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Superposition of Waves

In most of the realistic situations, a wave motion contains a mixture or group of waves of different frequencies travelling together in a medium. Typical examples are white light, or sound spoken by us: white light contains a continuous spectrum of wavelengths from about 3000A to 8000A, ordinary conversational sound contains a continuous spectrum of frequencies from about 200 Hz to 4000 Hz. We find that the superposition of several such wavelengths present in the medium produces a group of waves, called a wave group or a wave packet. It turns out that the velocity with which this wave group travels in the medium is, in general, different from the velocity of individual waves.

Let us consider two harmonic waves of equal amplitudes A and frequencies ω₁ and ω₂ = ω₁ + Δ ω travelling in the same direction (X-axis) in a medium where the wave velocity is v. At any point x and time t, the displacement of the medium is given by the superposition of two waves:

y = y₁ + y₂ = A cos ( ω₁ t – k₁ x ) + A cos (ω₂ t – k₂ x)