Wave Mechanics: Phase and Phase Difference

WM02 introduced the sinusoidal oscillator

u(t) = A cos(ωt + ϕ )

(1)

and defined the phase angle

𝜃(t) = ωt + ϕ.

(2)

The purpose of this entry is to make the idea of phase physically useful. Phase tells us where an oscillator is within its repeating cycle. Phase difference tells us how far one oscillator is advanced or delayed relative to another.

This lesson still considers Oscillation at One Point. Spatial phase, the term kx, wavelength, wavenumber, and traveling waves are introduced later in the series.

1 Phase as a coordinate within a cycle

For the cosine function, one full cycle corresponds to an increase of 2π radians in the phase angle. Thus the values

π- 3π- 0, 2, π, 2 , 2π

(3)

mark familiar locations within one cycle.

Figure. Phase may be pictured as an angular coordinate around a cycle. Advancing the phase by 2π returns to the same location in that cycle.

For

u (t) = A cos 𝜃(t),

(4)

the quarter-cycle points are

𝜃 = 0	: u = +A,	(5)
𝜃 =	: u = 0,	(6)
𝜃 = π	: u = −A,	(7)
𝜃 =	: u = 0,	(8)
𝜃 = 2π	: u = +A.	(9)

The two zero-displacement points are especially instructive. They have the same displacement, but they correspond to different phases and opposite directions of motion. Phase therefore carries more information about the state of an oscillation than displacement alone.

2 The phase advances uniformly for a sinusoid

WM02 defined

𝜃(t) = ωt + ϕ.

(10)

If ω is constant, then the phase increases uniformly with time. Over a time interval Δt, the phase advance is

Δ 𝜃 = ω Δt.

(11)

Using

2π ω = ---, T

(12)

we can also write

|------------| | Δt- | |Δ𝜃 = 2 π T . -------------

(13)

This equation gives a useful conversion between elapsed time and position within the cycle.

For example, after one quarter of a period,

Δt = T-, 4

(14)

Δ𝜃 = 2πT-∕4 = π-. T 2

(15)

A quarter period in time corresponds to a quarter cycle, or π∕2 radians, in phase.

3 Equivalent phases

Because cosine is periodic,

cos(𝜃 + 2π) = cos𝜃.

(16)

More generally,

|------------------------------------------| cos(𝜃-+-2πn-)-=-cos𝜃,-----n-=-0,±1,-±2,-...-

(17)

Therefore phase angles separated by an integer number of complete cycles are equivalent for a periodic sinusoid.

Figure. The phases ϕ and ϕ + 2π differ numerically, but they identify the same location within the repeating cycle. Phase is naturally interpreted modulo 2π.

For example,

5π-= π-+ 2π, 2 2

(18)

so 5π∕2 and π∕2 represent the same phase location.

Similarly,

π 3π − --= ---− 2π, 2 2

(19)

so −π∕2 and 3π∕2 are also equivalent modulo one cycle.

This does not mean that accumulated phase is always unimportant. In some problems we care about how many complete cycles have occurred. But when the question is only where the oscillator lies within its present cycle, phase can be reduced modulo 2π.

4 Comparing two sinusoidal oscillators

Consider two oscillators with the same angular frequency ω:

u₁(t)	= A₁ cos(ωt + ϕ₁),	(20)
u₂(t)	= A₂ cos(ωt + ϕ₂).	(21)

Their instantaneous phase angles are

𝜃₁(t)	= ωt + ϕ₁,	(22)
𝜃₂(t)	= ωt + ϕ₂.	(23)

Subtracting gives

𝜃2(t) − 𝜃1(t) = ϕ2 − ϕ1.

(24)

Because the ωt terms cancel, the difference is constant in time. We define

|--------------| -Δϕ-=--ϕ2-−-ϕ1.-

(25)

For same-frequency sinusoids, Δϕ is their constant phase difference.

The amplitudes A₁ and A₂ do not enter this expression. Two oscillators can have different amplitudes and still have a well-defined phase difference.

5 What phase lead and phase lag mean

Using the convention

u (t) = A cos(ωt + ϕ),

(26)

a larger phase constant means that the oscillator is further advanced through its cycle at the same clock time.

Thus, if

Δϕ = ϕ − ϕ > 0, 2 1

(27)

then oscillator 2 leads oscillator 1 by Δϕ under this sign convention.

Δ ϕ < 0,

(28)

then oscillator 2 lags oscillator 1 by the magnitude of that phase difference.

Figure. Two equal-frequency sinusoids separated by a constant phase difference. In the convention cos(ωt + ϕ), the curve with the larger phase constant is advanced in time and therefore leads.

The words “lead” and “lag” are meaningful only after the sign convention has been stated. Some disciplines write harmonic motion with a minus sign in the phase argument. The safest practice is therefore to inspect the actual equation rather than memorize a verbal sign rule without context.

6 Converting phase difference into a time shift

A constant phase difference can be expressed as an equivalent time shift. Suppose

u1(t) = A cos(ωt + ϕ1 )

(29)

and

u2 (t) = A cos(ωt + ϕ2).

