Beat Frequency Problems

1. The Phenomenon of Beats

When two sound waves of nearly equal (but not identical) frequencies are played simultaneously, our ears perceive a single sound whose loudness periodically increases and decreases. This periodic variation in loudness is what we call beats. The sound seems to "wobble" or throb.

This happens because the waves continuously drift in and out of phase with each other.

• When they are in phase (crest meets crest), they interfere constructively, and the sound is loud (waxing).
• When they fall out of phase (crest meets trough), they interfere destructively, and the sound becomes soft (waning).

To understand this both visually and audibly, please watch the following segment from a Khan Academy video.

The resulting wave looks like a high-frequency wave contained within a lower-frequency "envelope" that dictates its amplitude, as shown below.

2. Calculating Beat Frequency

The rate at which the loudness fluctuates is called the beat frequency. It is a very simple but powerful relationship: the beat frequency is equal to the absolute difference between the frequencies of the two interfering waves.

The formula is:
$f_{\text{beat}}=|f_1-f_2|$

Where:

• $f_{\text{beat}}$ is the beat frequency in Hertz (Hz), which corresponds to the number of beats you hear per second.
• $f_1$ and $f_2$ are the frequencies of the two individual waves.

The human ear can typically distinguish beats when the frequency difference is less than about 10-15 Hz. Beyond that, the throbbing is too rapid, and we start to hear two distinct tones.

Let's watch a quick video that focuses on solving problems using this formula.

3. Solving NEET-Style Problems

Now, let's put this into practice with the types of problems you'll face on the NEET.

Type 1: Finding an Unknown Frequency

This is the most common problem type. You are given a known frequency and the beat frequency, and you must determine the unknown frequency.

Example: A tuning fork with a known frequency of 440 Hz is sounded at the same time as a guitar string, and 4 beats per second are heard. What are the possible frequencies of the guitar string?

Solution:
We know $f_1=440$ Hz and $f_{\text{beat}}=4$ Hz.
Using the formula $f_{\text{beat}}=|f_1-f_2|$, we have:
$4=|440-f_2|$
This equation has two possible solutions for $f_2$:

1. $440-f_2=4⟹f_2=440-4=436$ Hz
2. $440-f_2=-4⟹f_2=440+4=444$ Hz

So, the guitar string could be vibrating at either 436 Hz or 444 Hz. Without more information, both are valid possibilities.

Type 2: Using a Second Measurement to Find the Unique Frequency

To find the single correct frequency, problems will often provide a second piece of information. This usually involves slightly changing the frequency of one of the sources and observing the new beat frequency.

A common way to change the frequency of a tuning fork is by loading it with wax or filing its prongs.

• Loading (adding wax): Increases the mass, which decreases the frequency.
• Filing (removing mass): Reduces the mass, which increases the frequency.

Let's work through a problem from your study material.

Let's solve it step-by-step:

1. Initial possibilities:

   • The known frequency is $f_{\text{known}}=256$ Hz.
   • The beat frequency is $f_{\text{beat}}=4$ Hz.
   • So, the possible frequencies for the unknown fork ($f_{\text{unknown}}$) are:
     • $f_{\text{unknown}}=256+4=260$ Hz
     • $f_{\text{unknown}}=256-4=252$ Hz

2. Analyze the second measurement:

   • The unknown fork is loaded with wax, which means its frequency decreases.
   • The new beat frequency is still 4 beats/sec.

3. Test the possibilities:

   • Case A: Assume the original frequency was 260 Hz.

     • When loaded, its frequency will decrease (e.g., to 259 Hz, 258 Hz, etc.).
     • Let the new frequency be $f_{\text{unknown}}^{\prime }$, where $f_{\text{unknown}}^{\prime }<260$ Hz.
     • The new beat frequency is $|f_{\text{unknown}}^{\prime }-256|$.
     • Is it possible for this new beat frequency to be 4 Hz? Yes. If the frequency drops from 260 Hz down to 252 Hz, the beat frequency with the 256 Hz fork would be $|252-256|=4$ Hz. This is consistent with the observation.

   • Case B: Assume the original frequency was 252 Hz.

     • When loaded, its frequency will decrease (e.g., to 251 Hz, 250 Hz, etc.).
     • The new frequency $f_{\text{unknown}}^{\prime }$ will be less than 252 Hz.
     • The new beat frequency is $|f_{\text{unknown}}^{\prime }-256|$. Since $f_{\text{unknown}}^{\prime }$ is getting even further away from 256 Hz, the difference will become larger than 4 Hz.
     • This contradicts the observation that 4 beats/sec are heard again.

4. Conclusion: The only possibility that fits all the data is that the original frequency was 260 Hz. Loading it with wax caused its frequency to drop to 252 Hz, reproducing the 4 Hz beat frequency.

This logical testing of hypotheses is the key to solving these problems.

4. Practice Problems

To reinforce your understanding, try these questions from the 'Concept Notes on Waves for NEET' (resource LINK).

• Exercise 1, Question 24 (page 328): A source X of unknown frequency produces 8 beats per second with a source of 250 Hz and 12 beats per second with a source of 270 Hz. The frequency of the source X is...
  • (a) 242 Hz
  • (b) 258 Hz
  • (c) 282 Hz
  • (d) 262 Hz
• Exercise 2, Question 66 (page 332): A tuning fork of frequency 512 Hz makes 4 beats per second with the vibrating string of a piano. The beat frequency decreases to 2 beats per sec when the tension in the piano string is slightly increased. The frequency of the piano string before increasing the tension was...
  • (a) 510 Hz
  • (b) 514 Hz
  • (c) 516 Hz
  • (d) 508 Hz
    (Hint: How does increasing tension affect the frequency of a string? Recall the formula $f\propto \sqrt{T}$.)

Conclusion

Today we've demystified the phenomenon of beats. It's a direct and audible consequence of wave superposition, and the problems related to it follow a clear, logical pattern.

Key Takeaways:

• Beats are the periodic "waxing and waning" of sound loudness when two waves with slightly different frequencies interfere.
• The beat frequency is the number of loudness cycles per second and is calculated with the simple formula: $f_{\text{beat}}=|f_1-f_2|$.
• Most NEET-style problems require you to determine an unknown frequency from two possibilities by analyzing how the beat frequency changes when one source's frequency is altered.
• Remember: Loading a tuning fork decreases its frequency; filing it increases its frequency.

Preview of the Next Lesson:
We have now covered the key concepts of superposition, standing waves, the Doppler effect, and beats. In our next lesson, we will have a comprehensive problem-solving session that combines all these topics. This will be a great opportunity to practice multi-concept NEET-style questions and solidify your understanding of the entire "Wave Interactions and Sound" module.