Superposition principle Waves The displacement due to two waves that pass 1.3 through the same point in space is the algebraic sum of displacements of the two waves. •Superposition principle •Interference The combination of waves is called interference. •Beats Constructive interference – the waves reinforce •Fourier analysis each other Destructive interference – the waves cancel each other.

Superposition Principle Constructive Interference • When two waves overlap in space the displacement of the wave is the sum of the Wave 1 individual displacements.

Wave 2

Superposition

distance → The two waves have the same phase

Destructive Interference Noise canceling headphones

Wave 1 Noise Wave 1

Wave 2 Wave 2 Anti-noise Superposition

Distance -> The two waves are out of phase (by 180o , or π)

1 Interference of sound waves. Interference due to path r2-r1 =0.13 m difference

path difference =δ =r2 –r1 r1 Wave 1 A Superposition of waves at A shows interference As the listener moves from position O to position P he r2 hears the first minimum in sound intensity. Find the Wave 2 due to path differences In phase at x=0 frequency of the oscillation. vsound =340 m/s

Condition for constructive interference δ = mλ 1 Condition for destructive interference δ =+()m λ 2 where m is any integer m = 0 + 1, + 2,….

Interference of sound waves Beats The two waves moving in the same direction have Phase shift due to path differences different frequencies and wavelengths constructive destructive constructive

x

When r2 –r1 =mλ Constructive Interference Time → The superposition shows maxima and minima in When r –r = (m+½) λ Destructive Interference 2 1 displacement amplitude. m is any integer

Beat Frequency tuning musical instruments

difference period T • two instruments play the same note. product of 2 cos functions • the beat frequency tell how different the high frequency -sum low frequency- difference frequencies are.

beat period • one player adjusts his instrument to minimize the beat frequency. y(t)=ω+ω A cos12 t A cos t • when the beat frequency => zero the ()t()tω−ω12 ω+ω 12 cosω+12 t cos ω t = 2cos cos instruments are in tune. 22

beat frequency = f1–f2

2 Interference in 2 dimensions Interference in light waves Screen White-maxima Black-minima Blue- zero

Water Waves show constructive and wave fronts in 2 Intensity profile for destructive interference dimensions (white light waves lines=maxima) (red=high intensity)

Fourier Analysis Fourier Analysis Any periodic wave can be built up of simple Transforms the signal from the time domain harmonic waves to the frequency domain Example- Square wave This is done mathematically using a Fourier Transform

Square wave is a sum of odd harmonics of sine

waves f1= f0 ,f3= 3fo ,f5= 5 fo ......

Superposition is responsible for many properties of waves.

Color is due to the superposition of light of different wavelengths.

Sound texture is due to the superposition of sounds with different frequencies.

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