(30)

Write

ϕ2 = ϕ1 + Δ ϕ.

(31)

Then

u₂(t)	= A cos(ωt + ϕ₁ + Δϕ)	(32)
	= A cos .	(33)

Therefore the corresponding time shift is

|----------| | Δϕ | |Δt = ---.| -------ω----

(34)

Using ω = 2π∕T gives another useful form:

|------------| | Δ-ϕ- | |Δt = 2π T. | -------------

(35)

This relation says that the fraction of a period represented by a phase difference is the same as the fraction of a full 2π phase cycle:

|----------| |Δt- Δ-ϕ-| |T = 2π .| ------------

(36)

For example, a phase difference of π∕2 is one quarter of a complete phase cycle, so it corresponds to one quarter of a period:

π∕2- T- Δt = 2π T = 4 .

(37)

7 Important special phase differences

Several phase differences occur repeatedly in physics and engineering.

Δϕ = 0: the oscillators are in phase.
|Δϕ| = π∕2: the oscillators are separated by one quarter cycle.
|Δϕ| = π: the oscillators are separated by one half cycle and are often called opposite in phase or antiphase.
|Δϕ| = 2π: the numerical phase difference is one full cycle, so the oscillators are again equivalent in phase.

Because phase is cyclic, a difference such as

3π- 2

(38)

can also be represented as

π- − 2.

(39)

The first description says “lead by three quarters of a cycle.” The second says “lag by one quarter of a cycle.” They describe the same relative phase modulo 2π.

When comparing phase differences, it is often convenient to choose a principal interval such as

− π < Δϕ ≤ π.

(40)

This convention selects the smaller signed angular separation, but other interval conventions are possible. The convention should be stated whenever ambiguity matters.

8 Worked example: phase lead and equivalent time shift

Consider

u₁(t)	= 3 cos(8πt),	(41)
u₂(t)	= 5 cos .	(42)

The amplitudes are different, but both oscillators have

ω = 8π rad/s.

(43)

Their phase constants are

π ϕ1 = 0, ϕ2 = --. 3

(44)

Thus

|--| |π-| Δϕ = ϕ2 − ϕ1 = -3-.

(45)

Under the +ϕ cosine convention, oscillator 2 leads oscillator 1.

The corresponding time lead is

Δt	=	(46)
	=	(47)
	= s .	(48)

The period is

T = 2π- = 1-s, 8π 4

(49)

and indeed

T- 1-- 6 = 24 s.

(50)

A phase lead of π∕3 is one sixth of a complete 2π cycle, so the time lead is one sixth of a period.

9 Worked example: reducing a phase difference

Suppose

Δ ϕ = 7π-. 4

(51)

Subtract one full cycle:

Δϕ_equiv	= − 2π	(52)
	= −	(53)
	= −.	(54)

Thus

|----------------------| |7π- π- | | 4 ≡ − 4 (mod 2π).| ------------------------

(55)

The relative phase can be described either as a lead of 7π∕4 or, more compactly, as a lag of π∕4.

10 What if the frequencies are different?

A constant phase difference requires equal angular frequencies. Consider

𝜃₁(t)	= ω₁t + ϕ₁,	(56)
𝜃₂(t)	= ω₂t + ϕ₂.	(57)

Their phase difference is

Δ𝜃(t)	= 𝜃₂(t) − 𝜃₁(t)	(58)
	= (ω₂ − ω₁)t + (ϕ₂ − ϕ₁).	(59)

Therefore

|--------------------------| |Δ 𝜃(t) = (ω2 − ω1)t + Δ ϕ0.| ----------------------------

(60)

If ω₁≠ω₂, the relative phase changes continuously with time. One oscillator gradually gains phase on the other.

Figure. When two angular frequencies differ, their relative phase drifts with time. Only equal-frequency sinusoids maintain a constant phase difference.

This distinction becomes important later in interference, beats, Fourier analysis, and wave propagation.

11 Phase is not amplitude

Amplitude and phase describe different features of a sinusoid.

For

u (t) = A cos(ωt + ϕ),

(61)

A tells how large the oscillation is;
ω tells how rapidly phase advances;
ϕ tells where the cycle is positioned relative to the chosen time origin.

Changing A stretches the graph vertically. Changing ϕ shifts the cycle horizontally in time without changing its amplitude or period.

This separation of roles will become increasingly important when several waves are superposed.

12 Common mistakes

Mistake: treating equal displacement as equal phase. A sinusoid normally passes through most displacement values twice per cycle, once in each direction.
Mistake: forgetting that phase is cyclic. Angles differing by 2πn represent the same location within a cycle.
Mistake: comparing phase constants when the frequencies are different and calling the result a permanent phase lead. If ω₁≠ω₂, the relative phase changes with time.
Mistake: converting phase directly to seconds without using ω. A phase angle becomes a time shift through Δt = Δϕ∕ω.
Mistake: stating “positive phase means lead” without specifying the sinusoidal sign convention. In this series the present result follows from cos(ωt + ϕ).