Problem 69 - Precalculus Mathematics for Calculus

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) The sum is \(A = C(1 + \cos \alpha), B = C \sin \alpha\). (b) \(f(t) = 20 \sin(\omega t + \frac{\pi}{6})\).

Step 1: Understanding the Problem

We have two functions, \(f_1(t) = C \sin \omega t\) and \(f_2(t) = C \sin(\omega t + \alpha)\). We need to add these functions and express their sum in a new form using trigonometric identities.

Step 2: Applying the Addition Formula for Sine

The sine addition formula states \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). We use this formula to rewrite \(f_2(t) = C \sin (\omega t + \alpha) = C(\sin \omega t \cos \alpha + \cos \omega t \sin \alpha)\).

Step 3: Express the Sum of Functions

Compute \(f(t) = f_1(t) + f_2(t)\): \[ f(t) = C \sin \omega t + C (\sin \omega t \cos \alpha + \cos \omega t \sin \alpha) \]. Simplifying gives \[ f(t) = C(\sin \omega t (1 + \cos \alpha) + \cos \omega t \sin \alpha) \].

Step 4: Formulate in Terms of A and B

From the expression \(f(t) = C(\sin \omega t (1 + \cos \alpha) + \cos \omega t \sin \alpha)\), we identify \(A = C(1 + \cos \alpha)\) and \(B = C \sin \alpha\). Thus, \(f(t) = A \sin \omega t + B \cos \omega t\).

Step 5: Substitution with Given Values

For part (b), substitute \(C = 10\) and \(\alpha = \frac{\pi}{3}\): \(\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}\) and \(\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\). Calculate \[ A = 10(1 + \frac{1}{2}) = 15 \] and \[ B = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \].

Step 6: Express in Amplitude and Phase Form

We seek \(f(t) = k \sin(\omega t + \phi)\): Identify \(k = \sqrt{A^2 + B^2}\) and \(\tan \phi = \frac{B}{A}\). Calculate \[ k = \sqrt{15^2 + (5\sqrt{3})^2} = 20 \]. Thus, \(\tan \phi = \frac{5\sqrt{3}}{15} = \frac{\sqrt{3}}{3}\), which gives \(\phi = \frac{\pi}{6}\).

Key Concepts

Addition Formula for Sine

The addition formula for sine is an essential trigonometric identity that simplifies the process of adding two sine functions. When you are dealing with functions like these in sound waves, understanding this formula is crucial. The formula states that \[\sin(a + b) = \sin a \cos b + \cos a \sin b\] In the context of the given problem, this formula helps to re-express the wave signals produced by the tuning forks. Each signal can be written in terms of sums and products of sine and cosine. For example, given \(f_2(t) = C \sin (\omega t + \alpha)\), you apply the addition formula:

• \(\sin(\omega t + \alpha)\) becomes \(\sin \omega t \cos \alpha + \cos \omega t \sin \alpha\).
• This shows the impact of both frequency and phase shift on the wave.

This conversion is vital to derive further expressions and helps us in understanding the interference pattern formed by these waves.

Interference in Sound Waves

When two sound waves interact, they create an interference pattern. This results from the superposition of the waves. In simple terms, interference is about how waves combine and affect each other.

In our example, each tuning fork produces a wave, \(f_1(t)\) and \(f_2(t)\). The sound you hear is from the combined effect of these waves, expressed as: \[f(t) = C \sin \omega t + C \sin (\omega t + \alpha) \] This creates a new wave. The result depends on the phase difference \(\alpha\) between the two waves.

• If \(\alpha\) is an integer multiple of \(2\pi\), the waves are in phase, amplifying the sound.
• If it's an odd multiple of \(\pi\), they are out of phase, causing destructive interference and reducing the sound.

Understanding interference is crucial in audio engineering and other fields where control over sound waves is required.

Amplitude and Phase Form

Expressing wave functions in terms of amplitude and phase shifts is a concise way to describe oscillations. It transforms the combined form of sine and cosine into a single sine function with a phase shift, which has both theoretical and practical benefits. Starting from the expression:

• \(f(t) = A \sin \omega t + B \cos \omega t\), with \(A = C(1 + \cos \alpha)\) and \(B = C \sin \alpha\).
• We express this as \(f(t) = k \sin(\omega t + \phi)\), where \(k\) is the amplitude and \(\phi\) is the phase shift.

The amplitude \(k\) is derived using: \[ k = \sqrt{A^2 + B^2} \] This gives a measure of the sound's intensity. The phase shift \(\phi\) is determined by: \[ \tan \phi = \frac{B}{A} \] This helps pinpoint the exact moment a wave hits its peak. Overall, using amplitude and phase representation provides a clear and simplified way to analyze complex waveforms. This concept is crucial in fields such as acoustics, electronics, and communications.