14.7 - BEATS
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Introduction to Beats
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Today, we will explore the phenomenon of beats in sound waves. Can anyone tell me what happens when two musical notes with slightly different frequencies are played together?
Do they create a new sound altogether?
Great question! Instead of creating a new sound, they interfere with each other in such a way that we hear the intensity of the sound fluctuating. This is called 'beats'.
So, what causes the sound to get louder or softer?
Good observation! As the two waves overlap, they can add together to produce a louder sound or cancel each other out leading to a softer sound.
How is the beat frequency calculated?
The beat frequency is given by the absolute difference between the two frequencies. So for waves of 11 Hz and 9 Hz, it's 2 Hz.
That sounds really cool!
Indeed! Understanding beats is not just important in physics; it's crucial in music as well. Musicians tune their instruments using beats!
To summarize, when two close frequencies interfere, we perceive fluctuations in volume, which we call beats. The frequency of these beats is the difference between the two original frequencies.
Practical Application of Beats
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How do you think musicians use beats to tune their instruments?
They listen for the beats to get them to match up?
Exactly! When two instruments are slightly out of tune, they produce beats. Artists will keep adjusting until the beats fade away.
What if the beat frequency becomes really large?
If the two frequencies are very far apart, the beats would occur less frequently or not be heard at all. So, they need to be close together.
How do they know when they are perfectly in tune?
When the beats disappear, indicating the frequencies match closely, they know they are in tune! It's fascinating how physics applies directly to music.
In summary, musicians use the concept of beats to perfect their tuning, aiming for that moment when beats vanish.
Mathematical Understanding of Beats
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Let's take a closer look at the formula for beat frequency. Who can recall what our formula is?
It's the absolute difference between the two frequencies!
Right! If we have two frequencies, ν1 and ν2, the beat frequency, νbeat, is calculated as |ν1 - ν2|. Can someone give me an example?
Sure! If one frequency is 430 Hz and the other is 425 Hz, the beat frequency would be 5 Hz.
That's correct! And if one string was slightly tightened, what would happen to the beat frequency?
Would it change? Maybe decrease or increase?
Exactly! If tightening increases the frequency, the beats would change accordingly. Understanding this relationship is key in both physics and music.
To wrap up, the beat frequency highlights the beautiful connection between music and mathematics through sound waves.
Introduction & Overview
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Quick Overview
Standard
The phenomenon of beats occurs when two harmonic sound waves with close but different frequencies are superimposed. This leads to the perception of alternating loud and soft sounds, with a frequency equal to the difference between the two original frequencies.
Detailed
Detailed Summary of Beats
In physics, 'beats' refer to the periodic variation in sound intensity resulting from the interference of two sound waves that have slightly different frequencies. When two harmonic waves of frequencies ω1 and ω2 (with ω1 > ω2) are heard together, they produce a new wave pattern characterized by beats. The resultant sound alternates between loud and soft at a frequency given by the difference between the two frequencies:
Beat Frequency Calculation
The beat frequency (νbeat) can be calculated as:
\[
u_{beat} = |
u_1 -
u_2| \]
where ν1 and ν2 are the frequencies of the two waves. For instance, if one wave is at 11 Hz and another at 9 Hz, the beat frequency will be:
\[
u_{beat} = |11 ext{ Hz} - 9 ext{ Hz}| = 2 ext{ Hz} \]
Thus, the auditory perception will show alternating sound intensities at this beat frequency.
The phenomenon of beats is effectively utilized in music tuning to check discrepancies between instruments. Musicians listen for beats, adjusting their instruments until these fluctuate minimally, indicating being in tune.
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Introduction to Beats
Chapter 1 of 3
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Chapter Content
‘Beats’ is an interesting phenomenon arising from interference of waves. When two harmonic sound waves of close (but not equal) frequencies are heard at the same time, we hear a sound of similar frequency (the average of two close frequencies), but we hear something else also.
Detailed Explanation
Beats occur when two sound waves that are nearly identical in frequency overlap. Instead of simply hearing a single sound, the listener perceives fluctuations in volume, known as beats. This happens because the two waves interfere with each other: they reinforce each other at some moments (constructive interference) and cancel each other out at others (destructive interference). This results in the sound alternating between loud and soft, creating a distinctive effect.
Examples & Analogies
Think of two musicians tuning their instruments. If one string is slightly out of tune, the sound will shift from loud to soft as they play together. This fluctuation in volume—like the 'wah-wah' effect—represents the beat frequency, making it easier for them to determine when the two sounds align perfectly.
Calculation of Beat Frequency
Chapter 2 of 3
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Chapter Content
We hear audibly distinct waxing and waning of the intensity of the sound, with a frequency equal to the difference in the two close frequencies. Artists use this phenomenon often. Since ω = 2πν, the beat frequency νbeat, is given by νbeat = ν1 – ν2.
Detailed Explanation
The beat frequency can be calculated using the formula νbeat = |ν1 - ν2|, where ν1 and ν2 are the frequencies of the two sound waves. The result gives the number of times per second the loudness varies. For example, if one wave has a frequency of 11 Hz and another has 9 Hz, the beat frequency is 2 Hz, meaning the sound will swell and dip twice per second.
Examples & Analogies
Imagine a radio tuned in between two stations. As you adjust the dial, you hear the music from both stations blending together and fading in and out. The fluctuation in how clear the sound is represents beats—the difference in frequency makes it sound like the volume is changing.
Example of Beats with String Instruments
Chapter 3 of 3
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Chapter Content
Example 14.6: Two sitar strings A and B playing the note ‘Dha’ are slightly out of tune and produce beats of frequency 5 Hz. The tension of the string B is slightly increased and the beat frequency is found to decrease to 3 Hz. What is the original frequency of B if the frequency of A is 427 Hz?
Detailed Explanation
In this example, String A is tuned to 427 Hz, resulting in a beat frequency of 5 Hz with String B. Since the beat frequency is the absolute difference between their frequencies, we know that originally νB was 422 Hz (427 Hz - 5 Hz). After increasing the tension on String B, its frequency increases, leading to a new beat frequency of 3 Hz. This indicates that νB is now slightly closer to that of String A.
Examples & Analogies
Think of tuning forks: when hitting two forks that are close in pitch, they create a very noticeable beating sound until you adjust one of them. This example demonstrates how musicians can tune their instruments by using beats to determine when they are in perfect harmony with each other.
Key Concepts
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Beats: Variations in sound intensity due to wave interference.
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Interference: Superposition of two or more waves to form a resultant wave.
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Beat Frequency: The frequency at which the loudness of the sound alternates, calculated as the difference of the two wave frequencies.
Examples & Applications
If two sound waves are at 440 Hz and 437 Hz, the beat frequency is 3 Hz.
When a guitarist tunes their strings, they listen for beats to ensure they are in harmony with themselves and their instruments.
Memory Aids
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Rhymes
Beats arise when sounds align, frequency differences intertwine.
Stories
Imagine two musicians tuning their instruments. When they play together, if they're slightly off, first it sounds strong and then weak, leading to a smooth harmony when in tune.
Memory Tools
To remember the beat frequency formula, think: 'Subtract to find the beat'.
Acronyms
B.F. = Between Frequencies; Remember `B` for Beats and `F` for Frequency!
Flash Cards
Glossary
- Beats
Periodic variations in sound intensity resulting from the interference of two sound waves with slightly different frequencies.
- Interference
The phenomenon that occurs when two waves superimpose to form a resultant wave.
- Frequency
The number of occurrences of a repeating event per unit of time, typically measured in Hertz (Hz).
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