Sinusoidal Waves frequency Problem

[JasonV]

Two sinusoidal waves of the same frequency are sent in the same direction along a taut string. One wave has an amplitude of 5.0 mm, the other 8.0 mm. (a) what phase difference between the two waves results in the smallest amplitude of the resultant wave? (b) what is that smallest amplitude? (c) what phase difference 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 (phase1-phase2)/2.

I have tried adding the two waves:

y1(x,t)=ym1 sin(kx-wt) and y2(x,t)=ym2 sin(kx-wt+phase difference)


5sin(kx-wt)+8sin(kx-wt+phase difference)

I do not know where to go next...my book gives an equation for when the amplitudes of the two waves are equal...here is how they derived the equation:


ym sin(kx-wt) + ym sin(kx-wt+phase difference)

sin α + sin β = 2sin1/2(α +β )cos1/2(α-β)

y'(x,t) = [2ymcos1/2phase] sin(kx-wt+1/2phase)

Since the problem has two waves with different amplitudes (5 and 8), i am not sure if i can use that equation. Please help me get started on this problem.

 

[rl.bhat]

You can wright two equation as y1 = 8sin(wt) and y2 = 5sin(wt + phi).
Resultant amplitude y = y1 + y2 = 8sin(wt) + 5sin(wt + phi). You can wright 8 = Rcos(theta) and 5 = Rsin(theta) where R is the amplitude of the resultant wave and theta is the additional phase difference which is equal to tan^-1(5/8). Wright the equation for Y. For maximum amplitude net phase diference should be zero, and for minimum it should be 180 degree.