Wave Mechanics: Wavenumber

WM04 introduced a spatially periodic profile u(x) and defined wavelength λ as the smallest positive distance over which the pattern repeats:

(1)

For a sinusoidal spatial pattern we wrote

(2)

The factor

(3)

appears so often in wave mechanics that it is given its own symbol. We define

(4)

The quantity k is called the angular wavenumber, or simply wavenumber in much of physics. It measures how rapidly phase changes with distance. In this Wave Mechanics series, the symbol k will always mean this angular spatial rate. This convention is standard in wave equations and sinusoidal wave notation [1, 2, 3].

WM05 is still a spatial-only lesson. No pattern is assumed to move. The translating disturbance F(x ∓ ct) is introduced in WM06, and the full sinusoidal traveling wave appears in WM07.

1 From one wavelength to one angular cycle

In WM02, one temporal cycle corresponded to an angular advance of 2π radians. The same idea applies in space.

If position increases by one wavelength,

(5)

then the spatial pattern advances by one complete phase cycle,

(6)

Therefore the phase advance per unit distance is

(7)

This is precisely the quantity we call k:

(8)

Equivalently,

(9)

This is the spatial counterpart of angular frequency:

(10)

The parallel structure is

This temporal–spatial symmetry is one of the main reasons angular frequency and angular wavenumber are so useful in wave mechanics.

2 Units of wavenumber

Because wavelength has units of length,

(11)

In physical discussion it is often more informative to say

(12)

For example,

(13)

means that phase increases by 8 radians for each meter of increasing x.

Formally, plane angle is a dimensionless quantity in SI and the radian is the coherent unit used for plane angle [4]. Therefore the dimensional unit of k can be written simply as m⁻¹. Keeping the word “radian” in the interpretation is nevertheless useful because it reminds us that k is an angular phase rate, not merely a count of cycles per meter.

3 Spatial phase

The sinusoidal spatial pattern can now be written more compactly as

(14)

The quantity

(15)

is the spatial phase.

The phase constant ϕ has the same meaning introduced in WM02 and WM03: it specifies where in the cycle the pattern begins at the chosen origin x = 0. At x = 0,

(16)

If ϕ = 0, the simple cosine profile begins at a maximum:

(17)

As x increases, the term kx advances the phase.

4 How distance maps into phase

For ϕ = 0, consider positions separated by fractions of one wavelength. Using

(18)

we obtain

x = 0	⇒ kx = 0,	(19)
x =	⇒ kx = ,	(20)
x =	⇒ kx = π,	(21)
x =	⇒ kx = ,	(22)
x = λ	⇒ kx = 2π.	(23)

This mapping is exactly analogous to the temporal relation

(24)

In space,

(25)

5 Short wavelength means large wavenumber

The relation

(26)

shows that k and λ are inversely related.

If wavelength decreases, the pattern completes more phase cycles within the same physical distance, so k increases. If wavelength increases, phase accumulates more slowly with distance, so k decreases.

Thus

(27)

and

(28)

Wavenumber is therefore a useful measure of spatial oscillation density.

6 Worked example 1: wavelength to wavenumber

Suppose

(29)

Then

k	=	(30)
	=	(31)
	≈ 7.85 rad/m.	(32)

Hence

(33)

A useful interpretation is that the spatial phase advances by about 7.85 radians for each meter.

7 Worked example 2: wavenumber to wavelength

Suppose

(34)

Using

(35)

we obtain

λ	=	(36)
	≈ 0.524 m.	(37)

Therefore

(38)

8 Worked example 3: reading a spatial sinusoid

Suppose x is measured in meters and

(39)

Comparing with

(40)

we identify

A	= 0.025,	(41)
k	= 4π rad/m,	(42)
ϕ	= .	(43)

The wavelength is

λ	=	(44)
	=	(45)
	= 0.50 m.	(46)

Thus

(47)

The phase at the origin is π∕6, so the pattern does not begin at the same point in its cycle as a zero-phase cosine.

9 The phase constant produces a spatial shift

Consider

(48)

A cosine maximum occurs whenever the phase equals 2πn. For the maximum nearest the origin, choose a convenient integer n and solve

(49)

This gives

(50)

For the branch with n = 0,

(51)

Thus a positive phase constant shifts the corresponding cosine maximum toward negative x.

This is the spatial counterpart of the time shift discussed in WM03.

10 Angular wavenumber versus reciprocal wavelength

The word wavenumber is not used identically in every field. In this series,

(52)

is the angular wavenumber used in wave mechanics.

A different quantity is the reciprocal wavelength

(53)

It counts spatial cycles per unit length rather than radians of phase per unit length. In spectroscopy, the unqualified term “wavenumber” commonly refers to this reciprocal-wavelength quantity and is often expressed in cm⁻¹ [5].

The two conventions differ by a factor of 2π:

(54)

This is analogous to the distinction between ordinary frequency f and angular frequency ω:

(55)

For clarity, this Wave Mechanics series will consistently use k for angular wavenumber.

11 Dimensional check of the cosine argument

The argument of a trigonometric function must represent a pure phase. In

(56)

we have

(57)

Therefore

(58)

with the phase interpreted in radians. The phase constant ϕ is also an angle, so the sum kx + ϕ is physically meaningful.

This dimensional check is valuable. An expression such as

(59)

would generally be meaningless because k and x have different dimensions.

12 Wavenumber is not wave speed

The symbol k tells us how rapidly a phase pattern changes in space. It does not tell us how rapidly that pattern moves through space.

At this point in the series we still have only

(60)

No time variable appears, so no propagation speed can yet be inferred.

Later, spatial phase kx will be combined with temporal phase ωt in an expression such as

(61)

Only then will the relation between phase evolution in space and time lead to wave speed.

13 Common mistakes

• Mistake: writing k = 1∕λ in a wave-mechanics equation that uses angular phase. In this series, k = 2π∕λ.
• Mistake: treating a larger k as a larger amplitude. Wavenumber controls horizontal spatial repetition; amplitude controls vertical scale.
• Mistake: forgetting that kx must be a phase. If x is in meters, then k must carry inverse-length units.
• Mistake: confusing wavenumber k with angular frequency ω. The former measures phase change per distance; the latter measures phase change per time.
• Mistake: assuming k alone gives a wave speed. A spatial profile without time dependence does not specify propagation.
• Mistake: confusing the symbol k with a spring constant. The same letter is used for different quantities in different contexts; units and equations identify which meaning is intended.

14 Summary

WM05 converts wavelength into an angular spatial rate. The defining relation is

(62)

with inverse relation

(63)

The spatial sinusoid can therefore be written as

(64)

and its spatial phase is

(65)

The central temporal–spatial analogy is

WM06 next replaces a fixed spatial profile with a translating disturbance and shows why functions of the form F(x − ct) and F(x + ct) represent motion in opposite directions.

References