Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Pressure on the eardrum If two tuning forks are struck simultaneously with the same force and are then held at the same distance from the eardrum, the pressure on the outside of the eardrum at time \(t\) is given by $$ p(t)=a \cos \omega_{1} t+a \cos \omega_{2} t, $$ where \(a, \omega_{1}\), and \(\omega_{2}\) are constants. If \(\omega_{1}\) and \(\omega_{2}\) are almost equal, a tone is produced that alternates between loudness and virtual silence. This phenomenon is known as beats. (a) Use a sum-to-product formula to express \(p(t)\) as a product. (b) Show that \(p(t)\) may be considered as a cosine wave with approximate period \(2 \pi / \omega_{1}\) and variable amplitude \(f(t)=2 a \cos \frac{1}{2}\left(\omega_{1}-\omega_{2}\right) t\). Find the maximum amplitude. (c) Shown in the figure is a graph of the equation $$ p(t)=\cos 4.5 t+\cos 3.5 t . $$

Step-by-Step Solution

Answer

(a) Use sum-to-product: \( p(t) = 2a \cos(\frac{\omega_1+\omega_2}{2}t) \cos(\frac{\omega_1-\omega_2}{2}t) \). (b) Approx. period \( 2\pi/\omega_1 \), max amplitude \( 2a \). (c) Simplified: \( 2 \cos(4t) \cos(0.5t) \).

Step 1: Apply the Sum-to-Product Formula

The given expression for pressure is \( p(t) = a \cos(\omega_1 t) + a \cos(\omega_2 t) \). Employ the sum-to-product identity: \( \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \). Set \( A = \omega_1 t \) and \( B = \omega_2 t \), and rewrite: \[ p(t) = 2a \cos\left(\frac{\omega_1 + \omega_2}{2} t\right) \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \] This expresses \( p(t) \) as a product of cosines.

Step 2: Identify Approximate Period and Variable Amplitude

The expression for \( p(t) \) is \[ p(t) = 2a \cos\left(\frac{\omega_1 + \omega_2}{2} t\right) \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \].Considering the cosine terms individually: - The term \( \cos\left(\frac{\omega_1 + \omega_2}{2} t\right) \) represents a cosine wave with approximate period \( \frac{2\pi}{\omega_1} \) because \( \omega_1 \) is close to \( \omega_2 \).- The term \( 2a \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \) provides the amplitude function \( f(t) = 2a \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \).

Step 3: Determine the Maximum Amplitude

The maximum amplitude occurs when the absolute value of the cosine term is 1. Therefore, \[ \text{Max amplitude} = 2a \times 1 = 2a \].Hence, the maximum amplitude of the pressure wave is \( 2a \).

Step 4: Analyze Graph for Specific Frequencies

Given the specific function \( p(t) = \cos 4.5t + \cos 3.5t \), follow a similar format:1. Convert it using the sum-to-product identity:\[ p(t) = 2 \cos\left(\frac{4.5 + 3.5}{2} t\right) \cos\left(\frac{4.5 - 3.5}{2} t\right) \]2. Simplify terms:\[ p(t) = 2 \cos(4t) \cos(0.5t) \]- Here the variation in amplitude is given by \( 2 \cos(0.5t) \), with a maximum amplitude of \( 2 \).

Key Concepts

Sum-to-Product Formula

The Sum-to-Product Formula is a valuable tool in trigonometry, particularly when dealing with expressions involving sums of cosine or sine functions. This identity is specifically helpful for converting a sum of cosines into a product, simplifying complex equations. To understand this better, consider the formula: \( \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \). Here, \(A\) and \(B\) are angles, and by using this identity, you can transform the sum of two cosines into a product of two separate cosine functions.
This transformation is beneficial for analyzing wave patterns and finding amplitudes of combined waveforms. For instance, in the exercise provided, this formula helps express the pressure on the eardrum function \(p(t) = a \cos(\omega_1 t) + a \cos(\omega_2 t)\) as the product: \( p(t) = 2a \cos\left(\frac{\omega_1 + \omega_2}{2} t\right) \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \). This expression simplifies the understanding of how two waves interact with each other to produce beats.

Amplitude Modulation

Amplitude Modulation, or AM, is a concept primarily used in communications, but it also plays a significant role in understanding how wave patterns work in physics. It involves varying the amplitude of a carrier wave to transmit information, but in the context of trigonometric identities and wave functions, it helps describe how the amplitude of a resultant wave changes over time.
In our function \( p(t) \), after applying the sum-to-product formula, we get: \( 2a \cos\left(\frac{\omega_1 - \omega_2}{2} t\right)\), which acts as the amplitude-modulating factor for the cosine wave \( \cos\left(\frac{\omega_1 + \omega_2}{2} t\right) \). This means that the amplitude of the overall wave fluctuates, causing the volume to rise and fall over time, similar to how an AM radio signal works. The maximum amplitude here is determined when this modulating cosine term equals its maximum value of 1, giving a peak amplitude of \(2a\).

• Amplitude modulation gives shape to the wave pattern by expanding or contracting the wave height.
• In our exercise, the modulation implies that the sound alternates between loud and soft, giving rise to the phenomenon of "beats."

Cosine Function

The Cosine Function is a fundamental component of trigonometry, describing a wave pattern often used to model oscillations and circular phenomena. The basic cosine wave, described by \( \cos(x) \), oscillates between -1 and 1, with a specific period and frequency depending on the context.
In the pressure function \( p(t) \), the cosine function forms the backbone of our wave analysis. By recognizing the separate cosine components \( \cos\left(\frac{\omega_1 + \omega_2}{2} t\right) \) and \( \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \), we see how the waveforms interact to create a combined oscillating pattern:

• \( \cos\left(\frac{\omega_1 + \omega_2}{2} t\right) \) indicates the main oscillating frequency of the wave, with its period approximately \(\frac{2\pi}{\omega_1}\).
• \( \cos\left(\frac{\omega_1 - \omega_2}{2} t\right) \) modulates this waveform, influencing how the sound intensity rises and falls.

Understanding these interactions is crucial when exploring more complex phenomena like beats, where multiple wave frequencies combine.

Beat Frequency

Beat Frequency is an amazing acoustic phenomenon observed when two close but distinct frequencies interfere with one another. This concept helps explain why tones sometimes appear to rapidly alternate in loudness, experiencing phases of increased or decreased sound intensity.
In the context of the provided exercise, beat frequency arises because the two tuning forks have slightly different frequencies denoted by \( \omega_1 \) and \( \omega_2 \). This means the resultant frequency difference, \( \frac{\omega_1 - \omega_2}{2} \), generates beats when these oscillations constructively and destructively interfere.
Key points to remember about beat frequency include:

• The rate or speed of the beat is determined by the exact frequency difference \( \omega_1 - \omega_2 \).
• The amplitude varies, resulting in moments of increased loudness (constructive interference) and silence (destructive interference).

This phenomenon is crucial within sound engineering and musical tuning, where precise adjustments ensure harmonics and pleasant tonal quality.