well, just plug and chug
sin(ωt + α) = sinωt cosα + cosωt sinα
so, you have
Csin ωt + C(sinωt cosα + cosωt sinα)
= Csin ωt + Csinωt cosα + Ccosωt sinα
=(C+cosα)sinωt + (Csinα)cosωt
Now, since ksin(ωt + φ)=ksinωt cosφ + kcosωt sinφ
you want to find k and φ such that
(C+cosα) = kcosφ
(C+sinα) = ksinφ
(C+cosα)^2 + (C+sinα)^2 = k^2
φ = arctan (C+sinα)/(C+cosα)
Now just put it all into a single formula
Interference
Two identical tuning forks are struck, one a fraction of a second after
the other. The sounds produced are modeled by f1(t) = C sin ωt and f2(t) = C sin(ωt + α).
The two sound waves interfere to produce a single sound modeled by the sum of these functions
f(t) = C sin ωt + C sin(ωt + α)
(a) Use the Addition Formula for Sine to show that f can be written in the form f(t) =
A sin ωt + B cos ωt where A and B are constants that depend on α
(b) Suppose C = 10 and α = π/3. Find constants k and φ so that f(t) = k sin(ωt + φ)
1 answer
1 answer