First slide

Superposition of waves

Question

Two waves
$y_{1} = A_{1} \sin (ωt - β_{1}), y_{2} = A_{2} \sin (ωt - β_{2})$
Superimpose to form a resultant wave whose amplitude is

Moderate

A

$\sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos (β_{1} - β_{2})}$
B

$\sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \sin (β_{1} - β_{2})}$
C

$A_{1} + A_{2}$
D

$| A_{1} + A_{2} |$

Solution

Phase difference between the two waves is
$φ = (ωt - β_{2}) = (ωt - β_{1}) = (β_{1} - β_{2})$
$∴$ Resultant amplitude $A = \sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos (β_{1} - β_{2})}$

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