRODUCTION

1.1 BACKGROUND

In today's world, engineering is an important area to meet modern times.

Development time is the time of the sophisticated and people can move fast without

any restrictions. As now, researchers may be studying equipment or materials

without destroying the material. In this analysis involves the simulation using

ABAQUS software. ABAQUS is a suite of software applications for Finite

Element Analysis (FEA) and computer-aided engineering.

ABAQUS is used in the automotive, aerospace, and industries products. The

software is popular with academic and research institutions due to the wide material

modeling capability, and the program's ability to be customized. Software ABAQUS

also provides a good collection of multiphysics capabilities, such as coupled

acoustic-structural, piezoelectric, and structural-pore capabilities, making it attractive

for production-level simulations where multiple fields need to be coupled. However,

this research is to use the software to investigate the relationship between material

stiffness with non linear acoustics effect using plates.

Stiffness is a property of a solid body. Stiffness can to classify as the

resistance of an elastic body to deformation by an applied force along a given degree

of freedom (DOF) when a set of loading points and boundary conditions are

prescribed on the elastic body. It is an extensive material property. The stiffness of a

structure is of principal importance in many engineering applications, so the modulus

of elasticity is often one of the primary properties considered when selecting a

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material. A high modulus of elasticity is sought when deflections are undesirable,

while a low modulus of elasticity is required when flexibility is needed.

While, non linear acoustic is to review, evaluate and analyze the relationship

between material stiffness with nonlinear acoustic. In this situation, two different

properties of structure will be placed elasticity properties investigated and different

properties of a material considered as a defect (damage). It is an essential element

either qualitatively or quantitatively to determine the presence of damage to prevent

any consequences that could lead to catastrophe. Nonlinear are used in the method

for crack detection in metallic structures. This method involved frequency

modulation, side bands, mode shape and changes in frequencies. Although all these

effects are crucial for incipient damage detection, their physical explanation is still

not well understood [4]. However, it is generally agreed that the interaction between

the high-frequency acoustical wave and the low-frequency modal excitation is

important for crack detection. The application of this method in real engineering

fields is very limited.

Damage in material can happen by many mechanisms. Damage is one that is

common in structural materials. As such, damage modeling has been a remarkably

active trend in the Engineering community since the 50s, so that it is largely beyond

our scope even to try to review the huge existing literature on this subject. This paper

covers methods of detecting damage (different properties) in the structure and

frequency response. Damage in structures can be defined as changes of material or

geometric properties in a structure that could affect the structure’s performance. In

solid material like metal, Jean Lemaitre (2005) [2] defined damage as the creation

and growth of micro voids or micro cracks that create discontinuities in a

homogeneous material.

Damage detection in structural materials is an important for public safety. It

is required that the neighborhood of the damage is known a priori and part of the

structure under inspection is readily accessible. Different techniques and methods for

damage detection are available in the literature which can be used and classified.

Example methods are visual, acoustic, magnetic field and eddy current techniques.

While for Nondestructive Test/Evaluation (NDT/NDE) methods, widely used NDT

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techniques are dye penetration, magnetic particle, eddy-current, radiography,

ultrasonic and acoustic emission, as summarized by Staszewski et. al. (2004) [3] and

Gdoutus [5].

In this technique the condition of a structure is determined and quantified by

inspecting changes in its global structural characteristics. Simulation is one of the

methods are categorized as Non-Destructive Testing (NDT) which is a method used

to detect or measure defects of a material or system without damaging the material

being tested. This method plays an important role in medical technology, quality

control and shelf life determination of a structure. It is widely used in the

manufacture of petrochemicals, power sources, transportation and civil engineering.

There are a variety of NDT techniques and these techniques should be used together,

depending on the material being tested. Almost all metal or non-metallic materials

tested by NDT techniques. For the sake of safety, reliability and operational life, it is

essential to monitor the health status of structural systems.

1.2 OBJECTIVE

Nonlinear acoustics will be used for damage detection in a structure. The main

focus will be in simulation of Finite Element Analysis (FEA) using ABAQUS

software. The ultimate objective of the research work presented to analyze the

Modulus Intensity (R value) behind these nonlinear acoustic modulations in the

presence of the crack. In order to achieve these objectives that project aims to:

i. To evaluate the effect of stiffness changes in small area that representing

fatigue crack on plate.

ii. To determine frequencies of stiffness versus natural frequency shifting

iii. To study relative and relation of material stiffness versus R value.

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1.3 OUTLINE OF THE THESIS

A brief summary of the thesis is presented in this section. The research work

undertaken consists of two major parts. Firstly, finite element modeling is used to

establish excitation frequencies for nonlinear acoustics. Secondly, analyze result to

perform and to investigate Modulus Intensity (R value) for nonlinear acoustic

modulations in the presence of the stiffness.

Damage detection by using nonlinear acoustic effects are reviewed in a chapter 2.

This chapter indicates the explanation of crack detection method and analytical

modeling method.

Finite Element Analysis (FEA) is described in chapter 3. Finite Element Analysis

focused to the uncrack and crack of the aluminum plate. Analysis of structure used

the ABAQUS software.

