Problem 107 - Precalculus Mathematics for Calculus

Question

Sound Beats When two pure notes that are close in frequency are played together, their sounds interfere to produce beats; that is, the loudness (or amplitude) of the sound alternately increases and decreases. If the two notes are given by $$f_{1}(t)=\cos 11 t \quad \text { and } \quad f_{2}(t)=\cos 13 t$$ the resulting sound is \(f(t)=f_{1}(t)+f_{2}(t)\) (a) Graph the function \(y=f(t)\) (b) Verify that \(f(t)=2 \cos t \cos 12 t\) (c) Graph \(y=2 \cos t\) and \(y=-2 \cos t,\) together with the graph in part (a), in the same viewing rectangle. How do these graphs describe the variation in the loudness of the sound?

Step-by-Step Solution

Answer

Graph the functions, verify the identity using the cosine sum formula, and observe the amplitude envelopes for beat patterns.

Step 1: Graph the Function

To graph the function \(y = f(t)\), start by understanding that \(f(t) = f_1(t) + f_2(t) = \cos(11t) + \cos(13t)\). Use graphing software or a calculator that supports trigonometric functions to plot \(y = \cos(11t) + \cos(13t)\) over a suitable range of \(t\), such as from \(-2\pi\) to \(2\pi\). Vary the range if necessary to observe the beat pattern, which manifests as alternating increases and decreases in amplitude.

Step 2: Verify the Identity

We need to verify that \(f(t) = 2\cos(t)\cos(12t)\). Use the trigonometric identity for the sum of cosines: \(\cos(A) + \cos(B) = 2\cos\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right)\). Here, \(A = 11t\) and \(B = 13t\), so:\[f(t) = 2 \cos\left(\frac{11t + 13t}{2}\right) \cos\left(\frac{11t - 13t}{2}\right) = 2 \cos(12t) \cos(t)\].This confirms the identity holds.

Step 3: Graph the Envelopes

To understand how the sound's loudness changes, graph \(y = 2 \cos(t)\) and \(y = -2 \cos(t)\) on the same plot as \(f(t)\). These two functions create an envelope around \(f(t)\) showing the maximum and minimum reaches of the sound wave. This pattern results from beats and illustrates how the amplitude of \(f(t)\) varies between these bounds, capturing the essence of beat-induced loudness modulation.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in simplifying and manipulating trigonometric functions. They help to reveal relationships between different trigonometric expressions. One of the most useful identities is the sum-to-product identity, particularly when dealing with sound waves and beats. To understand how two sounds create beats, consider the formula: \[ \cos(A) + \cos(B) = 2\cos\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right) \] This identity allows us to express the sum of two cosine functions in a product form. This transformation is crucial for analyzing the interference patterns created by sound waves.

• First, by setting \( A = 11t \) and \( B = 13t \), we calculate the averages and differences of these angles.
• The identity simplifies the addition of two close-frequency waves into a product, highlighting their interaction.
• This product shows how amplitude changes over time, a key factor in understanding sound beats.

Such identities not only simplify calculations but they also enable deeper insight into the function's behavior, such as visualizing sound wave interactions.

Graphs of Trigonometric Functions

Understanding the graphs of trigonometric functions is essential for visualizing phenomena like sound beats. Trigonometric graphs, such as those of cosine and sine functions, exhibit periodic behaviors which can model repetitive waves. When graphing a function like \( y = \cos(11t) + \cos(13t) \), we observe a complex waveform. The beat effect appears as the periodic variation in amplitude arises from the combination of slightly different frequencies.

• The resulting graph shows a wave with oscillating intensity, characteristic of beats.
• Graphing software helps in visualizing how these waves interact, creating areas of constructive and destructive interference.
• The key is to recognize the envelope patterns formed by the interactions.

Graphing alongside envelope functions like \( y = 2 \cos(t) \) and \( y = -2 \cos(t) \) further clarifies how the amplitude confines between these limits. This visual representation effectively illustrates the concept of beats, making it easier to comprehend how sound levels change.

Amplitude Modulation

Amplitude modulation is a vital concept in both physics and engineering, especially in understanding sound beats. It revolves around altering the amplitude of a wave, usually in response to another signal or frequency. When two sound waves of slightly different frequencies interfere, they create a new wave pattern, where the amplitude varies rhythmically, a process known as amplitude modulation. This is exactly what happens in sound beats.

• The modulation results from the interference of close-frequency sound waves.
• Variations in amplitude represent the beat frequency, which is the difference between the two original frequencies.
• This concept captures how sound intensity increases and decreases, affecting the perceived loudness.

The envelope created by the functions \( y = 2 \cos(t) \) and \( y = -2 \cos(t) \) visually captures the range within which the sound wave amplitude oscillates. Amplitude modulation thus provides a practical framework for understanding how slight frequency differences can produce significant acoustic effects.