Topic 9.511 min read
Beats
Beats are periodic changes in loudness produced when two sound waves of slightly different frequencies superpose. Their changing phase relationship alternates between constructive and destructive interference.
Choose Your Level
Beginner Level
Play two almost identical notes together. The sound repeatedly becomes loud, then soft, then loud again. Each loud-soft cycle is one beat.
Quick Example
Explanation: The frequencies differ by 3 Hz.
Recommended for Class 11
Overview
Beats are periodic changes in loudness produced when two sound waves of slightly different frequencies superpose. Their changing phase relationship alternates between constructive and destructive interference.
Learning Outcomes
- Beats require two sound waves with slightly different frequencies reaching the listener together.
- Constructive interference produces a loud maximum, while destructive interference produces a soft minimum.
- The gradual change in relative phase makes the resultant amplitude rise and fall periodically.
- Beat frequency is the number of loud-soft cycles heard each second.
- The beat frequency is the absolute frequency difference: fb = |f1 - f_2|.
- If more than about 10 beats occur each second, the human ear cannot distinguish them as separate beats.
- Beats are used to compare frequencies, find an unknown frequency, tune instruments, and in ultrasonic imaging and radar speed traps.
Intermediate Level
Formation of Beats
Consider two sound waves A and B with similar amplitudes and slightly different frequencies travelling through the same medium. The resultant wave C is formed by their superposition. Because the frequencies are not exactly equal, their relative phase changes continuously, so the two waves repeatedly move into and out of phase.
When crest overlaps crest, the waves interfere constructively and the resultant amplitude and sound intensity become large. A loud sound is heard. After a short interval, a crest of one wave overlaps a trough of the other; destructive interference reduces the resultant amplitude and a soft sound is heard. The repeating rise and fall of loudness is called beating.
| Relative phase | Interference | Resultant sound |
|---|---|---|
| In phase | Constructive | Maximum loudness |
| Out of phase | Destructive | Minimum loudness |
Beat Frequency
Beat frequency is the number of beats produced per second. It equals the absolute value of the difference between the frequencies of the two waves.
| Symbol | Meaning | SI unit |
|---|---|---|
| f_b | Beat frequency | hertz (Hz) |
| f1, f2 | Frequencies of the two sound waves | hertz (Hz) |
Two Nearby Frequencies
Explanation: The small frequency difference produces a slow, separately audible variation in loudness.
Applications of Beats
- Comparing two frequencies.
- Finding an unknown frequency by comparing it with a known frequency.
- Tuning musical instruments.
- Ultrasonic imaging.
- Radar speed traps.
Tuning a Musical Instrument
A musician sounds a piano string and a standard tuning fork together. Detectable beats show that their frequencies are different. The string tension is adjusted and the test is repeated. As the string approaches the tuning-fork frequency, the beat frequency decreases. When the beats disappear and f_b approaches 0 Hz, the two frequencies match and the string is in tune.
Example 9.3
Beat Frequency of Two Instruments
Two musical instruments produce frequencies of 750 Hz and 390 Hz. Calculate their beat frequency.
- f1=750Hz
- f2=390Hz
fb=|f1-f2|
fb=|750-390|
fb=360Hz
Final answer: The frequency difference is 360 Hz. This is far above 10 Hz, so the variations cannot be detected as separate beats by the human ear.
Common Mistakes
- Use the absolute difference, so beat frequency is never negative.
- Do not add the two frequencies when finding beat frequency.
- Beats require slightly different frequencies; identical frequencies give f_b=0 and no repeated loud-soft pattern.
- Beat frequency describes loudness variations per second, not the pitch of either original sound.
- A large numerical frequency difference may be calculated, but it will not be heard as separate beats.
Short-Question Preparation
Q. What are beats?
A. Beats are periodic variations between maximum and minimum loudness caused by the interference of two sound waves with slightly different frequencies.
Q. What is beat frequency?
A. It is the number of beats produced per second and is given by fb=|f1-f_2|.
Q. Why does loudness rise and fall during beats?
A. The relative phase of the two waves changes, producing alternating constructive and destructive interference.
Q. How are beats used to tune a piano string?
A. The string tension is adjusted until the beat frequency falls to zero, showing that the string and tuning fork have the same frequency.
Sticky Notes
- Beats require two sound waves with slightly different frequencies reaching the listener together.
- Constructive interference produces a loud maximum, while destructive interference produces a soft minimum.
- The gradual change in relative phase makes the resultant amplitude rise and fall periodically.
- Beat frequency is the number of loud-soft cycles heard each second.
- The beat frequency is the absolute frequency difference: fb = |f1 - f_2|.
- If more than about 10 beats occur each second, the human ear cannot distinguish them as separate beats.
- Beats are used to compare frequencies, find an unknown frequency, tune instruments, and in ultrasonic imaging and radar speed traps.
60-Second Revision
Two close sound frequencies repeatedly become in phase and out of phase. Constructive interference gives maximum loudness and destructive interference gives minimum loudness. The number of loud-soft cycles per second is fb=|f1-f2|. Musicians tune an instrument by reducing fb to zero.
Summary
Beats are periodic changes in loudness produced when two sound waves of slightly different frequencies superpose. Their changing phase relationship alternates between constructive and destructive interference.
Ask-the-Tutor Modes
- Explain beats with a loud-soft example.
- Show why phase changes create beats.
- Solve a beat-frequency numerical.
- Explain instrument tuning using beats.
- Test me with an MCQ.
Exam Focus
Exam Focus
Important points
- Definition and formation of beats.
- Beat-frequency formula fb=|f1-f_2|.
- Maximum loudness comes from constructive interference.
- Minimum loudness comes from destructive interference.
- Instrument tuning reduces beat frequency to zero.
Important formulas
- fb=|f1-f2|
Common exam traps
- Beat frequency is an absolute difference.
- Do not add the source frequencies.
Long question structure
Definition → formation → beat-frequency formula → hearing limit → applications → example
Common Mistakes
Common Mistakes
- Mistake
Use the absolute difference, so beat frequency is never negative.
- Mistake
Do not add the two frequencies when finding beat frequency.
- Mistake
Do not confuse beat frequency with the pitch of either original sound.
- Mistake
Identical frequencies give zero beat frequency.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Beats require two sound waves with slightly different frequencies reaching the listener together.
Constructive interference produces a loud maximum, while destructive interference produces a soft minimum.
The gradual change in relative phase makes the resultant amplitude rise and fall periodically.
Beat frequency is the number of loud-soft cycles heard each second.
Beats are periodic changes in loudness produced when two sound waves of slightly different frequencies superpose. Their changing phase relationship alternates between constructive and destructive interference.
- Definition
- Beats are periodic changes in loudness produced when two sound waves of slightly different frequencies superpose. Their changing phase relationship alternates between constructive and destructive interference.
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Exam Practice
Important Question
Q1. Define beats.
Important Question
Q2. Define beat frequency and write its formula.
Important Question
Q3. Explain how constructive and destructive interference form beats.
Important Question
Q4. Explain how beats are used to tune a musical instrument.
Important Question
Q5. Two sounds have frequencies 440 Hz and 437 Hz. Find their beat frequency.