# Principle of superposition, Reflection waves and Beats

• Two progressive waves propagating in the same or opposite direction superpose the resultant wave is represented by the algebraic sum of individual displacements.

y = y1 + y2

• When two sound waves of slightly different frequencies travelling in same direction super impose together the resultant sound waxes and wanes at regular intervals of time this waxing and wanning of sound is called beets. The phenomenon involved in beats & superposition of waves.
• Beat frequency = number of maxima heard per second = number of minima heard per second = number of beets per second = f1 ∼ f2 where f1 and f2 are frequencies of parent sounds.
• The apparent change in the frequency of the source of sound due to relative motion between the observer and the source of sound is called Doppler effect.

\tt n' = \left(\frac{V-V_{0}}{V-V_{s}}\right) n where

V = velocity of sound, V0 = velocity of observer, Vs = Velocity of source.

• Direction of velocity of sound is always from source to observer irrespective of their direction of motion V0, Vs are positive if they are in the direction of sound V0, Vs are negative if they are opposite to direction of sound.
• Doppler shift in radar \tt f' = \left(\frac{C + V}{C}\right) f where V is the speed of the plane f1 = Received frequency of radar.

### Beats View the Topic in this video From 0:19 To 12:51

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1. The net displacement y(x, t) is given by
y(x, t) = y1 (x, t) + y2 (x, t)

2. At those points where phase difference Φ = 0 or 2 nπ, i.e., Φ is an integar multiple of 2π (n = 1, 2, 3, ......), hence resultant amplitude is
a = a1 + a2 = amax