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Waves and Oscillations

Beats and Superposition of Waves

Beats are periodic variations in amplitude and intensity heard when two sound waves of slightly different frequencies interfere. The phenomenon results from the temporal superposition of harmonic waves traveling in the same direction.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Understanding beat frequency formula, numericals based on tuning forks, loading/filing of tuning forks, and identification of unknown frequencies.

PGT extension

Analytical treatment of amplitude modulation in acoustics, superposition of non-collinear or phase-shifted waves, transient beats, and Fourier perspective of beat phenomena.

Prerequisites

Principle of superposition of waves
Trigonometric identities for sum and difference of sines and cosines
Characteristics of sound waves (frequency, amplitude, wavelength)

Superposition of Two Waves of Nearly Equal Frequencies

When two sound waves of frequencies f₁ and f₂ (where f₁ ≈ f₂) and equal amplitudes A travel in the same direction, they superpose.
The resultant displacement at any point in space oscillates with an average frequency equal to the mean of the two frequencies: f_mean = (f₁ + f₂) / 2.
The amplitude of the resultant wave varies periodically with time, leading to alternate maxima (loud sound) and minima (faint sound).

Beat Frequency and Persistence of Hearing

The number of beats heard per second, known as the beat frequency, is given strictly by the absolute difference between the two component frequencies: f_beat = |f₁ - f₂|.
For human ears, the persistence of hearing is approximately 1/10th of a second (0.1 s); therefore, beats are clearly distinguishable only if the beat frequency does not exceed 10 Hz.
If the difference exceeds 10 Hz, the human ear perceives a continuous harsh sound or a chord rather than distinct pulsations.

Effect of Loading and Filing on Tuning Forks

Loading a prong of a tuning fork with wax increases its effective mass, which decreases its natural frequency of vibration.
Filing (or scraping) a tuning fork reduces its mass and stiffness, which increases its natural frequency of vibration.
These operations are standard diagnostic methods used in conjunction with beat counts to determine an unknown frequency relative to a standard fork.

Energy Conservation in Beats

At maximum intensity (constructive interference), the acoustic energy density is quadrupled relative to a single wave, while at minimum intensity (destructive interference), the energy density drops to zero.
The excess energy at the maxima is exactly compensated by the deficit at the minima, conserving total acoustic energy over a complete beat period.
The phenomenon demonstrates that intensity fluctuations are temporal rather than spatial standing variations.

Derivations

Mathematical Derivation of Beat Frequency

Conditions: Two harmonic waves of equal amplitude A and slightly different frequencies f₁ and f₂ traveling in the same direction.

Let the displacement of the first wave be y₁ = A sin(2πf₁t) and the second wave be y₂ = A sin(2πf₂t).
By the principle of superposition, the net displacement is $y = y_{1} + y_{2} = A [\sin(2\pif_{1}t) + \sin(2\pif_{2}t)]$.
Using the trigonometric identity sin C + sin $D = 2 \sin((C+D)/2) \cos((C-D)/2)$, we get $y = 2A \cos(2\pi((f_{1} - f_{2})/2)t) \sin(2\pi((f_{1} + f_{2})/2)t)$.
Let A_res(t) = 2A cos(2π((f₁ - f₂)/2)t) represent the time-varying amplitude and f_mean = (f₁ + f₂)/2 represent the carrier frequency.
Intensity is proportional to the square of the amplitude, so intensity maxima occur when cos(π(f₁ - f₂)t) = ±1, yielding a time interval between successive maxima of T_beat = 1 / |f₁ - f₂|.

Result: f_beat = |f₁ - f₂|

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Beat Frequency	$$f_{beat} = \|f_1 - f_2\|$$	f₁, f₂ = frequencies of the two interfering waves Hertz (Hz) or s^-1	\|f₁ - f₂\| ≤ 10 Hz for distinct human perception
Resultant Amplitude of Superposed Waves	$$A_{res} = 2A \cos\left(\pi(f_1 - f_2)t\right)$$	A = initial amplitude of each wave, t = time Meter (m)	Equal constituent amplitudes
Time Period of Beats	$$T_{beat} = \frac{1}{f_1 - f_2}$$	T_beat = time interval between two successive loud sounds Second (s)	f₁ ≠ f₂
Mean Frequency	$$f_{mean} = \frac{f_1 + f_2}{2}$$	f_mean = frequency of carrier oscillation Hertz (Hz)	Nearly equal frequencies
Maximum Intensity	$$I_{max} \propto (A_1 + A_2)^2$$	A₁, A₂ = amplitudes of component waves W/m²	Constructive interference
Minimum Intensity	$$I_{min} \propto (A_1 - A_2)^2$$	A₁, A₂ = amplitudes of component waves W/m²	Destructive interference
Loading Effect on Tuning Fork	$$f' < f$$	f' = new frequency after adding mass, f = original frequency Hz	Mass added to prong
Filing Effect on Tuning Fork	$$f'' > f$$	f'' = new frequency after filing, f = original frequency Hz	Mass removed from prong

