Wave Mechanics: The Sinusoidal Traveling Wave

WM02 introduced sinusoidal oscillation in time, WM05 introduced sinusoidal variation in space, and WM06 showed how an arbitrary shape can translate through space. WM07 now combines those ideas into one expression.

The result is the standard one-dimensional sinusoidal traveling wave

(1)

for propagation toward increasing x, and

(2)

for propagation toward decreasing x, when k > 0 and ω > 0.

Every symbol in these equations has already been introduced separately. The purpose of WM07 is to assemble them and understand what the complete phase means. These sinusoidal traveling-wave forms are standard in introductory wave mechanics [1, 2, 3, 4, 5].

WM07 remains kinematic. We do not yet derive the wave equation, and we do not yet make wave speed the main object of study. WM08 will use constant phase to derive the speed relations explicitly.

1 The pieces already in place

From WM02, a sinusoidal oscillation at one point can be written

(3)

From WM05, a sinusoidal spatial pattern can be written

(4)

The temporal angular frequency and spatial angular wavenumber are

(5)

From WM06, a shape F(x) moving toward positive x at constant speed c is written by replacing x with x − ct:

(6)

The sinusoidal traveling wave follows by applying this translation rule to the spatial sinusoid.

2 Translate the spatial sinusoid to the right

Start with the spatial profile

(7)

To move this profile toward increasing x while preserving its shape, replace x with x − ct:

(8)

Expanding the phase gives

(9)

At a fixed location x, the time-dependent phase changes at the angular rate kc. We identify that temporal angular rate with ω:

(10)

Therefore the translated sinusoid becomes

(11)

This is the standard right-moving sinusoidal wave form [3].

The important point is that no new kind of sinusoid has been invented. A spatial sinusoid has simply been translated in exactly the way developed in WM06.

3 Translate the spatial sinusoid to the left

For motion toward decreasing x, WM06 tells us to use x + ct:

u(x,t)	= A cos	(12)
	= A cos(kx + kct + ϕ).	(13)

Using the same identification ω = kc gives

(14)

Thus, with k > 0 and ω > 0,

(15)

This is the sinusoidal version of the WM06 rule for F(x ∓ ct).

4 The complete wave phase

For the right-moving wave, define

(16)

The disturbance is then simply

(17)

The phase now contains three contributions:

• kx tells how phase changes with position;
• −ωt tells how phase changes with time for the right-moving convention;
• ϕ sets the initial phase offset.

A wave therefore does not have one phase value everywhere. Its phase depends on both where and when the wave is observed.

For a left-moving wave, the corresponding phase is

(18)

5 Two views of the same traveling wave

The function u(x,t) contains both the spatial and temporal descriptions. Holding one independent variable fixed exposes the other.

At a fixed time t = t₀,

(19)

is a spatial snapshot. Its wavelength is determined by k:

(20)

At a fixed position x = x₀,

(21)

is a time history. Its period is determined by ω:

(22)

The ordinary frequency is therefore

(23)

This separation is one of the main reasons the earlier WM lessons introduced temporal and spatial periodicity independently before combining them.

6 What the amplitude means

The amplitude A is still the largest magnitude of the oscillating quantity about equilibrium. For displacement waves,

(24)

Changing A changes the vertical scale of the wave but does not by itself change wavelength, period, or propagation direction.

This distinction will become especially important when the series later introduces wave energy and power. Amplitude affects transported energy, but amplitude is not itself a propagation speed or frequency.

7 What the phase constant does

The phase constant ϕ shifts the location of the sinusoidal cycle at the chosen reference time and position. At t = 0,

(25)

At x = 0 for the right-moving form,

(26)

Changing ϕ therefore changes where in its cycle the wave begins relative to the chosen origin of space and time. It does not reverse the propagation direction.

Because cosine is 2π-periodic,

(27)

describe equivalent phase offsets, exactly as in WM03.

