Wave Mechanics: Wave Speed

WM06 introduced translating disturbances, and WM07 assembled the sinusoidal traveling-wave form

(1)

for propagation toward increasing x. WM08 now asks the quantitative question that was deliberately postponed:

How fast does a recognizable phase of the wave move through space?

The answer connects the temporal and spatial descriptions developed throughout WM01–WM07. For a sinusoidal wave, the propagation speed can be written in three equivalent forms,

(2)

These relations are standard results of elementary wave mechanics [1, 2, 3, 4]. The purpose of this lesson is to derive them from concepts already established in the series, not to introduce them as formulas to memorize.

WM08 is still primarily kinematic. It tells us how wave speed is encoded in k, ω, λ, T, and f. Later lessons will ask a different question: what physical properties of a medium determine that speed?

1 Tracking a point of constant phase

For the right-moving sinusoidal wave

(3)

define the complete phase

(4)

A crest, a trough, or any other corresponding point on successive cycles can be identified by holding the phase fixed. Let

(5)

where 𝜃₀ is constant. Then

(6)

Solving for position gives

kx	= ωt + 𝜃0 − ϕ,	(7)
x	= t + .	(8)

This is the equation of a straight line in an x–t diagram. Its slope is

(9)

Therefore the speed of a constant-phase feature is

(10)

This derivation uses only algebra and the constant-phase idea from WM07. No wave equation is required.

2 Direction and signed propagation velocity

It is useful to distinguish speed, which is nonnegative, from a signed one-dimensional propagation velocity.

For

(11)

we found

(12)

so the phase moves toward positive x. Its signed propagation velocity is

(13)

For the left-moving wave

(14)

constant phase gives

(15)

and therefore

(16)

If c denotes the positive speed magnitude, then in either direction

(17)

for positive k and ω.

Later in the series, dispersive waves will require a more careful distinction between phase velocity and group velocity. For the present one-dimensional sinusoidal wave, the constant-phase speed is the propagation speed we are studying.

3 One wavelength in one period

There is a second, highly physical route to the same result.

Consider a particular crest of a right-moving periodic wave. After one full period T, that crest has advanced by one wavelength λ. Therefore

(18)

The same statement can be verified directly from phase. Suppose two events are separated by

(19)

The change in phase is

Δ𝜃	= kλ − ωT	(20)
	= λ −T	(21)
	= 2π − 2π	(22)
	= 0.	(23)

Thus those two events lie on the same constant-phase track.

4 From period to frequency: deriving c = fλ

WM01 introduced

(24)

Substituting this into

(25)

gives

(26)

This relation can be read in words:

wave speed = cycles per second × distance per cycle.

The dimensional check is immediate:

(27)

OpenStax states the same fundamental relationship as v = λ∕T = λf for traveling waves [3].

5 Showing that ω∕k and fλ are the same quantity

From the definitions developed earlier,

(28)

and

(29)

Therefore

	=	(30)
	= fλ.	(31)

Equivalently, using ω = 2π∕T,

	=	(32)
	= .	(33)

Hence

(34)

6 Units of ω∕k

WM02 gave angular frequency units of radians per second, while WM05 gave angular wavenumber units of radians per meter. Therefore

(35)

The angular measure cancels, leaving the dimensions of speed.

This check is especially useful because ω and k can look abstract. Their ratio has an immediately familiar mechanical unit.

7 Frequency is not wave speed

A common misconception is that a higher-frequency wave must travel faster. The relation

(36)

shows why that conclusion does not follow from frequency alone. If two waves travel at the same speed, a higher frequency must be accompanied by a shorter wavelength:

(37)

For example, suppose the propagation speed is

(38)

Then a 1 Hz wave has

(39)

while a 2 Hz wave has

(40)

Both propagate at the same speed.

In many familiar approximately nondispersive situations, waves of different frequencies travel at nearly the same speed in a fixed medium. Feynman uses sound and Light as examples when introducing the distinction between nondispersive and dispersive propagation [4]. The later Wave Mechanics lessons on dispersion will revisit this point in detail.

8 Wave speed is not the local speed of the medium

WM06 emphasized that propagation of a disturbance does not require each material element to travel with the disturbance. The same distinction remains important here.

For a transverse wave on a string, a marked piece of string may move mostly up and down while a crest travels horizontally along the string. The quantity c describes the speed of the crest or other constant-phase feature, not the instantaneous transverse speed of that marked piece of string.

Thus two different velocities can appear in the same physical problem:

• the wave propagation speed, describing movement of the pattern;
• the local material velocity, describing motion of the medium at one location.

They should not be confused.

9 Kinematics versus dynamics: what determines c?

The relations derived in WM08 are kinematic. They tell us how the observed wave quantities must fit together:

(41)

They do not yet tell us what sets the numerical value of c for a physical system.

That is a dynamical question. For example, the speed of a mechanical wave can depend on properties such as Tension, inertia, stiffness, density, or compressibility. Standard wave treatments distinguish this medium-dependent physics from the kinematic relation among speed, wavelength, and frequency [1, 2, 3, 4].

Later in this series, the one-dimensional string wave equation will be derived from Newton’s law. At that point the speed will emerge from the properties of the string itself. WM08 therefore completes the kinematic foundation without prematurely assuming the governing PDE.

10 Worked example 1: frequency and wavelength

A periodic wave has

(42)

Its speed is

c	= fλ	(43)
	= (3.0 s−1)(0.80 m)	(44)
	= 2.4 m∕s .	(45)

The result says that a constant-phase feature advances 2.4 m each second.

11 Worked example 2: angular frequency and wavenumber

Suppose a right-moving sinusoidal wave has

(46)

Then

c	=	(47)
	=	(48)
	= 3.0 m∕s .	(49)

The radians cancel, leaving the expected units of speed.

We can verify the result using wavelength and frequency:

λ	= = = m,	(50)
f	= = ≈ 1.91 Hz.	(51)

Then

(52)

12 Worked example 3: infer wavelength at fixed speed

A wave travels through a system at

(53)

and has frequency

(54)

From

(55)

we obtain

λ	=	(56)
	=	(57)
	= 3.0 m .	(58)

If the frequency doubled while the propagation speed remained unchanged, the wavelength would be cut in half.

13 Common mistakes

• Mistake: assuming high frequency automatically means high wave speed. Speed depends on the product fλ, not on f alone.
• Mistake: using c = f∕λ. Dimensional analysis immediately rejects this because f∕λ does not have units of speed.
• Mistake: writing c = k∕ω. The correct angular form is c = ω∕k.
• Mistake: forgetting propagation direction. The speed magnitude is positive, but the signed phase velocity is positive for kx − ωt and negative for kx + ωt when k,ω > 0.
• Mistake: confusing the wave’s propagation speed with the local velocity of a material element.
• Mistake: assuming WM08 explains what physical property sets c. The present derivation is kinematic; the medium-dependent dynamics are developed later.

14 What WM08 completes

WM00–WM08 now provide a complete introductory language for one-dimensional traveling waves without beginning from the wave equation.

The central temporal quantities are

(59)

The central spatial quantities are

(60)

The right-moving sinusoidal traveling wave is

(61)

and the spatial and temporal descriptions are connected by

(62)

Every quantity in this equation has now been introduced separately and given a physical interpretation.

15 Connection to the next block

WM08 closes the first production block of the Wave Mechanics series. The next block begins with WM09, where the two-variable field u(x,t) is studied more formally.

The sequence will then introduce partial derivatives, coupled oscillators, the continuum limit, and finally derive the one-dimensional wave equation from the physics of a continuous medium.

The important order is

(63)

References