1

CHAPTER I

INTRODUCTION

1.1 Background

Have you ever watched a fishing boat at sea who are looking for fish? Then the ship

will remain at sea and only moves up and down to follow the movement of the waves, or

when we throw a stone into a pool of still water, a disorder that causes the particles of the

water we provide to vibrate or oscillate against the balanced point. Propagation of vibration

in the water cause waves in a puddle earlier. If we vibrate the end of the rope is stretched,

then the waves will propagate along the string. Rope waves and waves of water are two

common examples of simple waves we see in everyday life.

Wave motion in elastic media is due to the shift of a portion of the elastic medium

normal position. Due to the elastic properties of a medium, which is the source of the

disturbance wave motion is transmitted from one layer to the next layer. In the propagation,

the medium itself is not part moves together overall wave motion, all parts of the medium is

isolated in a limited way. For example, in a water wave an object (small objects) that floats

like a cork shows that the real movement of any part of the water is a little up and down and

backwards and forwards. However, the water waves move continuously along the water.

When these waves reach objects that float then the wave makes objects move the float,

which means that the waves move something (hereinafter referred to as energy) to the

object. This energy can be transmitted along a considerable distance by wave motion, in the

form of kinetic energy and potential energy of the material. Transmission occurs because

wave energy is conducted along a section of material to the next section, which is

characteristic of wave mechanics.

Waves carry energy in the direction of propagation. In the area through which the

vibration waves occur periodically. Wave is one of the best known natural phenomenon by

everyone. Most people do not notice the symptoms or characteristics possessed by the

waves. Small disturbance in the water could cause waves. Waves propagating in an elastic

medium or better known as wave mechanics. To transmit wave mechanics, we need a

material medium. However, we do not need a medium to transmit electromagnetic waves.

The properties of the medium that will define the rate of a wave through the medium, is

enersialnya and elasticity. All material medium, say air, water, and steel, have these

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properties and can transmit mechanical waves. Elastisitasnyalah which cause the restoring

force in any part of the medium is removed from the balanced position; inersialnya which

tells us how the parts were removed from this medium will reach the restoring forces. These

two factors together determine the rate of the wave. To that end, the authors draw up papers

Waves in Elastic Medium to further study the wave that propagates in the medium and its

formulation.

1.2 Formulation of Problem

1. What the meaning of Waves in Medium Elastic ?

2. What the meaning of General Equation of Walking Wave ?

3. What the meaning of Harmonic Wave ?

4. What the meaning of Energy Propagation and Waves Impedance ?

1.3 Purpose

1. Explain about Waves in Medium Elastic

2. Explain about General Equation of Walking Wave

3. Explain about Harmonic Wave

4. Explain about Energy Propagation and Waves Impedance

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CHAPTER II

CONTENT

2.1 Waves in Elastic Medium

Mechanical waves can propagate through the medium, if the medium is elastic. Elastic

understanding here is that if there are external forces, the medium is capable of expanding or

condenses, and after the external force is removed, the medium is able to restore or restore

the situation to normal.

2.1.1 In the spring wave

The mass of the spring element is located at a distance x from a point of reference,

when the waves propagate each unit of mass position changed from its equilibrium point.

Displacement of mass elements in x is expressed by 𝜓(𝑥), displacement element in x- x

with 𝜓(𝑥 − Δx), and displacement of mass elements at x+Δx expressed by𝜓(𝑥 + Δx) like

the picture below.

Gambar 2.1

a. Balance Condition (without waves)

b. No interference (waves)

From the picture above, can be written:

xxxkFL

xxxkFR (2.1)

Through Newton's Second Law, the equation of motion can be expressed as follows:

LR FFdt

xdm

2

2

4

xxxxxkdt

xdm

2

2

2

(2.2)

The first term of the second term on the right-hand side of equation (2.2) is expanded into a

Taylor series around x with Δx ≈ 0.

