Problem 69 - 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) A = C(1 + \cos\alpha), B = C\sin\alpha. b) f(t) = 10\sqrt{3}\sin (\omega t + \frac{\pi}{6}).

Step 1: Expand second sine term

Let's begin by expanding the second sine term using the sine addition formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). Here, our expression is \( f_2(t) = C \sin (\omega t + \alpha) \). Applying the formula, we have: \[ C \sin (\omega t + \alpha) = C(\sin \omega t \cos \alpha + \cos \omega t \sin \alpha) \]

Step 2: Write f(t) as a single function

Now, substitute the expanded form of \( f_2(t) \) back into the original function: \[ f(t) = C \sin \omega t + C(\sin \omega t \cos \alpha + \cos \omega t \sin \alpha) \] Simplifying it gives: \[ f(t) = C \sin \omega t + C \sin \omega t \cos \alpha + C \cos \omega t \sin \alpha \] Combine the sine terms to get: \[ f(t) = C \sin \omega t(1 + \cos \alpha) + C \sin \alpha \cos \omega t \]

Step 3: Identify constants A and B

We can now identify the constants \( A \) and \( B \) from the equation. By comparing with \ \( f(t) = A \sin \omega t + B \cos \omega t \): \[ A = C(1 + \cos \alpha), \quad B = C \sin \alpha \] So, the function \( f(t) = A \sin \omega t + B \cos \omega t \) is confirmed.

Step 4: Substitute given values for C and \( \alpha \)

Next, we use the given values \( C = 10 \) and \( \alpha = \frac{\pi}{3} \): \[ A = 10(1 + \cos \frac{\pi}{3}), \quad B = 10 \sin \frac{\pi}{3} \] Knowing that \( \cos \frac{\pi}{3} = \frac{1}{2} \) and \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), we substitute: \[ A = 10(1 + \frac{1}{2}) = 15, \quad B = 10 \frac{\sqrt{3}}{2} = 5\sqrt{3} \]

Step 5: Convert form to k \( \sin (\omega t + \phi) \)

We use the identity \( A \sin \omega t + B \cos \omega t = k \sin(\omega t + \phi) \) where \( k = \sqrt{A^2 + B^2} \) and \( \tan \phi = \frac{B}{A} \). Calculate \( k \): \[ k = \sqrt{15^2 + (5\sqrt{3})^2} = \sqrt{225 + 75} = \sqrt{300} = 10\sqrt{3} \] Calculate \( \phi \): \[ \tan \phi = \frac{5\sqrt{3}}{15} = \frac{\sqrt{3}}{3}\] Thus, \( \phi = \tan^{-1}(\frac{\sqrt{3}}{3}) \), which equals \( \frac{\pi}{6} \).

Step 6: Final Answer

The rewritten function is \( f(t) = 10\sqrt{3} \sin (\omega t + \frac{\pi}{6}) \).

Key Concepts

Addition Formula for Sine

The addition formula for sine is a fundamental trigonometric identity that simplifies expressions involving the sum of angles in sine terms. In its basic form, it is expressed as \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). This formula is particularly useful when you need to break down complex trigonometric expressions into more manageable components.

In the context of wave interference, using the addition formula allows us to expand a sine function of the form \( C \sin(\omega t + \alpha) \). This means we can express it as \( C(\sin \omega t \cos \alpha + \cos \omega t \sin \alpha) \).

• \( \sin \omega t \) and \( \cos \omega t \) are the fundamental trigonometric functions representing the oscillatory nature of waves.
• \( \cos \alpha \) and \( \sin \alpha \) act as scaling factors based on the angle \( \alpha \).

It is important to use the addition formula correctly to ensure the accurate manipulation of trigonometric functions in physics and engineering, especially when analyzing wave properties such as interference patterns.

Trigonometric Identities

Trigonometric identities are equations that express relationships between the trigonometric functions of an angle. These identities are extremely useful in simplifying expressions and solving problems that involve periodic components, such as waves and oscillations.

In wave interference, identities help us express a combination of sine and cosine functions in a simpler, single form. A common identity used is the linear combination of sine and cosine functions: \( A \sin \omega t + B \cos \omega t = k \sin(\omega t + \phi) \). In this expression:

• \( A \) and \( B \) are constants derived from the wave equations.
• \( k \), the amplitude, is computed as \( k = \sqrt{A^2 + B^2} \).
• \( \phi \), the phase shift, is given by \( \tan \phi = \frac{B}{A} \).

Trigonometric identities enable the transformation of complex wave interactions into simpler representations, aiding in better understanding and analysis of oscillatory systems.

Amplitude and Phase Shift

The concepts of amplitude and phase shift are crucial for understanding the behavior of periodic waveforms, especially in contexts such as sound waves, electromagnetic waves, or any system characterized by oscillations.

• Amplitude (\( k \)): In the context of wave functions, the amplitude is the peak value of the wave, indicating how "strong" or "intense" the wave is. It is calculated using the formula \( k = \sqrt{A^2 + B^2} \) where \( A \) and \( B \) come from the coefficients of sine and cosine terms.
• Phase Shift (\( \phi \)): This parameter indicates how much the wave is shifted horizontally from a reference point. The phase shift can be found using \( \tan \phi = \frac{B}{A} \), and provides insights into the synchrony or out-of-phaseness of waves interacting with each other.

Understanding amplitude and phase shift allows for better interpretation of the resultant waveform in tasks involving wave superposition, like examining sound interference from tuning forks. Adjusting these parameters in equations helps describe the exact nature and timing of wave peaks in relation to each other, which is vital in both theoretical and applied physics.