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)
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
To solve this problem, let's start with part (a) where we need to show that f(t) can be written in the form A sin(ωt) + B cos(ωt), where A and B are constants that depend on α.
Let's expand f(t) by applying the Addition Formula for Sine:
f(t) = C sin ωt + C sin(ωt + α)
Using the addition formula sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite f(t) as:
f(t) = C [sin ωt cos α + cos ωt sin α] + C sin ωt
Rearranging the terms:
f(t) = C sin ωt cos α + C cos ωt sin α + C sin ωt
Now, notice that C sin ωt cos α + C cos ωt sin α can be simplified using the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b):
f(t) = C sin(ωt + α) + C sin ωt
We can rewrite this as:
f(t) = A sin ωt + B cos ωt
Comparing this form to the desired form A sin(ωt) + B cos(ωt), we can see that A = C and B = C sin α. Therefore, A and B are constants that depend on α.
Moving on to part (b), where we are given C = 10 and α = π/3, we need to find constants k and φ such that f(t) = k sin(ωt + φ).
Using the result from part (a):
f(t) = A sin ωt + B cos ωt
Given that A = C = 10, and B = C sin α = 10 sin(π/3) = 10 * (√3/2) = 5√3, we have:
f(t) = 10 sin ωt + 5√3 cos ωt
To write f(t) in the desired form k sin(ωt + φ), we need to find k and φ. The trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b) can help us rewrite f(t):
f(t) = k sin(ωt + φ)
Comparing this form to the expression for f(t), we can equate the coefficients of sin ωt and cos ωt:
k sin ωt = 10 sin ωt
k cos ωt = 5√3 cos ωt
Since the coefficients of sin ωt are equal, k = 10.
Now, let's look at the coefficients of cos ωt:
k cos ωt = 5√3 cos ωt
To find the value of φ, we equate the coefficients:
5√3 = k cos φ
Substituting k = 10 and solving for cos φ:
5√3 = 10 cos φ
cos φ = (5√3) / 10 = √3 / 2
Therefore, φ = π/6.
Finally, we have:
f(t) = 10 sin(ωt + π/6)
So, the constants k and φ that make f(t) = k sin(ωt + φ) are k = 10 and φ = π/6.