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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 x being fixed. Contrarily, means to differentiate F with the variable x with the variable t 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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