Graphs and experiments

Displacement-Time Graph of Beats

Draw: A high-frequency sinusoidal wave oscillating within a slowly varying envelope bounded by positive and negative curves of 2A cos(πΔft).

Interpret: Clearly shows amplitude modulation where energy waxes and wanes periodically over time at frequency Δf.

Exam use: Identified in conceptual multiple-choice questions to distinguish spatial standing waves from temporal beats.

Acoustic Beats Demonstration (Quincke's Tube or Tuning Forks)

Draw: Two tuning forks mounted on resonance boxes side by side, one fitted with a removable small wax pellet.

Interpret: Striking both forks simultaneously produces a throbbing sound whose rate equals the frequency difference.

Exam use: Basis for practical viva questions on determining unknown tuning fork frequencies.

Intensity vs Time Graph

Draw: A squared cosine waveform showing peaks of intensity reaching 4I₀ and troughs dropping to zero, repeating with period T_beat.

Interpret: Demonstrates that maximum loudness is heard twice per beat cycle if calculated via energy envelope, but intensity peaks occur once per beat period T_beat = 1/Δf.

Exam use: Interpreting energy distribution in numerical problems.

Solved numericals

TGT core

Two tuning forks when sounded together produce 4 beats per second. The frequency of one fork is 256 Hz. When the other fork is loaded with a little wax, the beat frequency becomes 2 Hz. Find the original frequency of the second fork.

Let the unknown frequency of the second fork be f₂. Given f₁ = 256 Hz and f_beat = 4 Hz.
Therefore, f₂ = f₁ ± f_beat = 256 ± 4 = 260 Hz or 252 Hz.
When the fork is loaded with wax, its frequency decreases (f' < f₂).
The new beat frequency is given as 2 Hz. Let's test both cases with a decreased frequency.
If original f₂ = 260 Hz, loading decreases it to, say, 258 Hz. Then |256 - 258| = 2 Hz. This matches the condition.
If original f₂ = 252 Hz, loading decreases it to 250 Hz. Then |256 - 250| = 6 Hz, which does not match.
Thus, the original frequency is 260 Hz.

Answer: 260 Hz

PGT extension

Two sound sources emit waves given by y₁ = 5 sin(500πt) and y₂ = 5 sin(506πt). Calculate the beat frequency, the time interval between successive maxima of sound, and the maximum amplitude of the resultant wave.

Express the angular frequencies: ω₁ = 500π rad/s and ω₂ = 506π rad/s.
Calculate linear frequencies: f₁ = ω₁ / (2π) = 250 Hz and f₂ = ω₂ / (2π) = 253 Hz.
Beat frequency f_beat = |f₂ - f₁| = |253 - 250| = 3 Hz.
Time interval between successive maxima (beat period) T_beat = 1 / f_beat = 1 / 3 s.
Maximum amplitude of the resultant wave A_max = A₁ + A₂ = 5 + 5 = 10 units.

Answer: Beat frequency = 3 Hz, Time interva$l = 1/3 s$, Maximum amplitude = 10 units

TGT core

A tuning fork of frequency 512 Hz produces 5 beats per second when sounded with a vibrating piano string. What is the frequency of the piano string if filing the piano string reduces the beat frequency to 2 beats per second?

Let the frequency of the piano string be f. Given f_fork = 512 Hz and initial f_beat = 5 Hz.
Thus, $f = 512 \pm 5 = 517 Hz or 507 Hz$.
Filing a string reduces its mass/tension ratio and increases its tension/frequency parameter (or filing a tuning fork increases frequency; for a piano string, tuning adjustments that shorten or tighten increase frequency). Standard convention: filing a string increases its frequency (f' > f).
If original frequency f = 507 Hz, filing increases it towards 510 Hz, so new beat frequency |512 - 510| = 2 Hz.
If original frequency f = 517 Hz, filing increases it further away from 512 Hz, increasing the beat frequency above 5 Hz.
Therefore, the original frequency of the piano string is 507 Hz.

