Wave Mechanics: Superposition

WM00–WM08 developed the language of one-dimensional traveling waves. We can now describe a disturbance such as

(1)

or a second disturbance

(2)

WM09 asks what happens when two disturbances occupy the same region of space at the same time.

For a linear wave system, the answer is the principle of superposition:

(3)

The total disturbance is the algebraic sum of the individual disturbances at each position and time. This principle is central to wave mechanics and is the foundation of interference, beats, standing waves, Fourier methods, and normal-mode analysis [1, 2, 3].

Superposition is not a statement that all physical waves always add linearly. It applies when the governing response is linear, or when a physical system is being modeled within a regime where nonlinear effects are negligible. Later in this series the one-dimensional linear wave equation will provide a direct mathematical reason why sums of solutions are again solutions [4].

1 Point-by-point addition

The word superposition means that two disturbances can occupy the same place simultaneously and their instantaneous effects add.

Suppose, at one particular event (x₀,t₀),

(4)

and

(5)

Then the total displacement there is

(6)

The same addition is performed independently at every other point.

The disturbances do not need to have the same shape. They do not even need to be sinusoidal. If the system is linear, the instantaneous total is still the algebraic sum.

2 Interference is the visible consequence of superposition

When two or more waves overlap, their superposition produces a new pattern. This overlap phenomenon is called interference.

The word interference does not mean that the waves permanently damage or obstruct one another. In a linear medium, two pulses may overlap strongly, produce a temporary resultant disturbance, and then continue propagating with their original shapes.

This behavior distinguishes waves from colliding rigid objects. The material of a medium may move locally, but the wave patterns can pass through one another because the disturbance variables add rather than exclude one another.

3 Constructive interference

Consider two identical sinusoidal waves with the same amplitude, Wavenumber, angular frequency, and phase:

u1(x,t)	= A cos(kx − ωt),	(7)
u2(x,t)	= A cos(kx − ωt).	(8)

Their sum is

u(x,t)	= u1 + u2	(9)
	= 2A cos(kx − ωt).	(10)

Thus the resultant amplitude is

(11)

The waves are said to interfere constructively. Crest aligns with crest and trough aligns with trough.

4 Destructive interference

Now let the second wave be shifted in phase by π:

u1(x,t)	= A cos(kx − ωt),	(12)
u2(x,t)	= A cos(kx − ωt + π).	(13)

Because

(14)

we obtain

u(x,t)	= A cos 𝜃 − A cos 𝜃	(15)
	= 0.	(16)

For two equal-amplitude waves exactly π out of phase,

(17)

This is complete destructive interference.

OpenStax uses the same point-by-point addition picture to define superposition and interference, including the constructive and π-shifted destructive cases [3].

5 Partial interference and phase difference

Most overlapping sinusoidal waves are neither exactly in phase nor exactly opposite in phase.

Let

u1	= A cos 𝜃,	(18)
u2	= A cos(𝜃 + Δϕ),	(19)

where

(20)

and

(21)

Using the trigonometric identity

(22)

we obtain

u	= A cos 𝜃 + A cos(𝜃 + Δϕ)	(23)
	= 2A cos cos .	(24)

The magnitude of the resultant amplitude is therefore

(25)

This single expression contains the constructive and destructive cases:

Δϕ = 0	⇒ AR = 2A,	(26)
Δϕ = π	⇒ AR = 0.	(27)

At intermediate phase differences, the interference is partial.

The phase-difference dependence of the resultant amplitude is a standard result of interference theory [1, 2, 3].

6 Different amplitudes

Complete cancellation requires more than a phase difference of π; the two component amplitudes must also be equal.

Consider

u1	= A1 cos 𝜃,	(28)
u2	= A2 cos(𝜃 + Δϕ).	(29)

Expanding the second cosine gives

u

= (A1 + A2 cos Δϕ) cos 𝜃 − A2 sin Δϕ sin 𝜃.

