Wave Mechanics: Translating Disturbances

WM04 and WM05 described patterns that vary with position. In those lessons, we could draw a spatial profile such as u(x), but nothing in the mathematics said that the profile moved. WM06 adds the missing ingredient: time.

The central question is simple:

How do we write a mathematical function whose shape moves through space without changing form?

The answer is one of the most useful constructions in all of wave mechanics:

(1)

for motion toward increasing x, and

(2)

for motion toward decreasing x.

These forms describe a translated disturbance of arbitrary shape. The function F need not be sinusoidal. It may represent a pulse, a localized bump, or any other profile that preserves its shape while translating. This construction is standard in introductory treatments of traveling waves [1, 2, 3, 4].

WM06 deliberately stops before introducing the sinusoidal traveling-wave form A cos(kx−ωt + ϕ). That combination is the subject of WM07.

1 Start with a fixed shape

Suppose that at time t = 0 a disturbance has the spatial profile

(3)

Here F is simply a rule that assigns a disturbance value to each position. For example, F could describe a single localized pulse on a string.

The important point is that F describes the shape. The argument of F tells us where we are sampling that shape.

If the entire shape later moves to the right without stretching, compressing, or changing amplitude, then every recognizable feature of the profile must appear at a larger value of x as time increases.

2 Translation to the right

Assume the disturbance moves toward increasing x with constant speed c. After a time t, the shape has shifted a distance

(4)

To determine the disturbance now observed at position x, ask which point of the original profile has arrived there. That point began at

(5)

Therefore

(6)

This minus sign is sometimes surprising. The reason becomes clear when we track one identifiable feature of the shape.

3 Why the minus sign gives motion toward positive x

Let one particular feature of the profile correspond to a fixed argument ξ₀ of the function F. For the right-moving form,

(7)

Solving for the position of that feature gives

(8)

As t increases, x increases. Therefore the feature moves toward positive x.

Over a time interval Δt,

(9)

If ordinary derivative notation is familiar, the same statement can be written

(10)

No partial derivatives or wave equation are needed for this argument. We are only tracking the motion of one recognizable feature.

4 Translation to the left

Now consider

(11)

Again track a fixed feature by setting the argument equal to a constant:

(12)

Then

(13)

As time increases, the position decreases. Therefore

(14)

The two sign conventions are summarized by

(15)

5 A quick sign test

A reliable way to avoid memorizing the sign rule is to track a feature. Suppose the feature is initially located where the argument is zero.

For F(x − ct), the zero-argument feature satisfies

(16)

so

(17)

It moves right. For F(x + ct), the zero-argument feature satisfies

(18)

so

(19)

It moves left. This feature-tracking argument is more robust than trying to remember a verbal rule.

6 Dimensional check

The quantities inside the argument of F must be compatible. Since x is a position,

(20)

If c is a speed and t is a time, then

(21)

Therefore the combinations x−ct and x + ct are dimensionally meaningful. An expression such as F(x − c) would generally be invalid because position and speed do not have the same dimensions.

7 The shape can be arbitrary

Nothing in the translation argument required a sinusoid. For example, a localized pulse could be described by

(22)

where a sets the width of the pulse. A right-moving version is

(23)

Every feature of the pulse translates by the same distance ct. The profile is not required to be periodic.

This is conceptually important. Traveling-wave mathematics applies to pulses and other disturbances, not only to endlessly repeating sinusoids [3, 4].

8 Snapshot versus time history

Once a disturbance depends on both position and time,

(24)

there are two different one-dimensional views.

At a fixed time t = t₀,

(25)

is a spatial snapshot. It shows the shape across space at one instant.

At a fixed position x = x₀,

(26)

is a time history. It shows what one observer at one location measures as the disturbance passes.

These two views were previewed in WM00. WM06 is the first lesson in which both views belong to the same moving object.

9 Propagation is not necessarily material transport

A moving disturbance should not automatically be interpreted as the bulk motion of the material through which it travels. In a transverse wave on a string, for example, the disturbance may propagate horizontally while individual pieces of the string move mainly up and down around their local equilibrium positions.

This distinction is emphasized in standard wave treatments and is especially important once energy transport is introduced later in the series [3, 1].

10 Worked example 1: identify the direction

Consider

(27)

with x in meters and t in seconds. The form is F(x − ct), so

(28)

Therefore the disturbance moves toward positive x at

(29)

A feature initially at x = 2 m will be located after 4 s at

x	= 2 m + (3 m/s)(4 s)	(30)
	= 14 m.	(31)

11 Worked example 2: left-moving feature

Suppose

(32)

This is the form G(x + ct), so the disturbance moves toward negative x with speed

(33)

If a particular feature is at x = 7 m when t = 0, then after 1.5 s its position is

x	= 7 m − (5 m/s)(1.5 s)	(34)
	= −0.5 m.	(35)

Thus

(36)

12 Worked example 3: recover the original shape

Suppose a right-moving disturbance is

(37)

At t = 0,

(38)

At t = 3 s,

(39)

The profile at t = 3 s is therefore the original profile shifted 6 m toward positive x. The shape has not been redefined. Only its position has changed.

13 What is and is not assumed

The formulas F(x ∓ ct) make a specific assumption: the disturbance translates at constant speed while preserving its shape.

This is a useful idealization, but not every physical disturbance behaves this way forever. Real waves may attenuate, disperse, reflect, or change shape. Those effects require additional physics.

For WM06, however, shape-preserving translation is exactly the right starting point because it isolates the kinematics of propagation before introducing the wave equation or more complicated media.

14 Common mistakes

• Mistake: assuming the minus sign means motion toward negative x. For F(x − ct), tracking a fixed feature gives x = ξ₀ + ct, so the motion is toward positive x.
• Mistake: thinking F must be a sine or cosine. F can describe an arbitrary translated shape.
• Mistake: confusing the propagation of the disturbance with the motion of the material itself.
• Mistake: reading c as an amplitude. The quantity c has units of speed and controls horizontal translation with time.
• Mistake: writing F(x∓c) when time dependence is intended. The distance translated after time t is ct.
• Mistake: assuming that every physical wave must preserve its shape exactly. The forms in this lesson describe nondispersive shape-preserving translation.

15 Connection to the next lesson

WM05 gave the spatial sinusoid

(40)

WM06 has now shown how an arbitrary shape moves through expressions such as

(41)

In WM07 these two ideas will be combined. A sinusoidal profile will be made to translate, producing the familiar traveling-wave form

(42)

At that point every symbol in the expression will have been introduced separately before being assembled into a complete wave.

Summary

The central results of WM06 are

(43)

for shape-preserving motion toward positive x, and

(44)

for shape-preserving motion toward negative x.

Tracking a fixed feature gives

(45)

for the right-moving case and

(46)

for the left-moving case.

The disturbance may be periodic or nonperiodic. The function F describes its shape; the combination x ∓ ct translates that shape through space.

References

References