Superposition principle

[1], [2] Superposition principle/property says all linear systems, the net response by 2+ stimuli (like waves) is the responses' sum caused by each stimuli individually--so the the displacement at that point equals the displacements of individual waves' sum.

stimuli: what triggers behavioral/physical changes

So if input A's response is X and B's response is Y then input (A+B)'s response is (X+Y).

Any function F(x) satisfying the principle is a linear function. It can be defined by 2 simpler properties:

additivity:

and homogeneity:

for scalar 'a'.

The principle has many physics and engineering uses (phenomena like interference, diffraction and standing waves) as many physical systems are modeled as linear systems.

E.g. a beam can be modeled as a linear system with its input stimulus as load on the beam and its output response as its deflection.

The linear systems' cruciality is to be easier to analyze mathematically; many math methods, frequency-domain linear transform methods (Fourier and Laplace transforms, and linear operator theory are used).

As physical systems are often only roughly linear, superposition principle is an true behavior's approximations.

Rolling motion as superposition of 2 motions. Its wheel's rolling motion can be shown as a combination of 2 motions: translation without rotation and rotation without translation.

Superposition principle

Relation to Fourier analysis and similar methods

References

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Relation to Fourier analysis and similar methods

[1] By writing a general stimulus (in a linear system) as the stimuli's superposition of a simple form, often the response is easier to compute.

E.g. In Fourier analysis, the stimulus is written as the superposition of infinitely many sinusoids. Due to the principle, each of the sines can be analyzed separately and its individual response can be computed. (The response is itself a sinusoid, with same frequency as the stimulus, but often a different amplitude and phase.)

Superposition principle says the response to the original stimulus is the sum (or integral) of all individual sine responses.

It works for any wave type (sound waves, water surface waves, electromagnetic waves) in some conditions.

Individual wave displacements may be positive/negative. If the displacements are vectors, then the sum is calculated by vector addition.

References

[1] Wikipedia
[2] Superposition - Isaac Physics
[3] ODE | Principle of superposition - YT

Quizzes

[Q1] Flashcard
[Q2] Exercises (electrical engineering)

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Superposition principle

Relation to Fourier analysis and similar methods

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