TS EAMCET 5-Aug-2021 Shift 1 Question Paper

Given, x1=A‌cos‌ω‌t,x2=2A‌sin‌ω‌t
and x3=√2A‌cos(ωt+‌

π

4

)
According to superposition principle, resultant wave is given as
x‌‌=x1+x2+x3
‌‌=A‌cos‌ω‌t+2A‌sin‌ω‌t+√2A‌cos(ωt+‌

π

4

)
=‌‌√5A(‌

1

√5

‌cos‌ω‌t+‌

2

√5

‌sin‌ω‌t)+√2A‌cos(ωt+‌

π

4

)
=√5A(cos‌φ‌cos‌ω‌t+sin‌φ‌sin‌ω‌t)
‌+√2A‌cos(ωt+‌

π

4

)
x‌‌=√5A‌cos(ωt−φ)+√2A‌cos(ωt+‌

π

4

)‌‌...‌ (i) ‌Phase difference between above two waves,
φ′‌‌=(ωt+‌

π

4

)−(ωt−φ)⇒φ′=‌

π

4

+φ
⇒‌‌cos‌φ′‌‌=cos(‌

π

4

+φ)
‌‌=cos‌

π

4

‌cos‌φ−sin‌

π

4

‌sin‌φ
‌‌=‌

1

√2

×‌

1

√5

−‌

1

√2

⋅‌

2

√5

cos‌φ′‌‌=−‌

1

√10

. . . (ii)
From Eq. (i), amplitude of resultant wave,
A=√(√5A)2+(√2A)2+2√5A×√2A′‌cos‌φ′
=√5A2+2A2+2√10A2×(‌

−1

√10

) [from Eq.(ii)]
=√5A2=√5A