Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Hearing Beats in Music Musicians sometimes tune instruments by playing the same tone on two different instruments and listening for a phenomenon known as beats. When the two instruments are in tune, the beats disappear. The ear hears beats because the pressure slowly rises and falls as a result of the slight variation in the frequency. This phenomenon can be seen on a graphing calculator. (a) Consider two tones with frequencies of 220 and \(223 \mathrm{Hz}\) and pressures \(P_{1}=0.005 \sin 440 \pi t \quad\) and \(\quad P_{2}=0.005 \sin 446 \pi t\) respectively. A graph of \(P_{1}+P_{2}\) as \(Y_{3}\) felt by an eardrum over the 1 -second interval \([0.15,1.15]\) is shown here. How many beats are there in 1 second? (Graph can't copy) (b) Repeat part (a) with frequencies of 220 and 216 (c) Determine a simple way to find the number of beats per second if the frequency of each tone is given.

Step-by-Step Solution

Answer

(a) 3 beats, (b) 4 beats, (c) Beat frequency = |f1 - f2|.

Step 1: Understanding Beats

When two waves of slightly different frequencies interfere, they produce beats. The beat frequency is the absolute difference between the two frequencies. The number of beats per second, or the beat frequency, can be calculated using this difference.

Step 2: Calculate Beat Frequency for Part (a)

For part (a), the frequencies of the two tones are 220 Hz and 223 Hz, respectively. The beat frequency is \[ f_{beat} = |223 - 220| = 3\] Thus, there are 3 beats per second.

Step 3: Calculate Beat Frequency for Part (b)

For part (b), the frequencies of the two tones are 220 Hz and 216 Hz, respectively. The beat frequency is \[ f_{beat} = |220 - 216| = 4 \] Thus, there are 4 beats per second.

Step 4: General Formula for Beat Frequency

To find the number of beats per second, calculate the absolute difference between the two frequencies. If the frequencies are \( f_1 \) and \( f_2 \), then the beat frequency is \[ f_{beat} = |f_1 - f_2| \] This formula provides the number of beats per second experienced by the ear.

Key Concepts

Interference of Sound Waves

When two sound waves with slightly different frequencies collide, they can interference, creating an interesting auditory phenomenon called beat frequencies. As these sound waves overlap, they periodically strengthen or weaken each other, producing a distinctive pattern of sound that rises and falls. This pulsing pattern is perceived as beats by our ears. The strength of these beats, or their amplitude, increases when the waves are in phase and decreases when the waves are out of phase. This alteration in amplitude is what creates the characteristic "wah-wah" sound of beats. Understanding this concept is crucial for musicians who rely on this to fine-tune instruments, ensuring harmonious sound production.

Graphical Calculator

Graphical calculators offer a visual representation of mathematical problems, which can aid in understanding complex concepts like sound wave interference. By plotting functions representing sound waves, such as pressure as a sine function of time, users can visualize what happens when these waves combine. For instance, graphing the sum of two wave functions, like the pressures from two different tones, shows where the waves amplify each other (peaks) and where they cancel each other out (troughs). Students can use this graphical interpretation to see the formation of beats as the wave patterns fluctuate over time. This visual tool is indispensable in labs and classrooms to demonstrate theory in practice.

Frequency Difference

The frequency difference between two sound waves determines the beat frequency, or the number of beats heard each second. This is calculated by finding the absolute difference between the two frequencies. For example, if one wave has a frequency of 220 Hz and another has 223 Hz, the beat frequency is the absolute difference, which is 3 beats per second. This means that over one second, someone listening would perceive three fluctuations in volume. This idea forms a critical part of tuning instruments, as musicians listen for beats to gauge the closeness of two pitches.

Sound Waves in Music

Sound waves play a foundational role in music, with their frequencies determining pitch and their amplitudes, affecting volume. Different instrumentalists often play together, each producing unique sound waves that must harmonize for a pleasing auditory experience. When two tones are played simultaneously, any slight difference in frequency can lead to interference, producing beats. Musicians use this to their advantage: by adjusting their instrument's tuning until the beats vanish, they ensure the instruments are in tune. This is especially crucial in ensembles, where even minor tuning discrepancies can lead to noticeable dissonance. Understanding the interplay of sound waves is essential to mastering musical performance and creation.