Question

Alternating Current. Project: Beats: Beats occur when two sine waves of slightly different frequencies are combined. (a) Read about beats, and write a short paper on the subject. (b) Graph two sine waves, having frequencies of \(24 \mathrm{Hz}\) and \(30 \mathrm{Hz}\), for about 20 cycles. Then graph the sum of those two sine waves. (c) From the preceding graphs, what can you deduce about the frequency of the beats? (d) Try to produce beats, using a musical instrument, tuning forks, or some other device. (e) Name one practical use for beats.

Short answer

The beat frequency is the difference between the individual frequencies of the two waves. When graphed, the sum of waves of 24 Hz and 30 Hz results in a beat frequency of 6 Hz, which is calculated by counting the peaks of amplitude fluctuation in the combined wave.

Step-by-step solution

Understanding Beats

Beats are a phenomenon that occurs when two sound waves of nearly identical frequencies are played together. This results in a new sound wave that fluctuates in volume or amplitude. The beat frequency is the absolute difference between the frequencies of the two combining waves. When the waves are in sync (constructive interference), the amplitude increases, resulting in a loud sound. When the waves are out of sync (destructive interference), the amplitude decreases, causing a faint sound.

Graphing Individual Sine Waves

Graph the two sine waves of the given frequencies separately. For a frequency of 24 Hz, plot the sine wave with an equation that may look like: \(y_1 = \sin(2\pi \cdot 24t)\). Likewise, for a frequency of 30 Hz, the sine wave equation would be: \(y_2 = \sin(2\pi \cdot 30t)\). Here, \(t\) represents time in seconds. Plot both waves over a time period sufficient to show about 20 cycles, which would be slightly less than one second.

Graphing the Sum of the Sine Waves

Sum the two sine waves using the principle of superposition to get the combined wave: \(y = y_1 + y_2 = \sin(2\pi \cdot 24t) + \sin(2\pi \cdot 30t)\). Graph this equation over the same time span as used in Step 2 to show the resulting wave. The beats will be visible as the varying amplitude of the combined sine wave.

Determining the Beat Frequency

By examining the graph obtained in Step 3, determine the frequency of the beats. This is done by measuring the time between successive points of maximum amplitude (the loud parts of the beat) or minimum amplitude (the quiet parts of the beat). The beat frequency is the reciprocal of this time interval. Since the individual frequencies are 24 Hz and 30 Hz, the beat frequency is expected to be the difference between them, which is \(30 - 24 = 6\) beats per second, or 6 Hz.

Generating Beats

To try to produce beats, utilize any pair of sound sources that can generate tones of slightly different frequencies. Two options include tuning forks with near frequencies or a musical instrument like the piano or guitar. By playing two tones simultaneously and slightly altering one of the frequencies, beats can be heard as a wobbling sound.

Discussing Practical Use of Beats

One practical use of beats is in the tuning of musical instruments. When two tones are played together, the presence of beats can indicate that the instruments are not in perfect tune. As the two frequencies move closer towards unison, the speed of the beats decreases, and when they match, the beats disappear, indicating that the instruments are in tune.

Key concepts

Interference of Sine Waves

When two sine waves coincide, they interfere with each other. This process, known as interference, can be either constructive or destructive. Constructive interference occurs when the wave crests (or peaks) align, amplifying the resulting wave. Conversely, destructive interference happens when a crest meets a trough, dampening the resultant wave's amplitude.

The sum of these interferences can create a distinctive pattern, especially when the frequency of the waves is close but not identical. This produces a phenomenon known as beats, which are variations in sound intensity heard as a pulsing rather than a steady tone.

Understanding interference is critical since it explains not only the behavior of sound waves but also light waves, radio waves, and other forms of wave-based phenomena. When graphing two interfering sine waves, you visibly discern the areas of reinforcement and cancellation, which are fundamental to many aspects of physical and engineering sciences.

Beat Frequency

Beat frequency is the heart of understanding beats in sound waves. It's determined by subtracting the frequencies of the two interacting sine waves. For example, if you have two frequencies, one at 24 Hz and another at 30 Hz, their interaction creates beats that have a frequency of 6 Hz, which is the absolute difference between the two.

Listening to these beats is akin to hearing a rhythmic pulsing or throbbing in the sound. In practical scenarios, musicians harness the concept of beat frequency to fine-tune their instruments, ensuring that the beats slow down and ultimately disappear as the instrument approaches the correct pitch. Similarly, various scientific instruments use beat frequency to measure minute differences in frequencies with high precision.

Graphing Sine Waves

Graphing sine waves can seem daunting, but it's crucial for visualizing how sound waves behave and interact. When plotting a sine wave, you're effectively mapping out its amplitude over time. For the exercise's waves of 24 Hz and 30 Hz, plotting them over a sufficient time frame lets you see about 20 cycles, revealing the pattern of each wave.

To depict the cumulative effect when these waves combine, a new graph is created representing the sum of the two waves, highlighting the beats phenomenon. This visual aid is not only helpful for understanding acoustic phenomena but is also a powerful tool in engineering, physics, and even art such as music and digital sound creation.

Acoustic Tuning

Acoustic tuning is a practical application of beat frequency. When tuning a musical instrument, the tuner listens for beats created by playing a reference tone alongside the instrument's tone. As the instrument's pitch is adjusted to match the reference pitch, the beats per second (bps) will decrease. Upon reaching unison with the reference tone, the beats disappear; this indicates that the instrument is in tune.

This technique is widely used among musicians and technicians to ensure the instrument's notes are harmonious. Beyond music, acoustic tuning principles are beneficial in various fields, such as telecommunications and medical diagnostics, where precise frequency alignment is necessary.