Question

Two sinusoidal waves of the same frequency are to be sent in the
same direction along a taut string. One wave has an amplitude of
6.2 mm, the other 8.5 mm. **(a)** What phase
difference *φ*_{1} between the two waves results in
the smallest amplitude of the resultant wave? **(b)**
What is that smallest amplitude? **(c)** What phase
difference *φ*_{2} results in the largest amplitude
of the resultant wave? **(d)** What is that largest
amplitude? **(e)** What is the resultant amplitude if
the phase angle is (*φ*_{1} -
*φ*_{2})/2?

Answer #1

**let
A1 = 6.2 mm
A2 = 8.5 mm**

**a) when the waves are out of phase the resultant amplitude
will be smallest.**

**so, phi1 = 180 degrees (or) pi radians**

**b) smallest amplitue, A_smallest = A2 - A1**

**= 8.5 - 6.2**

**= 2.3 mm**

**c) when the waves are in phase the resultant amplitude
will be largest.**

**so, phi2 = 0 degrees (or) 0 radians**

**d) largest amplitue, A_largest = A1 + A2**

**= 6.2 + 8.5**

**= 14.7 mm**

**e) If phase diffrence, phi = (phi1 -
phi2)/2**

**= (180 - 0)/2**

**= 90 degrees (or) pi/2 radians**

**so, resultant amplitude, A = sqrt(A1^2 + A2^2 +
2*A1*A2*cos(phi))**

**= sqrt(6.2^2 + 8.5^2 + 2*6.2*8.5*cos(90))**

**= 10.5 mm**

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