The amplitude and phase of a wave that is formed by the superposition of two harmonic travelling waves, and , are: (Take the angular frequency of initial waves same as ) [2025]

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Q.
The amplitude and phase of a wave that is formed by the superposition of two harmonic travelling waves, $y_{1} (x, t) = 4 \sin (k x - ω t)$ and $y_{2} (x, t) = 2 \sin (k x - ω t + \frac{2 π}{3})$ , are: (Take the angular frequency of initial waves same as $ω$ ) [2025]

1 $[6, \frac{2 π}{3}]$

2 $[6, \frac{π}{3}]$

3 $[\sqrt{3}, \frac{π}{6}]$

4 $[2 \sqrt{3}, \frac{π}{6}]$

Ans.
(4)

$A = \sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos ϕ}$

Here, $ϕ = \frac{2 π}{3}$

$A = \sqrt{4^{2} + 2^{2} + 2 \times 4 \times 2 \cos \frac{2 π}{3}} = 2 \sqrt{3}$

$\tan α = \frac{2 \sin ϕ}{4 + 2 \cos ϕ} = \frac{1}{\sqrt{3}} \Rightarrow α = π / 6$

Solution

Q.
The amplitude and phase of a wave that is formed by the superposition of two harmonic travelling waves, $y_{1} (x, t) = 4 \sin (k x - ω t)$ and $y_{2} (x, t) = 2 \sin (k x - ω t + \frac{2 π}{3})$ , are: (Take the angular frequency of initial waves same as $ω$ ) [2025]

1 $[6, \frac{2 π}{3}]$

2 $[6, \frac{π}{3}]$

3 $[\sqrt{3}, \frac{π}{6}]$

4 $[2 \sqrt{3}, \frac{π}{6}]$

Ans.
(4)

$A = \sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos ϕ}$

Here, $ϕ = \frac{2 π}{3}$

$A = \sqrt{4^{2} + 2^{2} + 2 \times 4 \times 2 \cos \frac{2 π}{3}} = 2 \sqrt{3}$

$\tan α = \frac{2 \sin ϕ}{4 + 2 \cos ϕ} = \frac{1}{\sqrt{3}} \Rightarrow α = π / 6$

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Solution

Q. The amplitude and phase of a wave that is formed by the superposition of two harmonic travelling waves, y1(x,t)=4 sin (kx–ωt) and y2(x,t)=2 sin (kx–ωt+2π3), are: (Take the angular frequency of initial waves same as ω) [2025]

1 [6,2π3]

2 [6,π3]

3 [3,π6]

4 [23,π6]

Ans. (4) A=A12+A22+2A1A2 cos ϕ Here, ϕ=2π3 A=42+22+2×4×2 cos 2π3=23 tan α=2 sin ϕ4+2 cos ϕ=13⇒ α=π/6

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Q.
The amplitude and phase of a wave that is formed by the superposition of two harmonic travelling waves, $y_{1} (x, t) = 4 \sin (k x - ω t)$ and $y_{2} (x, t) = 2 \sin (k x - ω t + \frac{2 π}{3})$ , are: (Take the angular frequency of initial waves same as $ω$ ) [2025]

1 $[6, \frac{2 π}{3}]$

2 $[6, \frac{π}{3}]$

3 $[\sqrt{3}, \frac{π}{6}]$

4 $[2 \sqrt{3}, \frac{π}{6}]$

Ans.
(4)

$A = \sqrt{A_{1}^{2} + A_{2}^{2} + 2 A_{1} A_{2} \cos ϕ}$

Here, $ϕ = \frac{2 π}{3}$

$A = \sqrt{4^{2} + 2^{2} + 2 \times 4 \times 2 \cos \frac{2 π}{3}} = 2 \sqrt{3}$

$\tan α = \frac{2 \sin ϕ}{4 + 2 \cos ϕ} = \frac{1}{\sqrt{3}} \Rightarrow α = π / 6$