Answer: 507 Hz

Assumptions and limitations

Assumes the interfering waves travel in the exact same direction with comparable amplitudes for clear acoustic modulation.
Valid only when the frequency difference is small compared to the individual frequencies (Δf << f); otherwise, distinct beats merge into complex discordant sounds.

Common misconceptions

Common error: Beats are formed due to the interference of two waves traveling in opposite directions.

Correct: Beats are formed due to the superposition of two waves traveling in the same direction with slightly different frequencies.

Waves traveling in opposite directions form standing waves, not beats.

Common error: Two loud sounds are heard per beat period.

Correct: Only one maximum sound is heard per beat period, although amplitude reaches a maximum magnitude twice (once positive, once negative).

Intensity depends on the square of the amplitude, making intensity peaks occur once per envelope cycle.

Common error: Loading a tuning fork increases its frequency.

Correct: Loading a tuning fork increases its mass, which decreases its frequency.

Frequency is inversely proportional to the square root of mass (f ∝ 1/√m).

Quick revision

Beat frequency is simply the difference between the frequencies of two superposing waves: f_beat = |f₁ - f₂|.
Human ear can resolve beats effectively up to 10 Hz due to persistence of hearing.
Loading a tuning fork decreases frequency; filing increases frequency.

Must remember

Beat frequency is given by the absolute difference between two interfering frequencies: $f_{beat} = |f_1 - f_2|$.
The persistence of human hearing is approximately $0.1\text{ s}$, meaning beats above $10\text{ Hz}$ are not distinguishable as separate pulses.
Loading a tuning fork with wax increases its effective mass and decreases its natural frequency ($f' < f$).
Filing a tuning fork reduces its mass and stiffness, which increases its natural frequency ($f'' > f$).
Beats are produced by the superposition of two waves traveling in the same direction with nearly equal frequencies.

Glossary

Beat FrequencyThe number of intensity maxima heard per second, equal to $|f_1 - f_2|$.

Persistence of HearingThe retention of a sensory impression in the human ear for about $1/10\text{th}$ of a second.

LoadingAdding mass (such as wax) to the prongs of a tuning fork to lower its frequency.

FilingScraping or removing metal from the prongs of a tuning fork to raise its frequency.

Effect of Tuning Fork Modifications

Modification	Mass Change	Frequency Effect
Loading with wax	Increases mass	Decreases frequency ($f' < f$)
Filing the prongs	Decreases mass	Increases frequency ($f'' > f$)

Interference vs. Beats

Phenomenon	Direction of Waves	Frequency Condition
Stationary Interference / Standing Waves	Opposite directions	Equal frequencies
Beats	Same direction	Nearly equal frequencies

Practice Quiz

Choose an option to reveal the explanation.

1. What is the primary condition for the formation of beats when two sound waves superpose?

Explanation: Beats are formed due to the superposition of two waves traveling in the same direction with nearly equal frequencies ($f_1 \approx f_2$).

2. If two sound waves of frequencies $256\text{ Hz}$ and $260\text{ Hz}$ are sounded together, what is the beat frequency?

Explanation: Beat frequency is given by $f_{beat} = |f_1 - f_2| = |256 - 260| = 4\text{ Hz}$.

3. Up to what maximum beat frequency can the human ear distinctly distinguish individual beats due to the persistence of hearing?

Explanation: The persistence of human hearing is $0.1\text{ s}$, so beats are clearly distinguishable only if the beat frequency does not exceed $10\text{ Hz}$.

4. What happens to the natural frequency of a tuning fork when a small amount of wax is loaded onto its prongs?

Explanation: Loading a prong increases its effective mass, which decreases its natural frequency of vibration ($f' < f$).

5. Filing the prongs of a tuning fork has which of the following effects on its frequency?

Explanation: Filing or scraping removes mass and stiffness, which increases the natural frequency of vibration ($f'' > f$).

6. What is the carrier or mean frequency when two waves of frequencies $f_1$ and $f_2$ superpose?

Explanation: The resultant displacement oscillates with an average frequency equal to the mean of the two frequencies: $f_{mean} = (f_1 + f_2) / 2$.

7. How does the acoustic energy density behave at maximum intensity (constructive interference) in beats compared to a single wave?

Explanation: At maximum intensity, amplitude is $2A$, so intensity $I_{max} \propto (2A)^2 = 4A^2$, which quadruples the energy density relative to a single wave of amplitude $A$.

