Problem 107 - Algebra and Trigonometry

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\( and \)\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

(a) Graphs the sum of two cosines. (b) Uses trig identity to verify the formula. (c) Graphs show sound amplitude varies between envelopes.

Step 1: Graphing the Function y=f(t)

The given function is \(f(t) = \cos 11t + \cos 13t\). To graph this function, plot both \(f_1(t) = \cos 11t\) and \(f_2(t) = \cos 13t\), then graph their sum \(f(t)\) by calculating the value \(f(t)\) for a range of \(t\) values. A graphing calculator or software will help to visualize how the function changes over time, displaying the interference pattern that forms the beats.

Step 2: Verifying the Expression f(t) = 2 cos(t) cos(12t)

To verify that \(f(t) = 2 \cos t \cos 12t\), we use the trigonometric identity \( \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \). For \(f(t) = \cos 11t + \cos 13t\), set \(A = 13t\) and \(B = 11t\). This becomes:\[\cos 13t + \cos 11t = 2 \cos\left(\frac{13t + 11t}{2}\right) \cos\left(\frac{13t - 11t}{2}\right) = 2 \cos 12t \cos t\]Thus, the expression is verified.

Step 3: Graphing y=2 cos t and y=-2 cos t

Graph \(y = 2 \cos t\) and \(y = -2 \cos t\) alongside the graph of \(f(t)\) from step 1. These two graphs form an envelope that bounds the graph of the sound beats \(f(t)\). The amplitude of \(f(t)\) oscillates between \(2 \cos t\) and \(-2 \cos t\), indicating the variation in the loudness of the sound over time.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate different trigonometric functions to each other. One of the most useful identities is the sum-to-product identity, which helps in simplifying expressions involving sums of trigonometric functions.
In our exercise, the function is given as the sum of two cosine functions: \( f(t) = \cos 11t + \cos 13t \). To simplify this, we use the identity:

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

By applying it to \( A = 13t \) and \( B = 11t \), the transformation gives:

• \( f(t) = \cos 11t + \cos 13t = 2 \cos 12t \cos t \),

simplifying the interpretation of the wave function.
This identity is especially useful in identifying patterns and simplifying problems involving wave interference, such as sound waves, in this exercise.

Interference Patterns

Interference patterns occur when two or more waves overlap and combine to form a new wave pattern. This is a common phenomenon with sound waves, light waves, and other types of waves.
In sound, when two frequencies that are close together overlap, they create a phenomenon known as "beats." Beats are variations in sound amplitude, felt as changes in loudness. The periodic increases and decreases in sound intensity are the result of constructive and destructive interference.

• Constructive interference happens when wave peaks align, amplifying the sound.
• Destructive interference occurs when wave peaks and troughs align, reducing sound intensity.

In the exercise, the function \( f(t) = \cos 11t + \cos 13t \) represents such an interference pattern. The result is a new wave function, \( f(t) = 2 \cos t \cos 12t \), showcasing how two frequencies blend into a single pattern, with beats being the observable effect of these interference patterns.

Graphing Functions

Graphing functions is a visual method to understand mathematical relationships and patterns. In the context of waves, it helps us see how they fluctuate over time.
When graphing the function \( f(t) = \cos 11t + \cos 13t \), you'll observe the "beats" or variations in amplitude as peaks rise and fall.
By plotting \( y = 2 \cos t \) and \( y = -2 \cos t \), alongside the original function, you create an "envelope". This envelope visibly demonstrates the maximum and minimum reach of the wave's amplitude.

• The envelope curves \( 2 \cos t \) and \( -2 \cos t \) act as bounds for the graph of \( f(t) \).
• The varying graph of \( f(t) \) zigzags between these bounds, showing the amplitude changes over time.

Understanding these graphs assists in visualizing sound's loudness variation, a critical aspect in sound engineering and acoustics, allowing us to see the impact of interference at various points in time.