Problem 37 - Precalculus with Calculus Previews

Question

Sound Wave A note produced by a certain musical instrument results in a sound wave described by $$ f(t)=0.03 \sin 500 \pi t+0.03 \sin 1000 \pi t $$ where \(f(t)\) is the difference between atmospheric pressure and air pressure in dynes per square centimeter at the eardrum after \(t\) seconds. Express \(f\) as the product of a sine and a cosine function.

Step-by-Step Solution

Answer

\( f(t) = 0.06 \sin(750\pi t) \cos(250\pi t) \).

Step 1: Identify the Trigonometric Identity

The goal is to express the given function as a product of a sine and a cosine function. Observing both sinusoidal expressions \( \sin 500 \pi t \) and \( \sin 1000 \pi t \), we can employ the sum-to-product identity: \( \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \).

Step 2: Apply the Sum-to-Product Identity

We have \( A = 1000\pi t \) and \( B = 500\pi t \). Substitute these values into the identity: \[ f(t) = 0.03 \left( \sin(1000\pi t) + \sin(500\pi t) \right) = 0.03 \cdot 2 \sin \left( \frac{1000\pi t + 500\pi t}{2} \right) \cos \left( \frac{1000\pi t - 500\pi t}{2} \right). \]

Step 3: Simplify the Expression

Calculate \( \frac{1000\pi t + 500\pi t}{2} = 750\pi t \) and \( \frac{1000\pi t - 500\pi t}{2} = 250\pi t \). Substitute these into the expression: \[ f(t) = 0.06 \sin(750\pi t) \cos(250\pi t). \]

Step 4: Final Expression

Now, we express \( f(t) \) as desired: \( f(t) = 0.06 \sin(750\pi t) \cos(250\pi t) \). This is the equivalent expression of the sound wave in product form.

Key Concepts

Sum-to-Product Identity

The sum-to-product identity is a useful tool in trigonometry. It helps us convert the sum of two sinusoidal functions into a product form. This is particularly valuable when dealing with wave interference and beat frequencies. For sine functions, the identity is expressed as:

• \( \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \)

This conversion captures the combined effect of two waves in a form that is easier to analyze. In practice, when you see two sine terms being added, you can use this identity to simplify calculations and study the wave properties more deeply.
For example, in a sound wave context, this identity allows us to understand how different frequency components interact. It is particularly valuable when trying to see how different pitches mix and create what we perceive as harmonics.

Product-to-Sum Identities

Just as the sum-to-product identity helps simplify the sum of trigonometric functions into a product, the product-to-sum identities reverse the process. These identities are:

• \( \sin A \cos B = \frac{1}{2} \left( \sin(A+B) + \sin(A-B) \right) \)
• \( \cos A \sin B = \frac{1}{2} \left( \sin(A+B) - \sin(A-B) \right) \)
• \( \cos A \cos B = \frac{1}{2} \left( \cos(A+B) + \cos(A-B) \right) \)
• \( \sin A \sin B = \frac{1}{2} \left( \cos(A-B) - \cos(A+B) \right) \)

These identities are helpful when you have the product of sine and cosine functions that need to be expressed as a sum. This method is often used in signal processing to decompose complex waveforms into simpler parts.
By converting products to sums, we can analyze how two different frequencies interact. This can be essential in understanding phenomena like modulation and beat frequencies in sound engineering.

Sound Waves

Sound waves are a type of pressure wave traveling through a medium such as air. They consist of compressions and rarefactions, which we perceive as sound. In mathematics, sound waves are often expressed using sinusoidal functions due to their oscillatory nature.
When instruments produce sound, they generate waves that can be described by trigonometric functions like sine and cosine. Each wave corresponds to different sound properties such as frequency (pitch) and amplitude (loudness).
The study of sound waves involves looking at how these waves combine and interact. This is where trigonometric identities become useful, simplifying complex expressions. By using these identities, we gain insight into harmonics and resonance.
Understanding sound waves through trigonometry helps in various applications, from music technology to acoustics engineering.

Sinusoidal Functions

Sinusoidal functions, which include sine and cosine functions, are fundamental in modeling periodic phenomena. These functions accurately depict how variables oscillate over time, making them essential in studies of waves, particularly sound waves.
Key characteristics of sinusoidal functions include their amplitude, frequency, and phase shift:

• **Amplitude** represents the wave's height and is related to the volume in sound waves.
• **Frequency** determines how rapidly the wave oscillates, corresponding with the pitch of a sound.
• **Phase Shift** indicates how the wave aligns relative to the origin of time.

Sinusoidal functions enable us to replicate natural rhythmic patterns, such as the ebb and flow of tides or vibrations in a string. In sound engineering, these functions help us understand and manipulate sound properties effectively.
By examining sinusoidal functions, we can break down complex sound signals into simpler wave components, analyze them individually, and understand how different wavelengths contribute to the resulting waveforms we hear.