22

2

2

1x

dx

xdx

dx

xdxxx

22

2

2

1x

dx

xdx

dx

xdxxx

So this equation (2.2) become:

2

22

2

2

dx

xdx

m

k

dt

xd (2.3)

By considering the general equation of the wave

01

2

2

22

2

tvx

x , the wave

propagation speed is:

m

kxv (2.4)

Equation (2.4) can be written

xm

xkv

kv (2.5)

Modulus of elasticity K is a spring constant normalized. For example, there is a long spring

with spring constant k and λ. Spring length changes by Δ𝜆, so:

kF

kF

kF

5

It appears that the strain (strain)

a normalized scale. Modulus of elasticity K is a

constant that depends on the material and shape of the spring, rather than relying on the

length of the spring.

2.1.2 In the Ropes wave

A rope with tension T, one end is moved up and down, so that the wave propagates

rope. When considered part of the rope round Δx yang which is at a distance x, and deviate

so far ψ(x) from the balanced state, as shown below.

Gambar 2.2

Waves in Ropes

When the rope density, the mass per unit length is expressed by 𝜌, then through the

second law of Newton's equations of motion can be expressed with a rope elements:

dx

dxTxxT

dt

dx

2

2

(2.6)

dx

dxTx

dx

dtxT

dt

dx

2

2

(2.7)

xTdx

d

dx

d

dt

dx

2

2

(2.8)

2

2

2

2

dx

dT

dt

d (2.9)

From this equation (2.9) can be written:

Tv (2.10)

x

T

T

Δx

x

y

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2.1.3 Mechanical Wave Propagation

Mechanical wave propagation in the medium can be seen when observing a wave on a

rope. Suppose, chewy rope one end tied to a nail in the wall. If the rope quite taut and then

the other end of the rope is moved up and back down perpendicular to the rope

memanjangnaya once (single disorder), the rope will happen wave (pulse), as shown in

Figure (2.5a).

A piece of white cloth tied around the rope light will move up and back down through

time pulse. After the pulse passes a white cloth back to the balanced position without

experiencing a shift in the longitudinal direction of the rope. If the same end of the rope is

given periodic disturbances in the same shape, the strap-shaped pulses will occur at the

circuit and also referred to as a periodic wave (figure 2.5b). Waves on a string is called a

transverse wave, because the direction of motion of the particles of the medium (rope)

perpendicular to the direction of wave propagation.

In addition to transverse waves, mechanical waves can also be longitudinal waves. In the

longitudinal direction of the wave motion of the particles of the medium parallel to the

direction of wave propagation. In the spring tension may be raised both transverse waves

and longitudinal waves, depending on the direction of interference given in the spring. If the

disturbance is perpendicular to the direction of prolonged spring occurs transverse wave

(figure 2.5c), but if the spring prolonged disturbances will occur in the direction of the

longitudinal wave (figure 2.5d). Sound waves are longitudinal waves also.

To facilitate understanding of the mechanical wave propagation description, please

review the pulse wave propagation in the rope (figure 2.6). For the elastic strap, tension and

friction is negligible, the shape and the rate of pulses propagating on the string is fixed. As a

result of the disturbances given pulse propagating transverse (to the right). Deviation of the

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rope against the balanced position expressed by (as ordinate of each piece strap which is a

function of the direction of propagation x (as abscissa of each string section), as in Figure

2.6a.

Suppose at time t = 0, the deviation of the peak pulse expressed as a function = f(x,t)

respect to the origin O, and at time t after the pulse has shifted so far to the right vt (v pulse

propagation rate is fixed). Make a new axis are also shifted so far v.t (image 2.6b). if x

abscissa point of deviation of the peak pulse to the new origin O , then at time t the function

of the deviation of the peak pulse expressed in x will be equal to the pulse peak deviation of

the function expressed in x at time t = 0, because the shape of the pulse remains, namely

(x,t)=f (x’) respect to the origin O .

From Figure (2.6) shows that the x=x-vt, so that the deviation of the peak pulse current

function t to the origin O is

(x,t)=f(x – vt) (2.25)

It is commonly assumed that the wave function propagates to the right. Deviation seen

at the same propagating wave is a function of position x and time t.

In the same way that the wave propagates to the left, the wave function can be derived

as

(t)=f(x + vt) (2.26)

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2.1.4 General Equation in Walking Waves

To lower the general form of the wave equation, review the following rope waves.

