December 7th, 2009
PHY2048 Discussion-Fall ‘09
Quiz 12
Name:
UFID:
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 6.00 mm, the other 10.0 mm.
a) What phase difference φ1 between the two waves results in the smallest amplitude of the resultant
wave? What is the smallest amplitude?
To find the amplitude of the resultant wave, we need to express
the sinusoidal waves with phasors and add them vectorially.
Since the amplitude of the resultant wave is equal to the
magnitude of the sum of the two phasors, we have
ym’ = √(ymx’2+ymy’2) = √[(ym1+ym2cosφ)2+(ym2sinφ)2]
= √(ym12+ym22+2ym1ym2cosφ)
Therefore, the amplitude is smallest if
cosφ = -1 ⇒ φ = cos-1(-1) = π rad
The smallest amplitude is
ym’ = √(ym12+ym22-2ym1ym2) = √[( ym2-ym1)2] = |ym2-ym1| = 10-6 = 4.00 mm
b) What phase difference φ2 results in the largest amplitude of the resultant wave? What is the largest
amplitude?
The amplitude is largest if
cosφ = 1 ⇒ φ = cos-1(1) = 0 rad
The greatest amplitude is
ym’ = √(ym12+ym22+2ym1ym2) = √[( ym2+ym1)2] = |ym2+ym1| = 10+6 = 16.0 mm
c) What is the resultant amplitude if the phase difference is (φ1+φ2)/2
From a) and b), we have
(φ1+φ2)/2 = (π+0)/2 = π/2
Thus the resultant amplitude is
ym’ = √(ym12+ym22+2ym1ym2cos(π/2)) = √(ym12+ym22) = √(62+102) = 11.7 mm