8. What is the time period between two successive loud sounds (beats) if the beat frequency is $5\text{ Hz}$?

Explanation: The time period of beats is $T_{beat} = 1 / f_{beat} = 1 / 5 = 0.2\text{ s}$.

9. Which of the following statements about energy conservation in beats is correct?

Explanation: The excess energy at maxima is exactly compensated by the deficit at minima, conserving total acoustic energy over a complete beat period.

10. A tuning fork of unknown frequency gives $4\text{ beats/s}$ with a standard fork of frequency $384\text{ Hz}$. When loaded with a little wax, it gives $6\text{ beats/s}$. What was the original unknown frequency?

Explanation: Initial frequency is either $384 + 4 = 388$ or $384 - 4 = 380\text{ Hz}$. Loading decreases frequency. If original was $380$, loading decreases it further, increasing the difference from $384$, hence beats increase from $4$ to $6$. Thus, original frequency is $380\text{ Hz}$.

11. The resultant amplitude expression for two superposing waves of equal amplitude $A$ and frequencies $f_1$ and $f_2$ is proportional to:

Explanation: The amplitude envelope is given by $A_{res} = 2A \cos(\pi(f_1 - f_2)t)$.

12. Why are beats not heard when two independent violins play notes that differ by $30\text{ Hz}$?

Explanation: Because the beat frequency ($30\text{ Hz}$) exceeds $10\text{ Hz}$, the human ear perceives a continuous harsh sound or chord rather than distinct pulsations due to the persistence of hearing.

13. What is the minimum intensity for two superposing waves of equal amplitudes $A_1 = A_2 = A$?

Explanation: For equal component amplitudes, $I_{min} \propto (A_1 - A_2)^2 = (A - A)^2 = 0$.

14. In the context of amplitude modulation in acoustics (advanced PGT perspective), beat phenomena represent:

Explanation: Analytically, the wave can be viewed as an amplitude-modulated wave with carrier frequency $f_{mean}$ and envelope modulation frequency $|f_1 - f_2| / 2$ in terms of amplitude pulsation.

15. Which misconception about beats is explicitly corrected in wave physics?

Explanation: All listed options are common misconceptions corrected in the study notes: beats are formed in the same direction, only one loud sound is heard per beat cycle, and loading decreases frequency.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A tuning fork $A$ of frequency $250\text{ Hz}$ produces $5\text{ beats/s}$ with another tuning fork $B$. When fork $B$ is filed, the beat frequency becomes $2\text{ beats/s}$. What is the original frequency of fork $B$?

Explanation: Initial frequency of $B$ is $250 \pm 5 = 245$ or $255\text{ Hz}$. Filing $B$ increases its frequency ($f'' > f$). If $B$ was $245$, filing increases it towards $250$, so beats decrease from $5$ to $2$. If $B$ was $255$, filing increases it further away from $250$, increasing beats. Thus, original frequency of $B$ is $245\text{ Hz}$ (Model Question).

2. Two sources of sound $S_1$ and $S_2$ produce sound waves of frequencies $350\text{ Hz}$ and $356\text{ Hz}$ respectively. The number of beats heard per second is:

Explanation: Beat frequency $f_{beat} = |350 - 356| = 6\text{ Hz}$, meaning 6 beats are heard per second (Model Question).

3. A tuning fork gives $4\text{ beats/s}$ with $512\text{ Hz}$ standard fork. When a little wax is pasted on the tuning fork, it produces $2\text{ beats/s}$. Find its frequency before waxing.

Explanation: Initial frequency is $512 \pm 4 = 508$ or $516\text{ Hz}$. Waxing (loading) decreases frequency. If original was $508$, loading decreases it further, increasing the beat count. If original was $516$, loading decreases it towards $512$, reducing the beat count from $4$ to $2$. Thus, original frequency is $516\text{ Hz}$? Wait, let's re-verify: if original is $516$, loading decreases it to e.g. $514$, difference with $512$ becomes $2$. Correct! So frequency is $516\text{ Hz}$ (Model Question).

4. If the maximum intensity in a beat phenomenon is $9$ times the minimum intensity, what is the ratio of amplitudes of the two interfering waves ($A_1 / A_2$)?

Explanation: $I_{max} / I_{min} = (A_1 + A_2)^2 / (A_1 - A_2)^2 = 9 / 1$. Taking square root: $(A_1 + A_2) / (A_1 - A_2) = 3 / 1$. Solving gives $A_1 + A_2 = 3A_1 - 3A_2 \Rightarrow 2A_1 = 4A_2 \Rightarrow A_1 / A_2 = 2 / 1$ (Model Question).

