Problem 55 - Algebra and Trigonometry

Question

Interference Two identical tuning forks are struck, one a fraction of a second after the other. The sounds produced are modeled by \(f_{1}(t)=C \sin \omega t\) and \(f_{2}(t)=C \sin (\omega t+\alpha)\) The two sound waves interfere to produce a single sound modeled by the sum of these functions $$f(t)=C \sin \omega t+C \sin (\omega t+\alpha)$$ (a) Use the addition formula for sine to show that \(f\) can be written in the form \(f(t)=A \sin \omega t+B \cos \omega t,\) where \(A\) and \(B\) are constants that depend on \(\alpha .\) (b) Suppose that \(C=10\) and \(\alpha=\pi / 3 .\) Find constants \(k\) and \(\phi\) so that \(f(t)=k \sin (\omega t+\phi)\)

Step-by-Step Solution

Answer

(a) Use addition to get A, B expressions; (b) Find k=10√3, \( \phi=\pi/6 \).

Step 1: Applying the Sine Addition Formula

To simplify the expression \( f(t) = C \sin \omega t + C \sin(\omega t + \alpha) \), we use the sine addition formula: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). This gives us: \[C \sin(\omega t + \alpha) = C ( \sin \omega t \cos \alpha + \cos \omega t \sin \alpha )\]Substitute this back into the original equation for \( f(t) \): \[f(t) = C \sin \omega t + C ( \sin \omega t \cos \alpha + \cos \omega t \sin \alpha )\] Combine like terms to rearrange the equation:

Step 2: Combine Like Terms

Combine the like terms from the expression:\[f(t) = C \sin \omega t (1 + \cos \alpha) + C \cos \omega t \sin \alpha\]This can be rewritten as:\[f(t) = A \sin \omega t + B \cos \omega t\]where:\[A = C (1 + \cos \alpha), \quad B = C \sin \alpha\]

Step 3: Substitute Given Values to Find A and B

Given that \( C = 10 \) and \( \alpha = \pi /3 \), substitute these values into the expressions for \( A \) and \( B \):\[A = 10 (1 + \cos \pi/3)\]\[B = 10 \sin \pi/3\]We calculate these as:\[\cos \pi/3 = 1/2 \quad \Rightarrow \quad A = 10 (1 + 1/2) = 15\]\[\sin \pi/3 = \sqrt{3}/2 \quad \Rightarrow \quad B = 10 (\sqrt{3}/2) = 5 \sqrt{3}\]

Step 4: Express in the Form of a Single Sine Function

We want to rewrite \( f(t) \) in the form \( k \sin(\omega t + \phi) \). This can be achieved by using the identity: \[k\sin(\omega t + \phi) = k ( \cos \phi \sin \omega t + \sin \phi \cos \omega t)\] By comparing coefficients, we have: \[k \cos \phi = A = 15\] \[k \sin \phi = B = 5 \sqrt{3}\] To find \( k \), use \( k = \sqrt{A^2 + B^2} \): \[k = \sqrt{15^2 + (5 \sqrt{3})^2} = \sqrt{225 + 75} = \sqrt{300} = 10\sqrt{3}\]

Step 5: Determine Angle \( \phi \)

To find \( \phi \), use the ratio \( \tan \phi = \frac{B}{A} \):\[\tan \phi = \frac{5 \sqrt{3}}{15} = \frac{\sqrt{3}}{3}\]So \( \phi = \tan^{-1}(\frac{\sqrt{3}}{3}) \), which simplifies to \( \pi/6 \). Thus, \[f(t) = 10 \sqrt{3} \sin(\omega t + \pi/6)\]

Step 6: Confirm the Expressions

Thus, the constants \( k \) and \( \phi \) satisfy:\[k = 10 \sqrt{3}\]\[\phi = \frac{\pi}{6}\]This confirms the required expression for \( f(t) \).

Key Concepts

Sine Addition Formula

The sine addition formula is a powerful tool for simplifying and manipulating trigonometric expressions. It states that if you have two angles, say \( a \) and \( b \), then the sine of their sum, \( \sin(a + b) \), can be expressed as: - \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).
This formula is very useful in problems involving interfering waves or oscillations, like the one in this exercise. By applying the sine addition formula to the second term \( C \sin(\omega t + \alpha) \), we separate the wave component into terms involving both sine and cosine.
For our sound wave problem, this allows us to transform the expression into a more manageable form, eventually leading to a representation of the wave in terms of sine and cosine components, as detailed in the solution steps, which simplifies the calculation of constants \( A \) and \( B \).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They are essential in simplifying complex trigonometric expressions and solving equations.
The sine and cosine addition formulas are examples of such identities. In the given problem, identities like \( \sin(a + b) = \sin a \cos b + \cos a \sin b \) are crucial for breaking down sine terms into components that can be added or manipulated further.
Other useful identities include Pythagorean identities like \( \sin^2 \theta + \cos^2 \theta = 1 \). These can be used in conjunction with the above to find solutions.- Trigonometric identities not only simplify expressions but also play a key role in converting between different forms of trigonometric functions, crucial for correctly solving equations like those encountered in interference problems.

Sound Wave Interference

Sound wave interference occurs when two or more sound waves overlap and combine to form a new wave pattern. This interference can be constructive or destructive depending on how the waves align. - Constructive interference amplifies the sound when waves add up, leading to louder sounds. - Destructive interference reduces the sound when waves cancel each other out.
In this exercise, the waves from the two tuning forks interfere, and we're using trigonometric concepts to describe the resulting sound. The function \( f(t) = C \sin \omega t + C \sin(\omega t + \alpha) \) models this interference.
Understanding interference is crucial in designing musical instruments (like tuning forks) and in many areas of engineering, including acoustics, where wave behaviors are essential.

Complex Numbers in Trigonometry

Complex numbers provide a powerful way to analyze and solve trigonometric problems, especially those involving wave interference. By representing trigonometric functions in terms of complex numbers, we can apply algebraic methods to solve them efficiently.
A trigonometric expression like \( e^{i\theta} = \cos \theta + i \sin \theta \) (Euler's formula) shows how complex numbers and trigonometry are linked. This expression allows the use of exponential functions to handle trigonometric functions.- This connection is particularly useful when phasors (a type of vector used in physics, representing sinusoidally varying functions) are involved in interference problems.
These mathematical techniques make it easier to solve equations involving wave patterns and interference, such as those encountered in this exercise. By transitioning between real and imaginary parts, we can analyze amplitude and phase shifts without relying solely on geometric interpretations.