WAVE AND OPTICS PAPERWAVE FUNCTION IN A MEMBRANE, CAVITY, AND HALL

Arrange to fill the Wave and Optics task

Arranged by :

Ardiana Hanatan K2310013

(Physics Education

SBI 2010)

SEBELAS MARET UNIVERSITY

SURAKARTA

2012

WAVE FUNCTION IN A MEMBRANE, CAVITY, AND HALL

1. Wave function in a membrane

The wave function in a membrane is include in two

dimensional wave function. Two-dimensional wave

propagation is a membrane vibrations or waves on the

surface of the water. In this case the rectangular

membrane. In order to solve the problem of the vibrating

membrane or vibrate, it must determine the solution Ψ (x,

y, t) at the point x, y and time t ³ 0.

In this case it will apply the following assumptions :

1. Membrane mass per unit area is constant (homogeneous

membrane). Perfect and flexible membrane so thin that

no resistance against bending.

2. The membrane was stretched and then set limits on the

xy plane. Voltage per unit length T caused by

stretching the membrane in all points and in all

directions are the same, and did not change during the

motion.

3. Deflection Ψ (x, y, t) during the motion of the

membrane is relatively small compared to the size of

the membrane, and all of the angle of inclination is

small.

To lower the differential equations that govern the

motion of this membrane, visible forces acting on the

membrane. Because the membrane deflection and angle

inklinasinya small, then the voltage is tangential or

offensive membranes. So that the forces acting on the

sides of the small part is equal to T and T Dx Dy.

In accordance with the above picture, in which the

vertical component of the force is along the side

parallel to the plane is:

and,

To obtain the wave equation in the form of partial

differential equations used Newton's second law is the

resultant force acting on an object is proportional to

mass and acceleration, which is calculated at a point

between y and y + Dy where r is the mass of extensive

membrane unity, while Dx and Dy is the widest part of the

membrane. So according to Newton's second law:

or

Then the second resultant vertical component is :

where y1 and has a value between and. Similarly, the

horizontal component of the resultant force acting on the

other two sides :

and,

According to Newton's second law is the resultant

force acting on an object is proportional to mass

and acceleration are calculated at a point

between x and x + Dx, where r is the union of broad

2.

2.

2.

2.

masses of the membrane, while Dx and Dy is the widest

part of the membrane. So according to Newton's second law

is :

or,

Then the horizontal components of the resultant force

acting on the other two sides are :

Where and lies between x and x + Dx. According to

Newton's second law, the number of the forces is given in

equation (2.4) and (2.8) together with mass and

acceleration , then:

2.

2.

2.

Equation (3.26) is a two-dimensional wave equation or

membrane wave equation in the form of second order

partial differential equations. Since T stated voltage is

always positive and r the density of states which is

positive, then it is taken to indicate that the

constant is always a positive value.

2. Wave function in a cavity

Understanding electromagnetic wave phenomena is very

important to be able to design RF cavities such as for

atmospheric microwave plasma torch, microwave vacuum

oscillator/amplifier, and charged-particle accelerator.

This chapter deals with some electromagnetic wave

equations to show applications to develop the analytic

design formula for the cavity. For the initial and crude

design parameter, equivalent circuit approximation of

radial line cavity has been used. The properties of

resonator, resonant frequency, quality factor, and the

3.2

parallel-electrodes gap distance have been considered as

design parameters.

The rectangular cavity is introduced for atmospheric

microwave plasma torch as a rectangular example, which

has uniform electromagnetic wave distribution to produce

wide area plasma in atmospheric pressure environment. The

annular cavity for klystrode is introduced for a

microwave vacuum oscillator as a circular example, which

adapted the grid structure and the electron beam as an

annular shape which gives high efficiency compared with

conventional klystrode. Some simulation result using the

commercial software such as HFSS and MAGIC is also

introduced for the comparison with the analytical

results.

Microwave circuits are built of resonators connected

by waveguides and coaxial lines rather than of coils and

condensers. Radiation losses are eliminated by the use of

such closed elements and ohmic loss is reduced because of

the large surface areas that are provided for the surface

currents. Radio-frequency energy is stored in the

resonator fields. The linear dimensions of the usual

resonator are of the order of magnitude of the free-space

wavelength corresponding to the frequency of excitation.

A simple cavity completely enclosed by metallic walls can

oscillate in any one of an infinite number of field

configurations. The free oscillations are characterized

by an infinite number of resonant frequencies

corresponding to specific field patterns of modes of

oscillation. Among these frequencies there is a smallest

one,

where the free-space wavelength is of the order of

magnitude of the linear dimensions of the cavity, and the

field pattern is unusually simple; for instance, there

are no internal nodes in the electric field and only one

surface node in the magnetic field.

The oscillations of such a cavity are damped by

energy lost to the walls in the form of heat. This heat

comes from the currents circulating in the walls and is

due to the finite conductivity of the metal of the walls.

The total energy of the oscillations is the integral over

the volume of the cavity of the energy density,

,where E and H are the electric and magnetic field

vectors, in volts/meter and ampereturns/ meter,

respectively. The cavity has been assumed to be empty.