(30)

This combination is itself a sinusoid of the same k and ω. Its amplitude is

(31)

Two useful checks are

Δϕ = 0	⇒ AR = A1 + A2,	(32)
Δϕ = π	⇒ AR = |A1 − A2|.	(33)

Thus unequal waves shifted by π interfere destructively but do not, in general, cancel completely.

7 Superposition does not create or destroy the component waves

The resultant disturbance can be larger or smaller than either component at a given event. This does not mean that the individual waves have ceased to exist as useful components of the description.

In a linear system we may write

(34)

throughout the overlap. After localized pulses separate, the original pulse shapes can reappear. For continuous sinusoidal waves, the component waves can likewise be regarded as continuing through one another while the observable disturbance is their sum.

Energy accounting requires more care than simply adding instantaneous amplitudes, so energy and power are treated later in the series.

8 Why linearity matters

The principle of superposition is fundamentally a statement about a linear model.

Suppose a later governing equation has a linear operator L such that

(35)

If

(36)

then linearity means

(37)

Therefore the sum is also a solution. Feynman explicitly demonstrates this for the linear wave equation [4].

WM09 does not yet derive the wave equation; that comes in a later block. The argument above is included only to show where the superposition principle will ultimately come from mathematically.

Real physical systems can become nonlinear at sufficiently large amplitude or under other conditions where the restoring response is not proportional to the disturbance. In such regimes, simple superposition may fail. OpenStax makes the same distinction between linear and nonlinear waves [3].

9 A preview of counter-propagating-wave superposition

A particularly important future application occurs when equal sinusoidal waves move in opposite directions:

u1	= A cos(kx − ωt),	(38)
u2	= A cos(kx + ωt).	(39)

Their sum is

(40)

This expression no longer has the form of a single wave translating rigidly to one side. It is the mathematical seed of a standing wave. Nodes, antinodes, resonance, and normal modes are intentionally deferred to later lessons.

10 Worked example 1: point-by-point addition

At one position and time, two disturbances have values

(41)

The total displacement is

u	= u1 + u2	(42)
	= 4.0 mm − 1.5 mm	(43)
	= 2.5 mm .	(44)

The signs matter because superposition is an algebraic, not arithmetic, addition.

11 Worked example 2: equal waves with a phase difference

Two equal sinusoidal waves have amplitude

(45)

and phase difference

(46)

The resultant amplitude is

AR	= 2A	(47)
	= 2(4.0 mm) cos	(48)
	= 8.0 mm	(49)
	= 4 mm	(50)
	≈ 6.93 mm .	(51)

The result lies between zero and the fully constructive value 2A = 8.0 mm.

12 Worked example 3: destructive interference with unequal amplitudes

Let

(52)

Then

AR	=	(53)
	= mm	(54)
	= mm	(55)
	= 2.0 mm .	(56)

The waves interfere destructively, but the cancellation is incomplete because their amplitudes are unequal.

13 Common mistakes

• Mistake: adding amplitudes without considering sign or phase. Superposition adds the instantaneous disturbances, not merely their positive amplitude magnitudes.
• Mistake: assuming destructive interference always gives zero. Complete cancellation requires matching amplitudes and the appropriate phase difference.
• Mistake: thinking two pulses bounce off one another like rigid objects. In an ideal linear system, the component disturbances pass through and the observable overlap is their sum.
• Mistake: assuming superposition is universal. It is a property of linear wave models and can fail in nonlinear regimes.
• Mistake: treating a large resultant amplitude as evidence that the component waves have merged permanently. The decomposition into linear components remains valid while the system remains linear.

14 What WM09 adds to the wave-mechanics language

The first block established how one wave is described. WM09 adds the rule for combining multiple linear waves:

(57)

From this single principle follow the basic ideas of constructive interference, destructive interference, partial interference, and the temporary overlap of traveling pulses.

For equal sinusoidal components with phase difference Δϕ,

(58)

and for unequal amplitudes,

(59)

These results will reappear throughout later wave mechanics, especially in standing waves, Fourier expansions, modal analysis, acoustics, optics, and quantum mechanics.

15 References

References