8 The sine form is equally valid

A sinusoidal traveling wave is often written with sine instead of cosine:

(28)

Since sine and cosine differ only by a phase shift, this represents the same class of physical waves. For example,

(29)

Therefore the choice between sine and cosine is usually a matter of convenient phase convention, not different physics [3, 1].

9 Dimensional check of the phase

The argument of a trigonometric function must be a dimensionless angular quantity. In the wave phase

(30)

we have

(31)

so kx is an angular phase. Likewise,

(32)

so ωt is also an angular phase. The phase constant ϕ is an angle, so all three terms are compatible.

An expression such as

(33)

is incomplete unless the coefficient of t supplies the required inverse-time units.

10 Worked example 1: read the wave parameters

Consider

(34)

where x is in meters, t is in seconds, and u is in meters.

Comparing with

(35)

gives

A	= 0.030 m,	(36)
k	= 4π rad/m,	(37)
ω	= 10π rad/s,	(38)
ϕ	= .	(39)

The minus sign on the temporal term means the wave propagates toward positive x.

The wavelength is

λ	=	(40)
	=	(41)
	= 0.50 m .	(42)

The period is

T	=	(43)
	=	(44)
	= 0.20 s .	(45)

The frequency is

(46)

WM08 will use k and ω together to determine the propagation speed.

11 Worked example 2: evaluate the disturbance at one event

Let

(47)

with u in centimeters, x in meters, and t in seconds. At

(48)

the phase is

𝜃	= π(1.0) − 2π(0.25) +	(49)
	= π − +	(50)
	= .	(51)

Therefore

u(1.0, 0.25)	= 2.0 cos cm	(52)
	= − cm	(53)
	≈−1.73 cm .	(54)

The calculation illustrates an important interpretation: a traveling-wave formula assigns a definite disturbance value to every event (x,t).

12 Worked example 3: determine the propagation direction

Compare

(55)

and

(56)

For positive k and ω, u₁ moves toward increasing x and u₂ moves toward decreasing x.

A reliable check is to follow a fixed phase value. For the first wave,

(57)

As time increases, x must increase to keep the phase fixed. For the second wave,

(58)

so x must decrease as time increases.

13 Constant phase and the next lesson

A crest, trough, or any other recognizable point on a perfect sinusoid can be identified by a particular constant phase value. For a right-moving wave, that condition is

(59)

Following this condition through an x–t diagram traces the motion of that wave feature.

This provides an important bridge. WM06 tracked an arbitrary feature of a translated shape. WM07 identifies the corresponding feature by its phase. WM08 will turn that feature-tracking argument into the explicit propagation speed relation.

14 Common mistakes

• Mistake: reading kx − ωt as motion toward negative x. With positive k and ω, the minus sign corresponds to propagation toward positive x.
• Mistake: treating k and ω as interchangeable. k controls spatial phase change; ω controls temporal phase change.
• Mistake: confusing wavelength with period. λ = 2π∕k is a length; T = 2π∕ω is a time.
• Mistake: assuming ϕ changes the propagation direction. The phase constant shifts the cycle relative to the chosen origins but does not reverse motion.
• Mistake: forgetting that u(x,t) can be examined either as a spatial snapshot or as a time history.
• Mistake: assuming sine and cosine describe different types of waves. They differ only by phase convention.
• Mistake: trying to derive the wave equation from the sinusoidal form here. WM07 is deliberately kinematic; the PDE is derived later from the physics of a continuous medium.

15 What WM07 has accomplished

The first seven lessons have now assembled the full one-dimensional sinusoidal traveling-wave language:

(60)

The quantities have separate physical roles:

(61)

The spatial and temporal scales are

(62)

For positive k and ω, replacing −ωt with +ωt reverses the direction of propagation.

16 Connection to WM08

WM07 has shown how a sinusoidal profile translates and how constant phase identifies a moving feature. WM08 will ask the natural quantitative question:

How fast does a point of constant phase move?

Starting from

(63)

WM08 will derive the propagation-speed relations and connect the temporal and spatial descriptions through f, λ, ω, and k.

References