The total energy W in a particular mode decreases

exponentially in time according to the expression,

,where and Q is a quality factor of the mode

which is defined by

The fields and currents decrease in time with the factor

,

The principle, or fundamental, mode of oscillation of

such cavity, and the one with the longest free-space

wavelength , has electric and magnetic fields that do not

depend on the azimuthal angle defining the half plane

though both the axis and the point at which the fields

are being considered. In addition, the electric field is

zero only at wall farthest apart from the gap and the

magnetic field is zero only at the gap. In this mode the

magnetic field is everywhere perpendicular to the plane

passing through the axis the electric field lies in that

plane. Lines of magnetic flux form circles about the axis

and lines of electric flux pass from the inner to the

outer surfaces.

In the principle mode of radial-line cavity only Ez ,

and Hz are different from zero and these quantities are

independent of φ . (see Fig. 2 for cylindrical

coordinates and dimensions of the cavity). The magnetic

field automatically satisfies the conditions of having no

normal component at the walls.

The cavities in which RF interaction phenomena

happens with charged particles almost always have a

narrow gap, that is, the depth of the gap d (see Fig. 2),

is small compared with the radius r0 of the post.If the

radius of the post is much less than onequarter of the

wavelength, and if the rest of the cavity is not small,

the electric field in the gap is relatively strong and

approximately uniform over the gap. It is directed

parallel to the axis and falls off only slightly as the

edge of the gap is approached. On the other hand, the

magnetic field increases from zero at the center of the

gap in such a manner that it is nearly linear with the

radius. In a radial-line cavity the electric field

outside the gap tends to remain parallel to the axis,

aside from some distortion of the field that is caused by

fringing near the gap; it is weaker than in the gap and

tends to become zero as the outer circular wall is

approached.

The magnetic field, on the other hand, increases from

its value at the edge of the gap and has its maximum

value at the outer circular wall. It is seen that,

whereas the gap is a region of very large electric field

and small magnetic field, the reentrant portion of the

cavity is a region of large magnetic field and small

electric field. The gap is the capacitive region of the

circuit, and the reenetrant portion is the inductive

region. Charge flows from the inner to the outer

conducting surface of the gap by passing along the inner

wall, across the outer end. The current links the

magnetic flux and the magnetic flux links the current, as

required by the laws of Faraday, Biot and Savart.

a. The capacitance in Cavity

b. The inductance in cavity

c. The resonance frequency in cavity

3. Wave function in a hall

The wave function in a hall is same with the three

dimensional wave mechanics. However, the generalization

to a particle moving in three dimensions is fairly

straightforward. A massive particle moving in three

dimensions has a complex wavefunction of the form :

(84

4)

where is a complex constant, and . Here, the

wavevector, k, and the angular frequency, , are related

to the particle momentum, P, and energy, T :

(84

5)

(84

6)

respectively, the three-dimensional version of

Schrödinger's equation is easily shown to take the form :

(84

7)

where the differential operator

(84

8)

is known as the Laplacian. The interpretation of a three-

dimensional wavefunction is that the probability of

finding the particle between x and x+dx, between y and

y+dy, and between z and z+dz, at time :

(84

9)

Moreover, the normalization condition for the

wavefunction becomes :

(85

0)

Incidentally, it is easily demonstrated that

Schrödinger's equation, preserves the normalization

condition, of a localized wavefunction. Heisenberg's

uncertainty principle generalizes to :

(85

1)

(85

2)

(85

3)

Finally, a stationary state of energy is written as :

(85

4)

Where the stationary wavefunction, , satisfies :

(85

5)

As an example of a three-dimensional problem in wave mechanics,

consider a particle trapped in a square potential well of

infinite depth which is such that

(85

)

Within the well, the stationary wavefunction, ,

satisfies

(85

7)

subject to the boundary conditions

(85

8)

and

(85

9)

Since outside the well. Let us try a seperable

wavefunction of the form

(86

0)This expression automatically satisfies the boundary

conditions. The remaining boundary conditions, aresatisfied provided

(86

1)(86

2)(86

3)Where nx, ny, and nz are (independent) positive integers.

Substitution of the wavefunction :

(86

4)

Thus, it follows from Equations that the particle energy

is quantized, and that the allowed energy levels are

(86

5)

The properly normalized stationary wavefunctions

corresponding to these energy levels are

As is the case for a particle trapped in a one-dimensional

potential well, the lowest energy level for a particle trapped

in a three-dimensional well is not zero, but rather

(86

7)

Here,

(86

8)

is the ground state (i.e., the lowest energy state) energy in

the one-dimensional case.

Now, it is clear, that distinct permutations of nx,

ny, and nz which do not alter the value of nx2+ny

2+nz2 also

do not alter the energy. In other words, in three

dimensions it is possible for distinct wavefunctions to

be associated with the same energy level. In this

situation, the energy level is said to be degenerate. The

ground state energy level, 3E1, is non-degenerate, since

the only combination of (nx, ny, nz) which gives this

energy is (1,1,1). However, the next highest energy

level, 6E1, is degenerate, since it is obtained when (nx,

ny, nz) take the values (2,1,1), or (1,2,1) or (1,1,2). In

fact, it is not difficult to see that a non-degenerate

energy level corresponds to a case where the three mode

numbers (i.e., nx, ny, nz) all have the same value, whereas a

three-fold degenerate energy level corresponds to a case

where only two of the mode numbers have the same value,

and, finally, a six-fold degenerate energy level

corresponds to a case where the mode numbers are all

different.