The principle of superposition of waves states that when two or more waves interact, the resulting displacement at any point is the algebraic sum of the individual displacements produced by each wave. In other words, when waves overlap, they combine to form a new wave, and the resulting displacement at a given location is determined by adding up the displacements of the individual waves.
Mathematically, if two waves are described by the equations:
y₁ = A₁sin(k₁x - ω₁t + φ₁) y₂ = A₂sin(k₂x - ω₂t + φ₂)
where A₁ and A₂ are the amplitudes, k₁ and k₂ are the wave numbers, ω₁ and ω₂ are the angular frequencies, φ₁ and φ₂ are the phase angles, x is the position, and t is the time, then the superposition principle states that the resulting wave y, at any given point and time, is given by:
y = y₁ + y₂ = A₁sin(k₁x - ω₁t + φ₁) + A₂sin(k₂x - ω₂t + φ₂)
This principle applies to waves in various contexts, such as sound waves, electromagnetic waves, and water waves. It allows us to analyze and understand the behavior of complex wave phenomena by considering the individual contributions of different waves and their subsequent combination.