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Classical Quantization

None of the foregoing discussion allows us to uniquely specify any wavelike solution to the WAVE EQUATION, because nowhere have we given any BOUNDARY CONDITIONS forcing the wave to have any particular behaviour at any particular point. This is not a problem for the general phenomenology discussed so far, but if you want to actually describe one particular wave you have to know this stuff.

Figure: The first three allowed standing waves in a "closed box" (e.g. on a string with fixed ends).

Boundary conditions are probably easiest to illustrate with the system of a taut string of length  $L$  with fixed ends, as shown in Fig. 14.4.^14.6Fixing the ends forces the wave function  $A(x,t)$  to have nodes (positions where the amplitude is always zero) at those positions. This immediately rules out traveling waves and restricts the simple sinusoidal "modes" to standing waves for which  $L$  is an integer number of half-wavelengths:^14.7

$$\lambda_n \; = \; {2 L \over n} , \quad n = 1,2,3,\cdots$$ (14.19)

Assuming that   $c = \omega/k = \lambda \nu =$ const, the frequency  $\nu$  [in cycles per second or Hertz (Hz)] of the $n^{\rm th}$ mode is given by   $\nu_n = c/\lambda_n$  or

$$\nu_n \; = \; n \, {c \over 2L} , \quad n = 1,2,3,\cdots$$ (14.20)

For a string of linear mass density  $\mu$  under tension  $F$  we can use Eq. (16 to write what one might frivolously describe as THE GUITAR TUNER'S EQUATION:

$$\nu_n \; = \; {n \over 2L} \, \sqrt{F \over \mu} , \quad n = 1,2,3,\cdots$$ (14.21)

Note that a given string of a given length $L$ under a given tension $F$ has in principle an infinite number of modes (resonant frequencies); the guitarist can choose which modes to excite by plucking the string at the position of an antinode (position of maximum amplitude) for the desired mode(s). For the first few modes these antinodes are at quite different places, as evident from Fig. 14.4. As another "exercise for the student" try deducing the relationship between modes with a common antinode - these will all be excited as "harmonics" when the string is plucked at that position.

Exactly the same formulae apply to sound waves in organ pipes if they are closed at both ends. An organ pipe open at one end must however have an antinode at that end; this leads to a slightly different scheme for enumerating modes, but one that you can easily deduce by a similar sequence of logic.

This sort of restriction of the allowed modes of a system to a discrete set of values is known as QUANTIZATION. However, most people are not accustomed to using that term to describe macroscopic classical systems like taut strings; we have a tendency to think of quantization as something that only happens in QUANTUM MECHANICS. In reality, quantization is an ubiquitous phenomenon wherever wave motion runs up against fixed boundary conditions.

Footnotes

. . . fig:StandWaves.^14.6

The Figure could also describe standing sound waves in an organ pipe closed at both ends, or the electric field strength in a resonant cavity, or the probability amplitude of an electron confined to a one-dimensional "box" of length $L$.

. . . half-wavelengths:^14.7

Note that the $n^{\rm th}$ mode has $(n-1)$ nodes in addition to the two at the ends.