Wave Mechanics: Oscillation in Space

WM01–WM03 described quantities that vary with time at one point. The basic object was

u = u(t).

(1)

We now make a different simplification. Freeze time and ask how a quantity varies from place to place. The basic object becomes

|----------| |u = u (x ).| -----------

(2)

This is a spatial profile or spatial snapshot. It may repeat as x changes, just as a periodic time history repeats as t changes. The distance required for one complete spatial repetition is called the wavelength.

This lesson does not yet describe propagation. A curve that varies with x is not automatically moving. Wavenumber is introduced in WM05, translating disturbances in WM06, and the full sinusoidal traveling wave in WM07.

1 From a time history to a spatial profile

A graph of u(t) answers the question

How does the quantity change as time passes at one fixed place?

A graph of u(x) answers a different question:

How does the quantity change from place to place at one fixed instant?

For a stretched string, for example, imagine taking a photograph at one instant. Each horizontal position x has some transverse displacement u(x). The photograph records the string shape at that instant.

Figure. A spatial snapshot u(x). The horizontal axis is position, not time. The repeated distance between corresponding points is the wavelength λ. Nothing in this picture alone says that the pattern is moving.

The independent variable has changed from time to position. That single change creates a parallel set of ideas:

Temporal description	Spatial description

t	x
u(t)	u(x)
period T	wavelength λ
seconds	meters

This temporal–spatial symmetry becomes one of the organizing ideas of wave mechanics.

2 Spatial periodicity

A time-periodic function satisfies

u(t + T ) = u(t).

(3)

A spatially periodic function obeys the analogous relation

|----------------| u-(x +-λ) =-u(x).-

(4)

The symbol λ is the Greek letter lambda. It denotes a distance in space.

If the pattern is periodic, moving one wavelength to the right brings us to an equivalent location in the repeating pattern.

More generally,

u(x + nλ ) = u(x), n = 0,±1, ±2, ...

(5)

because any integer number of complete spatial cycles returns to the same pattern state.

3 Wavelength

The wavelength is the smallest positive spatial distance over which the complete pattern repeats. We write

|--------------------------------| -λ-=-one-complete--spatial-period.-

(6)

Its SI unit is the meter:

[λ ] = m.

(7)

Depending on scale, useful units can also include centimeters, millimeters, micrometers, or kilometers.

For a sinusoidal-looking pattern, wavelength can be measured between any two corresponding points in adjacent cycles, for example

crest to next crest,
trough to next trough,
upward zero crossing to next upward zero crossing,
any point to the next point having the same displacement and the same local direction of change with x.

The last statement is the spatial counterpart of the period-measurement rule from WM01. Merely finding the same value of u is not enough because most values occur more than once per cycle.

4 Period and wavelength are analogous, not identical

Period and wavelength both measure repetition, but they measure it along different independent variables.

Figure. Temporal and spatial periodicity are mathematically parallel. The upper graph repeats after a time T; the lower graph repeats after a distance λ. The two quantities have different physical dimensions and must not be interchanged.

Period satisfies

u(t + T ) = u(t),

(8)

while wavelength satisfies

u (x + λ) = u(x).

(9)

Their units make the distinction unavoidable:

[T]	= s,	(10)
[λ]	= m.	(11)

A statement such as “the wavelength is 0.5 s” is dimensionally wrong. Likewise, “the period is 2 m” is wrong.

5 Amplitude remains a vertical scale

The meaning of amplitude does not change when we switch from a time graph to a spatial graph. For a profile centered on u = 0,

A = max |u(x)|.

(12)

Amplitude measures the size of the dependent variable. Wavelength measures the horizontal repetition distance.

These are independent features. Two spatial patterns can have the same amplitude but different wavelengths.

Figure. Two periodic spatial profiles with the same amplitude but different wavelengths. The shorter wavelength repeats more frequently in space, but its vertical scale need not be larger or smaller.

Thus a short wavelength does not mean a large amplitude, and a long wavelength does not mean a small amplitude.

6 A simple sinusoidal spatial pattern

A convenient example of a periodic spatial pattern is

( x ) u (x ) = A cos 2π -- . λ

(13)

This equation is useful because increasing x by one wavelength changes the cosine argument by 2π:

u(x + λ)	= A cos	(14)
	= A cos	(15)
	= A cos	(16)
	= u(x).	(17)

So the function has spatial period λ exactly as required.

