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(wt) and f2(t) = C sin(wt + Alpha). The two sound waves interfere to produce a single sound modeled by the sum of these functions
f(t) = C sin(wt) + C sin(wt + Alpha)
a) Use the addition Formula for Sine to show that f can be written in the form f(t) = A sin(wt) + B Cos(wt), where A and B are constants that depend on Alpha.
b) Suppose that C = 10 and Alpha = pi/3. Find constants k and Ö so that f(t) = ksin(wt+Ö)
a) To show that f(t) can be written in the form f(t) = A sin(wt) + B cos(wt), where A and B are constants that depend on Alpha, we use the addition formula for sine:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
Comparing this formula with f(t) = C sin(wt) + C sin(wt + Alpha), we can see that the first term Csin(wt) matches with the sin(A)cos(B) term, and the second term Csin(wt + Alpha) matches with the cos(A)sin(B) term.
So, we can rewrite f(t) as:
f(t) = Csin(wt) + Csin(wt + Alpha)
= Csin(wt)cos(Alpha) + Ccos(wt)sin(Alpha)
Now, we can recognize A = Ccos(Alpha) and B = Csin(Alpha), where A and B are constants that depend on Alpha. Therefore, we can rewrite f(t) as:
f(t) = Asin(wt) + Bcos(wt)
b) Given C = 10 and Alpha = pi/3, we can find A and B.
A = Ccos(Alpha) = 10cos(pi/3) = 10(1/2) = 5
B = Csin(Alpha) = 10sin(pi/3) = 10(√3/2) = 5√3
So, f(t) = 5sin(wt) + 5√3cos(wt)
Now, we need to find k and Ö so that f(t) = ksin(wt + Ö). Comparing this with f(t) = 5sin(wt) + 5√3cos(wt), we see that k = √(5^2 + (5√3)^2) = √(25 + 75) = √100 = 10.
To find Ö, we can use the formula tan(Ö) = B/A. Thus,
tan(Ö) = (5√3) / 5
Ö = arctan(√3)
Therefore, f(t) = 10sin(wt + arctan(√3)).
a) To show that f(t) can be written in the form f(t) = A sin(wt) + B cos(wt), where A and B are constants that depend on Alpha, we can use the addition formula for sine.
The addition formula for sine states that sin(A + B) = sin(A)cos(B) + sin(B)cos(A).
Let's simplify f(t) using this formula:
f(t) = C sin(wt) + C sin(wt + Alpha)
= C [sin(wt) cos(Alpha) + sin(Alpha) cos(wt)] + C sin(wt) (applying the addition formula for sine)
= C sin(wt) cos(Alpha) + C sin(Alpha) cos(wt) + C sin(wt)
Now, we can see that A = C sin(Alpha) and B = C cos(Alpha).
Therefore, f(t) can be written as f(t) = A sin(wt) + B cos(wt), where A = C sin(Alpha) and B = C cos(Alpha). The constants A and B depend on the value of Alpha.
b) Given that C = 10 and Alpha = pi/3, we can find the constants k and Ö such that f(t) = k sin(wt + Ö).
From part a), we know that A = C sin(Alpha) = 10 sin(pi/3) = 10 * sqrt(3)/2 = 5 sqrt(3).
To find k, we notice that k is the magnitude of the amplitude of the function f(t), which is given by the coefficient of the sine term. In this case, the coefficient of the sine term is A = 5 sqrt(3).
Therefore, k = 5 sqrt(3).
To find Ö, we note that the phase shift is given by the coefficient of wt in the function f(t). In this case, the coefficient of wt is 0.
Therefore, Ö = 0.
Thus, f(t) = k sin(wt + Ö) becomes f(t) = 5 sqrt(3) sin(wt) when C = 10 and Alpha = pi/3.
a) To show that f(t) can be written in the form f(t) = A sin(wt) + B cos(wt), we will use the addition formula for sine:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
Given the functions f1(t) = C sin(wt) and f2(t) = C sin(wt + Alpha), we can rewrite f(t) as:
f(t) = C sin(wt) + C sin(wt + Alpha)
Using the addition formula for sine, we have:
f(t) = C [sin(wt)cos(Alpha) + cos(wt)sin(Alpha)] + C sin(wt)
Expanding this expression, we get:
f(t) = C sin(wt)cos(Alpha) + C cos(wt)sin(Alpha) + C sin(wt)
Rearranging the terms, we have:
f(t) = C sin(wt) + C sin(wt)cos(Alpha) + C cos(wt)sin(Alpha)
Factoring out C sin(wt), we get:
f(t) = C sin(wt) [1 + cos(Alpha)] + C cos(wt)sin(Alpha)
Now, let A = C [1 + cos(Alpha)] and B = C sin(Alpha). Therefore, we can rewrite f(t) as:
f(t) = A sin(wt) + B cos(wt)
where A and B are constants that depend on Alpha.
b) Given C = 10 and Alpha = pi/3, we can find the constants k and Ö to write f(t) in the form f(t) = k sin(wt + Ö).
From part (a), we know that:
A = C [1 + cos(Alpha)] = 10 [1 + cos(pi/3)] = 10 [1 + 1/2] = 10 * (3/2) = 15
B = C sin(Alpha) = 10 sin(pi/3) = 10 * (sqrt(3)/2) = 5sqrt(3)
Now, comparing f(t) = A sin(wt) + B cos(wt) with f(t) = k sin(wt + Ö), we can find k and Ö as follows:
k = sqrt(A^2 + B^2) = sqrt((15)^2 + (5sqrt(3))^2) = sqrt(225 + 75) = sqrt(300) = 10sqrt(3)
Ö = atan2(B, A) = atan2(5sqrt(3), 15) = pi/6
Therefore, f(t) can be written as f(t) = 10sqrt(3) sin(wt + pi/6).