5. Two waves are represented by $y_1 = A \sin(200\pi t)$ and $y_2 = A \sin(208\pi t)$. How many beats per second are produced?

Explanation: Angular frequencies are $\omega_1 = 200\pi \Rightarrow f_1 = 100\text{ Hz}$, $\omega_2 = 208\pi \Rightarrow f_2 = 104\text{ Hz}$. Beat frequency = $|104 - 100| = 4\text{ Hz}$ (Model Question).

6. Which of the following conditions is mandatory for observing distinct beats between two sound notes?

Explanation: Due to persistence of hearing, beat frequency must not exceed $10\text{ Hz}$, and the waves must travel in the same direction (Model Question).

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What is the time interval between successive maxima of sound intensity when two waves of frequencies $400\text{ Hz}$ and $405\text{ Hz}$ interfere?

Explanation: Beat frequency $f_{beat} = |400 - 405| = 5\text{ Hz}$. Time interval $T_{beat} = 1 / 5 = 0.2\text{ s}$.

2. If a tuning fork is filed, its frequency increases because:

Explanation: Filing removes metal, reducing the mass and stiffness of the prongs, which increases the natural frequency ($f'' > f$).

3. Two tuning forks $A$ and $B$ produce $4\text{ beats/s}$. Fork $A$ has a frequency of $256\text{ Hz}$. When fork $B$ is loaded with a little wax, $2\text{ beats/s}$ are heard. The original frequency of $B$ was:

Explanation: Frequency of $B$ is $256 \pm 4 = 252$ or $260\text{ Hz}$. Loading $B$ decreases its frequency. If $B$ was $260$, loading decreases it towards $256$, reducing beats from $4$ to $2$. Thus, original frequency of $B$ is $260\text{ Hz}$.

4. The phenomenon of beats demonstrates which wave characteristic?

Explanation: Beats are a temporal manifestation of the principle of superposition and interference of waves.

5. What is the carrier frequency when two sound waves of $320\text{ Hz}$ and $324\text{ Hz}$ superpose?

Explanation: Carrier frequency is the mean frequency: $f_{mean} = (320 + 324) / 2 = 322\text{ Hz}$.

6. At the minimum intensity point of a beat cycle, what is the acoustic energy density?

Explanation: At destructive interference (minimum intensity for equal amplitudes), the resultant amplitude is zero, dropping the energy density to zero.

7. Why can two human voices singing slightly out of tune produce a wavering sound?

Explanation: Slightly out of tune voices have nearly equal frequencies, producing audible beats (wavering sound).

8. If the frequency difference between two waves is $15\text{ Hz}$, what will a listener perceive?

Explanation: Since $15\text{ Hz}$ exceeds the $10\text{ Hz}$ persistence limit of hearing, individual beats are not resolved; a continuous harsh sound or chord is perceived.

9. Which of the following equations represents the resultant amplitude of two superposed waves?

Explanation: The amplitude envelope of the superposed waves varies as $A_{res} = 2A \cos(\pi(f_1 - f_2)t)$.

10. A tuning fork of unknown frequency gives $5\text{ beats/s}$ with a $300\text{ Hz}$ fork and $5\text{ beats/s}$ with a $310\text{ Hz}$ fork. What is the unknown frequency?

Explanation: With $300\text{ Hz}$, frequency is $300 \pm 5 = 295$ or $305\text{ Hz}$. With $310\text{ Hz}$, frequency is $310 \pm 5 = 305$ or $315\text{ Hz}$. The common frequency is $305\text{ Hz}$.

Beats and Superposition of Waves

Scope and prerequisites

TGT core

PGT extension

Prerequisites

Superposition of Two Waves of Nearly Equal Frequencies

Beat Frequency and Persistence of Hearing

Effect of Loading and Filing on Tuning Forks

Energy Conservation in Beats

Derivations

Mathematical Derivation of Beat Frequency

Formula sheet

Graphs and experiments

Displacement-Time Graph of Beats

Acoustic Beats Demonstration (Quincke's Tube or Tuning Forks)

Intensity vs Time Graph

Solved numericals

Two tuning forks when sounded together produce 4 beats per second. The frequency of one fork is 256 Hz. When the other fork is loaded with a little wax, the beat frequency becomes 2 Hz. Find the original frequency of the second fork.

Two sound sources emit waves given by y1 = 5 sin(500πt) and y2 = 5 sin(506πt). Calculate the beat frequency, the time interval between successive maxima of sound, and the maximum amplitude of the resultant wave.

