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1-2: A Brief Review on Wave Motions

Before going

to the main points of

Schroedinger's wave mechanics,

let us briefly review

general wave motions.

[A Simple Wave Motion]

For simplicity,

here we consider

wave motions

in a one-dimensional space.

We can, for instance,

imagine the sounds

(oscillations)

traveling along

the strings of a guitar

or a harp.

The amplitude of

the oscillation

of a string

is a function of position

x and time t

being represented

by F(x,t)

as shown in the

following figure.

In the above figure,

right-going sine-waves

are drawn.

Sine-wave is one

of the simplest wave motions,

which is written as

The solid curve

shows

the wave at

the time t=0

and the dashed curve

that for a time

a little bit later.

Let the wavelength

of these waves be

and the frequency

Then

where

k is called

the wave number

and

the angular frequency.

The propagating speed

of the wave, c,

is given by

[An Ordinary Wave Equation]

It is known well that

the wave equation

which the wave F(x,t)

traveling in a string

obeys can be written

The symbol

denotes partial differentiation,

which means the variation

rate of a function of

more than or equal

to two variables

with respect only

to a specified variable.

For example,

denotes the differentiation

of the function F

with respect to the

variable t

with the variable

being fixed.

Contrarily,

means to differentiate F

with the variable x

with the variable

being fixed.

The constant c

in Eq. (4)

is the speed of

the wave propagation

which depends on the

density of the string

and on the string tension.

Needless to say,

the sine-wave (1)

satisfies the wave equation (4).

Namely, the sine-wave (1)

is a special solution

of Eq. (4).

The above-mentioned

Eq. (4)

is a wave equation for the wave motion

propagating in

a one-dimensional space.

In the case of

a three-dimensional space

like the wave motions in the air

(sound wave) or electromagnetic waves

in a vacuum (light),

the three-dimensional coordinates

(x, y, z)

are needed.

Accordingly,

the wave F

in this case

must be a function

of four variables,

x, y, z

and t,

expressed as

F(x, y, z, t),

for which the ordinary

wave equation is written

where the constant c

is the speed of propagation

of the wave.

[The Principle of Superposition]

Since the wave equation

(4) or (5)

is quite general,

we cannot determine

the form the wave

only by this equation.

If the form of the wave

at the time t = 0

that is called the initial condition

is given,

then the later movement

of the wave would be

determined by

the wave equation

(4) or (5).

These wave equations

have an essential property

indispensable to a wave.

That is the principle of superposition.

Let us explain

this in detail below.

Suppose that a function

F(x,t)

is a wave satisfying

the wave equation (4).

And suppose another function

G(x,t)

to be

a different wave

satisfying the same

wave equation (4).

Namely, we suppose that

Then the sum of these two waves,

F + G,

which is sometimes called

the superposition

of two waves,

also satisfies

the same wave equation

We can therefore say that

the superposition of two or more waves satisfying the same wave equation is also another wave satisfying the same wave equation.

This law is called

the principle of superposition.

We can say in other words that

wave motions satisfy

the principle of superposition.

[Example 1 (Standing wave)]

Both of a right-going

sine-wave

and a left-going sine-wave

satisfy the one-dimensional

wave equation (4).

Accordingly, the superposed one

of these two waves

also satisfies

the same wave equation (4).

This wave proceeds

neither right nor left,

i.e., it is a

standing wave.

The waves in a string

whose both ends are fixed

are of this type.

(See the following figure).

The normal oscillations of a string of length L.

In order that both

of the ends of the string

are zero-points

of oscillation,

the wavelength

should be one of

2L,

2L/2,

2L/3,

. . . .

[Example 2 (Interference)]

Suppose two sine-waves

with different wave numbers

(or wavelengths)

If we

superpose these two waves

by calculating F + G,

they interfere

with each other

to strengthen

at some positions

(constructive interference)

or weaken at other positions

(destructive interference).

These are the interference

phenomena,

an example of which

is illustrated

in the following figure.

The interference is considered

an evidence of wave nature;

i.e.,

the principle of superposition

is certainly satisfied.

Interference of two sine-waves

The two sine-waves,

A and B, have

slightly different wavelengths.

The thick solid curve

in C is

the superposition

of the sine-waves,

A and B.

An interference

(beat) is seen

in the curve C.

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