Two waves having equations x1=a sin (ω t+ varphi1), x2=a sin (ω t+ varphi2) If in the resultant wave the frequency and amplitude remain equal to those of superimposing waves. Then phase difference between them is

Question

Two waves having equations x1=a sin (ω t+ varphi1), x2=a sin (ω t+ varphi2) If in the resultant wave the frequency and amplitude remain equal to those of superimposing waves. Then phase difference between them is (A) (π/6) (B) (2 π/3) (C) (π/4) (D) (π/3)

Tardigrade Admin · Accepted Answer

Superposition of waves does not alter the frequency of resultant wave and resultant amplitude ⇒ a2=a2+a2+2 a2 cos varphi=2 a2(1+ cos varphi) ⇒ cos varphi=-1 / 2= cos 2 π / 3 ∴ varphi=2 π / 3

Q. Two waves having equations
$x_{1} = a sin (ω t + φ_{1}), x_{2} = a sin (ω t + φ_{2})$
If in the resultant wave the frequency and amplitude remain equal to those of superimposing waves. Then phase difference between them is

$\frac{π}{6}$

$\frac{2 π}{3}$

$\frac{π}{4}$

$\frac{π}{3}$

Superposition of waves does not alter the frequency of resultant wave and resultant amplitude
$\Rightarrow a^{2} = a^{2} + a^{2} + 2 a^{2} cos φ = 2 a^{2} (1 + cos φ)$
$\Rightarrow cos φ = - 1/2 = cos 2 π /3$
$∴ φ = 2 π /3$

Q. Two waves having equations x1​=asin(ωt+φ1​),x2​=asin(ωt+φ2​) If in the resultant wave the frequency and amplitude remain equal to those of superimposing waves. Then phase difference between them is

6π​

32π​

4π​

3π​