Next chapter discussed the modal analysis of the aluminum plate that conducted

to know the mode shape of the structure. This chapter consists of the result uncrack

and cracked plate structure of the mode shape and frequency response function.

Result analysis can to get after modal analysis is being described. Relation

between material stiffness against modulation intensity (R Value) can to look. Last

chapter is a conclusion and recommendation of the thesis. Scope of this study is only

limited the analytical analysis using Finite Element Method with use of ABAQUS

software.

1.4 PROBLEM STATEMENT

The mechanism of the nonlinear acoustic effect is not quite established R.B

Jenal (2010) had assumed that one of the mechanisms is due to the behavior of the

fatigue crack surfaces interaction. From the above hypothesis this study is aimed to

relate the relation of the material stiffness surfaces interaction with the high

frequency vibration.

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CHAPTER 2

DAMAGE DETECTION BY USING NONLINEAR ACOUSTIC EFFECT

2.1 INTRODUCTION

There are many ways that we can use to analyze whether the product can be

safely used or whether it is in good condition or not. Non-destructive testing is the

best method. This is because, it not only saves cost but also the products used for the

analysis can also be used again. An examples of the ways that we commonly as

magnetic particle, dye penetration, eddy current, radiography, ultrasound and others

against summarized by Staszewski et. al. (2004) and Gdoutus. However, the overall

methodology used to test the different deformities.

Magnetic particle is a method of using the principle that a flaw in magnetic

material produces distortion in an induced magnetic field. The method is easy, fast

and economical to apply, but similarly to the dye penetration method it can only be

used to detect cracks or damage near the applied surface.

Dye penetration is a method of applying colour or fluorescent dye onto the

cleaned surface of a component to detect any surface flaws. After applying the dye

onto any surface and applying a post-penetrant material such as chalk, flaws will

appear as coloured lines. It is a fast method for damage detection and can detect

small cracks, but it is only capable of detecting surface flaws.

Eddy-current is a method of using the change of impedance in a coil caused

by the eddy current from a tested conductor surface. A coil with alternating current is

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placed near the conductor surface to induce an eddy current. The sensitivity of the

method to the defect is dependent on the penetration depth of the eddy current into

the conductor and is influenced by the frequency of the alternating current, the

magnetic permeability and electrical conductivity of the conductor, and the geometry

of the coil and conductor. Therefore the method is highly sensitive for defects near

the conductor surface but it is difficult to relate the defect size to the impedance

change and the impedance change is also affected by others factors.

Radiography is the oldest NDT method and uses X- or γ-ray to detect a

defect. The X-ray is transmitted to a tested material and the emerging radiation is

measured. If the material contains defects or variations in its thickness or density, the

emerged radiation intensity will not be uniform. This method is suitable for detecting

volumetric defects. However it is important to know the orientation of the defect

beforehand to get the best effect.

Ultrasound is a method of transmitting ultrasonic waves into a test material.

Any defects and boundaries in the specimen will reflect a pulse wave and the

reflection waves are measured. By using the reflection wave data, the defect size and

location can be estimated. It is a very effective method for detecting defects in most

positions, gives a quick response, is economical, is applicable to thick material, and

is highly portable in-situ. However, it is very difficult to distinguish between cracks

and other types of defect and it has limited application to certain specimen geometry.

We can see an example as follows:

Figure 2-1 : specimen with impact damage area (2)

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The above diagram (figure 2-1) shows an example of the damage that we can

analyze occurred in a surface structure. Damage occurred in the middle. It may be

due to various factors.

2.2 CRACK DETECTION METHOD

Detection of damage in a structure can be done in various ways. Each of the

methods used would have its own advantages and disadvantages. However, damage

detection can to divide two main categories: experimental and analysis. This part will

present analytical modeling examples by previous researchers using nonlinear

acoustics methods for defects like stiffness material.

2.2.1 ANALYTICAL MODELLING METHOD

In the analysis of the structure, a mathematical formula can also be used for

the detection of non-linear effects caused by wave distortion when interacted with

nonlinearity or non-uniformity of its passage medium. Mathematical modeling

method can to use to show non linear wave effect graph.

Figure 2-2: step simulation to analysis of specimen

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That graph can to show via of non linear phenomena of sideband. To make a

structural analysis of ABAQUS software should be used first to obtain the response

frequency and nonlinear wave effect. Figure 2-2 show step simulation of the

specimen. So we can see the relationship between the stiffness of the (R value). From

this, they derived the sideband amplitude ratio over the fundamental frequency f0

amplitude, R value, as:

2 │Ho ( f 0 ) – Hc ( f 0 )│ 2 B1 R = – ––––––––––––––––– = – – ……………....... Eq. 2.1

│Ho ( f 0 ) + Hc ( f 0 )│ B0

Where HO and HC are transfer function values when the crack is fully open and

closed

HO C BO = ––––––– is mean value of the modulation signal 2

HO - HC BO = ––––––– is the peak-to-peak variation around the modulation 2

The transfer functions were computed by using a standard analytical formula written

as

U inU jn

H i j (w) = ∑ ––––––––––––––––––––––––– ………………………………………………….. Eq. 2.2

n wn2 + i wnw / Qn - w2

Where H i j (w) denotes transfer function value at frequency w between points i and j.