Suppose that the elastic strap, weight and friction force can be ignored and the rope along

the x axis.

Gambar 2.7 Konfigurasi elemen tali dalam proses gerak.

To formulate the equations of motion rope we review the forces acting on a rope

element in general position (no equilibrium), as shown in Figure 2.7. The forces acting on

the element can be expressed as follows:

Fx= T2 cos 2- T1 cos 1 (2.27a)

Fy= T2 sin 2- T1 sin 1 (2.27b)

Fy=(T2 cos 2) tan 2- (T1 cos 1 )tan 1 (2.28)

with T1 and T2 states tension force acting on the two ends of the rope elements.

Because elements of this rope is not moving in the direction of x, then ax = 0, and

according to Newton's second law, equation (2.28) will produce

T2 cos 2 = T1 cos 1 = To (2.29a)

To is the style with tension straps in the equilibrium position. By defining f (x) is a style of

unity length, then the equation (2:28) can be expressed as

f(x)dx = To(tan2 − tan1)

xdxxx

T

0 (2.29b)

then the description of the Taylor series around x:

9

.......!2 2

22

xxxdxxxx

dx

xxdx

x

(2.30)

third row so the tribe can be ignored, then equation (2:29) becomes: (dx)2 1 third row so

the tribe can be ignored, then equation (2:29) becomes:

f(x)=

2

2

xTo

(2.31)

Furthermore, based on Newton's second law, can be written

f(x)dx=(dm)a=2

2

)(t

dx

so

f(x) =2

2

t

(2.32)

Match the equation (2.31) and (2.32) over immediate results:

This equation is commonly written in standard form for the one-dimensional wave:

with

oTv (2.35)

Although derived for transverse waves of string, wave equation equation (2:34) above

applies in general to all kinds of one-dimensional wave, either transverse or longitudinal,

and does not depend on the type medium. The effect of wave type and the type of medium

is only found in the statement and v.

2.2 Walking Wave Equation Solutions and Characteristics

Waves that propagate to the right or the left in general satisfy the equation

(x,t)=f(xvt) (2.36)

which is essentially a solution of equation (2:34). And to prove the statement in

equation (2.36), the write x-vt

(2.37)

so

fx

ff

ff

f

xx

f

2

2

and,

(2.38)

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so

fvt

ffv

f

tt

f

2

2

2

dan,

(2.39)

Based on equations (2:38) and (2:39) clearly f (x, t) satisfies the equation (2:34), ie

2

2

x

f

-

2

2

2

1

t

f

v

=0

so also it can be proved that f (x + vt) also satisfy the same wave equation. So the

complete solution wave equation (2:34), the general shape

(x,t)=f(x-vt) + g(x+vt) (2.40)

so also it can be proved that f (x + vt) also satisfy the same wave equation. So the

complete solution wave equation (2.34), the general shape

Review the first term of equation (2.40) more thoroughly, ie f (x-vt). If you want to

follow a fixed part (hereinafter referred to as phase) of the wave with time t, then the

equation we consider a fixed value of (Say, the peak pulse described above).

Mathematically this means that we are reviewing how x varies with t if (x-vt) has a fixed

value. We immediately see when t greater then x must be greater to maintain (x-vt) remains.

Thus, the equation (x,t)=f(x-vt) declare a wave turns to the right (x greater with increasing

time t). While the second term of equation (2.40), g (x + vt), stating wave to the left (x gets

smaller with increasing time t).

Phase velocity of waves obtained easily. For waves that propagate to the right, obtained

x-vt=constant (2.41)

Differentiation with respect to time will give

v dt

dx vatau0v

dt

dx (2.42)

The above description is generic, applicable to oscillatory motion in any direction. In

other words, the shape of the wave equation and v in the above interpretation applies for

transverse waves (oscillatory motion perpendicular to the direction of wave propagation

as a rope), as well as for longitudinal waves (oscillatory motion parallel to the direction

of penjalarannya, like waves in the air).

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2.3 Harmonic Wave

The simplest solution for the wave equation is a harmonic wave, which is the source of

interference in the form of motion in harmony. For example, if one end of a spring is hung

on a place, while at the other end of the suspended load (look for the pictured below).

