BELIEVE   ME   NOT!    - -     A   SKEPTICs   GUIDE  

. . . surface.^14.1

Technically speaking, I couldn't have picked a worse example, since water waves do not behave like our idealized example - a cork in the water does not move straight up and down as a wave passes, but rather in a vertical circle. Nevertheless I will use the example for illustration because it is the most familiar sort of easily visualized wave for most people and you have to watch closely to notice the difference anyway!

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. . . SHM).^14.2

Note that ${\displaystyle e^{+i \omega t}}$ would have worked just as well, since the real part is the same as for ${\displaystyle e^{-i \omega t}}$. The choice of sign does matter, however, when we write down the combined time and space dependence in Eq. (4), which see.

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. . . vector;^14.3

The name ``wave vector'' is both apt and inadequate - apt because the term vector explicitly reminds us that its direction defines the direction of propagation of the wave; inadequate because the essential inverse relationship between k and the wavelength $\lambda$ [see Eq. (1)] is not suggested by the name. Too bad. It is at least a little more descriptive than the name given to the magnitude k of $\Vec{k}$, namely the ``wavenumber.''

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. . . that^14.4

In general $\Vec{k} \cdot \Vec{r} = x k_x + y k_y + z k_z$. If $\Vec{k} = k \, \Hat{\imath}$ then k_x = k and k_y = k_z = 0, giving $\Vec{k} \cdot \Vec{r} = k \, x$.

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. . . wave^14.5

Don't try this with an electromagnetic wave! The argument shown here is explicitly nonrelativistic, although a more mathematical proof reaches the same conclusion without such restrictions.

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. . . . ^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.

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. . . half-wavelengths:^14.7

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

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. . . have^14.8

I have avoided complex exponentials here to avoid confusion when I get around to calculating the transverse speed of the string element, v_y. The acceleration is the same as for the complex version.

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. . . lives,^14.9

Indeed, we are made of waves, as QUANTUM MECHANICS has taught us!

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. . . shakes.^14.10

Note also that any of s, ds, P or dP can be either positive or negative; we merely illustrate the math using an example in which they are all positive.

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. . . correct.^14.11

I should probably show you a few wrong guesses first, just to avoid giving the false impression that we always guess right the first time in Physics; but it would use up a lot of space for little purpose; and besides, ``knowing the answer'' is always the most powerful problem-solving technique!

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. . . outgoing^14.12

One can also have ``incoming'' spherical waves, for which Eq. (38) becomes

$$A(x,t) \; = \; A_0 \; { e^{i(k r + \omega t)} \over r } .$$

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. . . Cartesian^14.13

The LAPLACIAN operator can also be represented in other coordinate systems such as spherical ( $r,\theta,\phi$) or cylindrical ( $\rho,\theta,z$) coordinates, but I won't get carried away here.

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. . . as^14.14

The LAPLACIAN operator can also be thought of as the inner (scalar or ``dot'') product of the GRADIENT operator   $\Vec{\nabla}$  with itself:   $\nabla^2 = \Vec{\nabla} \cdot \Vec{\nabla}$,  where

$$\Vec{\nabla} \; = \; \Hat{\imath} {\partial \over \partial x} . . . . . . over \partial y} \; + \; \Hat{k} {\partial \over \partial z}$$

in Cartesian coordinates. This VECTOR CALCULUS stuff is really elegant - you should check it out sometime - but it is usually regarded to be beyond the scope of an introductory presentation like this.

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. . . frequencies.^14.15

If the wavelength $\lambda$ increases (so that the wavenumber $k = 2\pi/\lambda$ decreases), then the frequency $\omega$ must decrease to match, since the ratio $\omega/k$ must always be equal to the same propagation velocity c.

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. . . on.^14.16

One can detect a history of proponents of different bands claiming ever higher (and therefore presumably ``better'') frequency ranges . . . .

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. . . particles.^14.17

He was actually correct, but it is equally true that light consists of waves. If you are hoping that these apparently contradictory statements will be reconciled with common sense by the Chapter on QUANTUM MECHANICS, you are in for a disappointment. Common sense will have to be beaten into submission by the utterly implausible facts.

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. . . slowly.^14.18

Boy, is this ever Aristotelian!

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. . . LAW:^14.19

SNELL'S LAW is normally expressed in terms of the INDEX OF REFRACTION n in each medium:

$$n \; \sin(\theta) \; = \; n' \; \sin(\theta') ,$$

where (we now know) the INDEX OF REFRACTION is the ratio of the speed of light in vacuum to the speed of light in the medium:

$$n \; \equiv \; {c_0 \over c} .$$

The reason for inventing such a semicircular definition was that when Willebrord Snell discovered this empirical relationship in 1621 he had no idea what n was, only that every medium had its own special value of n. (This is typical of anything that gets the name ``index.'') I see no pedagogical reason to even define the dumb thing.

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. . . continuous^14.20

A ``crest'' doesn't turn into a ``trough'' just because the propagation velocity changes!

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. . . origin.^14.21

This is sort of like the mathematical assertion that all lines look straight if we look at them through a powerful enough microscope.

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. . . light.^14.22

``Corpuscles'' are hypothetical particles of light that follow trajectories Newton called ``rays,'' thus starting a long tradition of naming every new form or radiation a ``ray.''

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