Wave addition
Wave addition occurs when two or more waves meet at the same point in space at the same time. The result is obtained by adding their displacements point by point. This phenomenon is called superposition.
Principle of Superposition
When two or more waves overlap, the resulting displacement at any point is the algebraic sum of the displacements of the individual waves at that point. This means that if two waves are in phase (their peaks and troughs align), they will constructively interfere, resulting in a larger amplitude. Conversely, if they are out of phase (the peak of one wave aligns with the trough of another), they will destructively interfere, leading to a smaller amplitude or even cancellation.
Mathematically, if we have two waves represented by the functions \(y_1(x, t)\) and \(y_2(x, t)\), the resultant wave \(y(x, t)\) can be expressed as:
$$ y(x, t) = y_1(x, t) + y_2(x, t) $$
total displacement = displacement of wave 1 + displacement of wave 2
Types of Interference
Constructive Interference
This happens when the crests of one wave line up with the crests of the other, or valleys align with valleys. The displacements have the same sign, so the resulting wave is larger.
Destructive Interference
This happens when a crest aligns with a valley. The displacements have opposite signs, so the resulting wave’s amplitude decreases.
Instructions:
Move sliders \(a\), \(b\) and \(c\) (angular frequencies) or \(i\), \(j\) and \(k\) (wavenumbers) to change the shape of the red, green and blue waves. The violet wave is the sum of the other three waves.
$$ \sin\left( \omega x - k t \right) $$