A tuning fork of frequency 512 Hz produces 5 beats per second when sounded with a vibrating piano string. What is the frequency of the piano string if filing the piano string reduces the beat frequency to 2 beats per second?

Assumptions and limitations

Common misconceptions

Quick revision

Must remember

Glossary

Effect of Tuning Fork Modifications

Interference vs. Beats

Practice Quiz

1. What is the primary condition for the formation of beats when two sound waves superpose?

2. If two sound waves of frequencies $256\text{ Hz}$ and $260\text{ Hz}$ are sounded together, what is the beat frequency?

3. Up to what maximum beat frequency can the human ear distinctly distinguish individual beats due to the persistence of hearing?

4. What happens to the natural frequency of a tuning fork when a small amount of wax is loaded onto its prongs?

5. Filing the prongs of a tuning fork has which of the following effects on its frequency?

6. What is the carrier or mean frequency when two waves of frequencies $f_1$ and $f_2$ superpose?

7. How does the acoustic energy density behave at maximum intensity (constructive interference) in beats compared to a single wave?

8. What is the time period between two successive loud sounds (beats) if the beat frequency is $5\text{ Hz}$?

9. Which of the following statements about energy conservation in beats is correct?

10. A tuning fork of unknown frequency gives $4\text{ beats/s}$ with a standard fork of frequency $384\text{ Hz}$. When loaded with a little wax, it gives $6\text{ beats/s}$. What was the original unknown frequency?

11. The resultant amplitude expression for two superposing waves of equal amplitude $A$ and frequencies $f_1$ and $f_2$ is proportional to:

12. Why are beats not heard when two independent violins play notes that differ by $30\text{ Hz}$?

13. What is the minimum intensity for two superposing waves of equal amplitudes $A_1 = A_2 = A$?

14. In the context of amplitude modulation in acoustics (advanced PGT perspective), beat phenomena represent:

15. Which misconception about beats is explicitly corrected in wave physics?

PYQ Patterns

1. A tuning fork $A$ of frequency $250\text{ Hz}$ produces $5\text{ beats/s}$ with another tuning fork $B$. When fork $B$ is filed, the beat frequency becomes $2\text{ beats/s}$. What is the original frequency of fork $B$?

2. Two sources of sound $S_1$ and $S_2$ produce sound waves of frequencies $350\text{ Hz}$ and $356\text{ Hz}$ respectively. The number of beats heard per second is:

3. A tuning fork gives $4\text{ beats/s}$ with $512\text{ Hz}$ standard fork. When a little wax is pasted on the tuning fork, it produces $2\text{ beats/s}$. Find its frequency before waxing.

4. If the maximum intensity in a beat phenomenon is $9$ times the minimum intensity, what is the ratio of amplitudes of the two interfering waves ($A_1 / A_2$)?

5. Two waves are represented by $y_1 = A \sin(200\pi t)$ and $y_2 = A \sin(208\pi t)$. How many beats per second are produced?

6. Which of the following conditions is mandatory for observing distinct beats between two sound notes?

Timed Topic Test

1. What is the time interval between successive maxima of sound intensity when two waves of frequencies $400\text{ Hz}$ and $405\text{ Hz}$ interfere?

2. If a tuning fork is filed, its frequency increases because:

3. Two tuning forks $A$ and $B$ produce $4\text{ beats/s}$. Fork $A$ has a frequency of $256\text{ Hz}$. When fork $B$ is loaded with a little wax, $2\text{ beats/s}$ are heard. The original frequency of $B$ was:

4. The phenomenon of beats demonstrates which wave characteristic?

5. What is the carrier frequency when two sound waves of $320\text{ Hz}$ and $324\text{ Hz}$ superpose?

6. At the minimum intensity point of a beat cycle, what is the acoustic energy density?

7. Why can two human voices singing slightly out of tune produce a wavering sound?

8. If the frequency difference between two waves is $15\text{ Hz}$, what will a listener perceive?

9. Which of the following equations represents the resultant amplitude of two superposed waves?

10. A tuning fork of unknown frequency gives $5\text{ beats/s}$ with a $300\text{ Hz}$ fork and $5\text{ beats/s}$ with a $310\text{ Hz}$ fork. What is the unknown frequency?

Two sound sources emit waves given by y₁ = 5 sin(500πt) and y₂ = 5 sin(506πt). Calculate the beat frequency, the time interval between successive maxima of sound, and the maximum amplitude of the resultant wave.