How Does a Phase Difference Affect Amplitude in Co-Directional Waves?

[dnoi]

Homework Statement

Two waves traveling in the same direction are identical except for a phase difference. Show that if the amplitude of the sum of the waves is 1.5 times larger than the amplitude of the individual waves, then the phase difference must be

delta(phi) = 2 arccos (3/4)

 

[Shooting Star]

Let y1 and y2 be the displacements at a certain point due to the two waves. If the phase diff is 'b', then,

y1 = a*sin(wt),
y2 = a*sin(wt + b).

The rest is algebraic. (Hint: add the two eqns.)

 

[dnoi]

thanks!

now i tried...

a*sin(wt)+a*sin(wt+b) = 2sin(wt+b/2)cos(b/2)

substituted 1.5a for 2cos(b/2)

a*sin(wt)+a*sin(wt+b) = (3/2)a*sin(wt+b/2)

is this the right way to go? or maybe i just made some algebraic mistakes

 

[Shooting Star]

thanks!
a*sin(wt)+a*sin(wt+b) = 2sin(wt+b/2)cos(b/2)

Where is the 'a' on the RHS?

substituted 1.5a for 2cos(b/2)

...for 2*a*cos(b/2)...OK?

 

[dnoi]

oh yes. thank you.