U i/j n is mode shape value at location i/j for mode n

wn is natural frequency at mode n

Qn is damping factor at mode n

They then used Equations 2-1 and 2-2 in two modeling methods to quantify the

nonlinear effects. The method is reacceptance finite element method.

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2.2.2 FINITE ELEMENT METHOD (FEM)

The finite element method (FEM) originated from the need for solving

complex elasticity and structural analysis problems. The finite element method

(FEM) (its practical application often known as finite element analysis (FEA)) is a

numerical technique for finding approximate solutions specializations of the

mechanical engineering such as automotive industries commonly use integrated FEM

in design and development of the structure. In a structural simulation, FEM helps

tremendously in producing stiffness and strength visualizations and also in

minimizing weight, materials, and costs.

FEM is also popular method for analyzing structural vibration responses.

With rapid development in computing and software technology, this method has

become more reliable and problem solving has become faster. The results from this

analysis also showed clearly the dependence of R value on the fundamental

frequency and damping factor may also affect sensitivity of the nonlinear acoustics

method. The fundamental frequency resulted from the FEM. It also shows the

sensitivity of the R value.

Figure 2-3: Ratio of first sideband amplitude over fundamental frequency

amplitude, R value, against the ultrasound frequency resulted (R.B Jenal- fatigue

crack detection using nonlinear acoustic – analysis of vibro-acoustic modulations) :

Figure 2-3 : (a) receptance analysis (Duffour)

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Receptance analysis is an analytical method of decomposing a system into a

few sub-systems to analyze a system’s vibration responses. As presented by Duffour

et al., this method was used to model a beam schematically described in Figure 2-4.

The beam was divided into three sections, where the cracked section was represented

by a very short beam (no.2) with a reduced cross-section area compared to the other

two sections (nos 1 and 3). The cross-section area was dependent on the crack depth.

Figure 2-4: A cracked beam is decomposed into three plain beams where the cracked

section is represented by a short reduced cross section beam (Duffour)

First they calculated the natural frequencies and mode shape coefficients of

each beam section by using standard theory for the longitudinal vibration of uniform

bars. The results were substituted into Equation 2-2 and then the results from each

section were assembled by using a coupling formula for linear system in series.

Finally the R values were determined by using Equation 2-1 within a frequency

range.

The results from this analysis showed that the sensitivity of using the

nonlinear acoustic effect for damage detection is extremely dependent on the

fundamental frequency and it should be close to the resonances of the specimen.

Damping factors for the specimen also may affect this method sensitivity.

Figure 2-5 : (a) Finite Element Method (Duffour)

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Figure 2-5 (b) and (c): Ratio of first sideband amplitude over fundamental frequency

amplitude, R value, against the ultrasound frequency resulted ((R.B Jenal- fatigue

crack detection using nonlinear acoustic – analysis of vibro-acoustic modulations) :

Figure 2-5 : (b) Average of R value against crack size with various damping factors

(Duffour)

2.3 NON-LINEAR ACOUSTICS

Non-linear acoustics is a branch of physics dealing with sound waves being

distorted. The amplitude dependence is due to the nonlinear response of the medium

in which the frequency propagates, and not to the nonlinear behavior of the sound

source. According to the linear theory of acoustics, increasing the level of a source

by 10 dB results in precisely the same sound field as before, just 10 dB more intense.

Linear theory also predicts that only frequency components radiated directly by the

source can be present in the sound field.

2.3.1 SURFACE ACOUSTIC WAVES

Surface waves take many forms in nature, science, and technology. They

include ultrasonic surface waves at the interface between a solid on one hand and

vacuum, gas, liquid or another solid on the other. A common feature of all kinds of

surface waves is that most of energy is localized near the surface, within a depth of

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about one wavelength. Instead of propagating thought-out the whole three-

dimensional medium, the energy remains localized at the surface and spreads out

primarily in the two-dimensional (2D) interface region.

Waves illustrated the principal features of elastic surface waves. The particle

motion can be easily visualized when a waves passes a leaf floating on the waves

surface. The leaf moves to and fro, but also up and down around its original position.

The radius of the particle orbit is equal to the wave amplitude (H.Peter - Surface

acoustics waves materials science).

Elastic surface waves, usually called surface acoustic waves (SAWs) were

discovered in 1885 by Lord Rayleigh. SAWs depend on the elastic forces acting

between the constituent atoms. The internal forces of medium, or stress, are assumed

to depend only on the deformation of the material, or strain , measured relative to the

undisturbed state. In the bulk of an elastic material, the longitudinal and transverse

waves modes are independent and propagate with different velocities, but in surface

waves the two modes are coupled. Due to the asymmetry of the elastic forces at the

surface, the motion normal to the surface may be different from that in the direction

of wave propagation along the surface. Consequently, in the elastic medium, the

particle motion is elliptical polarized. The depth dependence of the particle

displacement and polarization are illustrated in figure 2.6.

Figure 2-6: particles move in ellipses in a surface acoustic wave. The amplitudes

decrease exponentially with depth (H.Peter )