This form contains the quantity

2π ---, λ

(18)

which measures angular phase change per unit distance. WM05 gives that quantity its own symbol, k, and develops its physical meaning as wavenumber. For now, it is enough to notice that one wavelength corresponds to one angular cycle of 2π.

7 The smallest positive repeat distance

As with time periodicity, a pattern that repeats after λ also repeats after 2λ, 3λ, and so on. Wavelength normally means the smallest positive repeat distance.

Suppose a pattern satisfies

u (x + 0.40 m ) = u(x)

(19)

for every x, and no smaller positive distance has this property. Then

|------------| -λ-=-0.40-m.-|

(20)

The pattern also repeats after 0.80 m and 1.20 m, but those are two and three wavelengths rather than new wavelengths.

Figure. Points separated by one wavelength occupy equivalent locations in a repeating spatial pattern. The marked points have the same displacement and the same local orientation of the curve.

8 Equal displacement does not necessarily mean one wavelength

Consider the sinusoidal spatial profile. The curve can pass through the same value of u twice within one wavelength, once while increasing with x and once while decreasing with x.

Therefore a distance between two equal-displacement points may be only a fraction of a wavelength.

For a reliable measurement of λ, compare corresponding pattern states: crest to crest, trough to trough, or a zero crossing with the same crossing direction.

This is directly analogous to the temporal result from WM01 and WM03: equal displacement does not necessarily identify the same phase state.

9 Worked example 1: reading wavelength from repeated crests

Suppose adjacent crests of a spatial profile occur at

x = 0.35 m, x = 1.10 m. 1 2

(21)

Because adjacent crests are corresponding points in neighboring cycles,

λ	= x₂ − x₁	(22)
	= 1.10 m − 0.35 m	(23)
	= 0.75 m .	(24)

If another crest occurs at 1.85 m, the same spacing is confirmed:

1.85m − 1.10m = 0.75 m.

(25)

10 Worked example 2: counting spatial cycles

A periodic pattern has wavelength

λ = 0.25 m.

(26)

How many complete wavelengths fit in a distance of 2.0 m?

The number of spatial cycles is

L N = --. λ

(27)

Thus

N	=	(28)
	= 8 .	(29)

The meter units cancel, leaving a dimensionless cycle count.

11 What a spatial snapshot does not tell us

A single function u(x) describes shape in space. It does not tell us how that shape changes with time.

From one frozen profile alone we cannot determine whether the pattern

is stationary,
moves to the right,
moves to the left,
changes shape,
or belongs to a standing-wave pattern.

Time dependence requires a second independent variable. Eventually we will write

u = u (x,t).

(30)

But introducing both variables too early hides the separate meanings of period and wavelength. WM04 keeps the spatial idea isolated so that the later combination is easier to interpret.

12 Common mistakes

Mistake: calling a graph of u(x) a time history. Its horizontal axis is position.
Mistake: treating wavelength and period as interchangeable. Wavelength has dimensions of length; period has dimensions of time.
Mistake: measuring between arbitrary equal-displacement points. Corresponding points must represent the same location within the spatial cycle.
Mistake: assuming a shorter wavelength means a larger amplitude. Wavelength and amplitude describe independent geometric features.
Mistake: assuming that a sinusoidal spatial profile is automatically a traveling wave. Propagation requires time dependence.

13 Summary

WM04 replaces the one-point temporal description u(t) with a spatial profile

|----------| -u-=-u-(x-).|

(31)

A spatially periodic function satisfies

|----------------| u-(x +-λ) =-u(x),-

(32)

where λ is the smallest positive repeat distance, or wavelength.

The central temporal–spatial analogy is

Temporal	Spatial

t	x
T	λ
u(t + T) = u(t)	u(x + λ) = u(x)
seconds	meters

A useful sinusoidal example is

( ) x- u (x ) = A cos 2π λ .

(33)

WM05 will turn the spatial angular rate 2π∕λ into a named quantity, wavenumber, and show why spatial phase is naturally measured in radians per meter.

"Wave Mechanics: Oscillation in Space" is owned by bloftin.
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