If the load is then given deviation from the balanced position and then released, the

load will move up and down because of a simple harmonic motion. If the load is connected

undergo simple harmonic motion rope taut enough, then the burden will be a source of

interference on the ropes, and consequently will form sinusoidal wave

The wave function of a sinusoidal wave moving in the direction of the positive x with speed

v, the shape is expressed as

vtxkvtxftx sin, 0

Or we can write the equation as follows:

tkxtx sin, 0

Where 𝜔 = 𝑘𝑣 , 𝑘 = 2𝜋𝜆⁄ , 𝜔 = 2𝜋𝑣 , and 𝑣 = 1

𝑇⁄

So the equation above for tkxtx sin, 0 we can write as the equation below:

𝜓(𝑥,𝑡) = 𝜓0𝑠𝑖𝑛 [2𝜋 (𝑥

𝜆−

1

𝜆)]

2.4 Energy Propagation and Wave Impedance

2.4.1 Energy flow

Motion of wave is a propagation local disturbance events. Interference at a certain

point means inputs energy at the point of the medium. Therefore, the motion of the wave

energy propagation is also a process that starts from the location of the original disturbance.

For the formulation of concrete, re-review the sinusoidal waves that propagate to the right

a

b

v

m

m

Equalibrium

state

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on the ropes due to a sinusoidal force at the left end of the rope. Furthermore, take an

arbitrary point Q as the figure below:

For medium to the right of Q, the energy input is received in the form of work done by

the tension force in the rope to his left. Given that the motion of a point Q only take place in

the vertical direction, the effective driving force in question is expressed by:

xTF

TF

TF

TF

0

0 tan

tancos

sin

Instantaneous power (energy flow) is given to the right of the rope is:

tFP

0

Wave (x-vt) that receives power from the left and propagated to the right will satisfy the

equation

tvt

1

With this, the amount of energy propagation is expressed by the equation:

2

0

2

0

xvT

tv

TP

2.4.2 Wave Impedance

If there is a wave that propagates on the ropes and getting interference outside force,

then the response against external disturbance force F, the rope will acquire the velocity of

the local oscillation. For the medium is resistive, the response is linear magnitude of

response to a specific external force F depends on the characteristics of the medium

constant, Z (Z is called the wave impedance). According to equation

θ

Q Q v

x X

(x,t)

13

Z

F

t

With F satisfying the equation for waves propagating to the right. Substituting F into

the price of the above equation is obtained impedance wave propagating to the right,

t

x

oTZ

Value of Z is determined by the characteristics of the wave and the medium in

question. It is clear that for a particular style, speed oscillation caused by the disorder is

inversely proportional to Z. For waves on a string, equation above can be rewritten as

follows:

v

TZ o

Equation above can be analogous to Ohm's law, R = V / I, so by analogy is then

referred to as the wave impedance Z.

14

CHAPTER III

CLOSING

3.1 Conclusion

From the above discussion, it can be drawn some conclusions as follows:

Elastic medium is a medium that if there are external forces, the medium is capable of

expanding or condenses, and after the external force is removed, the medium is able to

restore or restore the situation to normal. Mechanical wave propagation in the medium

can be seen when observing a wave on a rope. Wave function can be derived as (t)=f(x

+ vt). General wave equation, equation which states that the rate of waves commonly

written in standard form for the one-dimensional wave

oTv

.

Wave equation solution

and its characteristics, as well as the shape of the wave equation v above interpretation

applies for transverse waves (oscillatory motion perpendicular to the direction of wave

propagation as a rope), as well as for longitudinal waves (oscillatory motion parallel to

the direction of penjalarannya, like waves in the air) Harmonic wave, which is the source

of interference in the form of motion in harmony. For example if sebauh per (spring)

hanging on one end somewhere, while on the other end hung load. Each happened wave

energy propagation will occur starting from the location of the original disturbance while

the wave impedance (Z) is determined by the characteristics of the wave and the medium

is concerned, to a certain style, speed oscillation caused by the disorder is inversely

proportional to Z.