Physics - Mechanics and Properties of Matter - How to Solve: Waves (String Waves, Sound Waves, Doppler Effect, Beat Frequency) – NEET UG Physics Guide

How to Solve: Waves (String Waves, Sound Waves, Doppler Effect, Beat Frequency) – NEET UG Physics Guide

Introduction

Mastering waves unlocks 10-12 marks in NEET Physics—enough to push you from a 150 to a 160+ score. These concepts explain why ambulances sound different as they pass, how musical instruments produce notes, and even how ultrasound imaging works in medicine.

WHAT YOU NEED TO KNOW FIRST

1. Simple Harmonic Motion (SHM) – Displacement, velocity, acceleration relationships.
2. Basic Trigonometry – Sine and cosine functions, phase difference.
3. Speed, Distance, Time – Fundamental kinematics.

KEY TERMS & FORMULAS

1. String Waves (Transverse Waves on a String)

Key Terms: - Wavelength (λ) – Distance between two consecutive crests/troughs. - Frequency (f) – Number of waves per second (Hz). - Wave Speed (v) – Speed at which the wave travels along the string. - Tension (T) – Force stretching the string. - Linear Mass Density (μ) – Mass per unit length of the string (kg/m).

Formulas:
1. Wave Speed on a String [ v = \sqrt{\frac{T}{\mu}} ] - v = wave speed (m/s) - T = tension in the string (N) - μ = linear mass density (kg/m) MEMORISE THIS

1. Fundamental Frequency (First Harmonic) of a String [ f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} ]
2. L = length of the string (m)

3. f₁ = fundamental frequency (Hz) MEMORISE THIS

4. Frequency of nth Harmonic [ f_n = n \cdot f_1 = \frac{n}{2L} \sqrt{\frac{T}{\mu}} ]

5. n = harmonic number (1, 2, 3, ...) MEMORISE THIS

2. Sound Waves (Longitudinal Waves)

Key Terms: - Compression – Region of high pressure. - Rarefaction – Region of low pressure. - Intensity (I) – Power per unit area (W/m²). - Intensity Level (β) – Loudness in decibels (dB).

Formulas:
1. Speed of Sound in Air [ v = 331 + 0.6T \quad \text{(T in °C)} ] - v = speed of sound (m/s) - T = temperature in Celsius MEMORISE THIS

1. Intensity of a Sound Wave [ I = \frac{P}{A} ]
2. P = power (W)

3. A = area (m²) Given on exam sheet

4. Intensity Level (Decibels) [ \beta = 10 \log_{10} \left( \frac{I}{I_0} \right) ]

5. β = intensity level (dB)
6. I = intensity (W/m²)
7. I₀ = threshold of hearing (10⁻¹² W/m²) MEMORISE THIS

3. Doppler Effect

Key Terms: - Source (S) – Object emitting sound. - Observer (O) – Person hearing the sound. - Relative Motion – Movement of source or observer.

Formula: [ f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right) ] - f' = observed frequency (Hz) - f = actual frequency (Hz) - v = speed of sound (m/s) - v₀ = speed of observer (m/s) - vₛ = speed of source (m/s) MEMORISE THIS

Sign Rules: - Observer moving toward source: Use +v₀ in numerator. - Observer moving away: Use -v₀ in numerator. - Source moving toward observer: Use -vₛ in denominator. - Source moving away: Use +vₛ in denominator.

4. Beat Frequency

Key Terms: - Beats – Periodic variation in loudness when two waves of slightly different frequencies interfere. - Beat Frequency (f_b) – Difference between two frequencies.

Formula: [ f_b = |f_1 - f_2| ] - f_b = beat frequency (Hz) - f₁, f₂ = frequencies of two waves (Hz) MEMORISE THIS

STEP-BY-STEP METHOD

Step 1: Identify the Type of Wave Problem

• String Waves? → Use tension, length, mass density.
• Sound Waves? → Use speed of sound, intensity, decibels.
• Doppler Effect? → Check if source/observer is moving.
• Beat Frequency? → Two close frequencies interfering.

Step 2: List Given Data & What’s Asked

• Write down all given values (e.g., T = 50 N, μ = 0.01 kg/m, L = 1 m).
• Circle what the question is asking (e.g., "Find the fundamental frequency").

Step 3: Choose the Correct Formula

• Match the question to the right formula (e.g., if asked for frequency, use ( f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} )).

Step 4: Plug in Values & Solve

• Substitute numbers carefully.
• Check units (e.g., tension in N, length in m).

Step 5: Verify the Answer

• Does the answer make sense? (e.g., frequency should be in Hz, not kHz unless specified).
• Recheck calculations if the answer seems off.

WORKED EXAMPLES

Example 1 – Basic (String Waves)

Question: A string of length 1 m and mass 0.01 kg is under a tension of 100 N. Find its fundamental frequency.

Solution:
1. Identify: String wave → fundamental frequency.
2. Given: - L = 1 m - m = 0.01 kg → μ = m/L = 0.01 kg/m - T = 100 N
3. Formula: ( f_1 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} )
4. Plug in: [ f_1 = \frac{1}{2 \times 1} \sqrt{\frac{100}{0.01}} = \frac{1}{2} \sqrt{10000} = \frac{1}{2} \times 100 = 50 \text{ Hz} ]
5. Answer: 50 Hz

What we did and why: We used the fundamental frequency formula for a string, calculated linear mass density, and substituted values carefully. The answer is reasonable (50 Hz is a typical guitar string frequency).

Example 2 – Medium (Doppler Effect)

Question: A car moving at 20 m/s emits a sound of frequency 500 Hz. What frequency does a stationary observer hear as the car approaches? (Speed of sound = 340 m/s)

Solution:
1. Identify: Doppler effect → source moving toward observer.
2. Given: - f = 500 Hz - v = 340 m/s - vₛ = 20 m/s (source speed) - v₀ = 0 (observer stationary)
3. Formula: ( f' = f \left( \frac{v}{v - v_s} \right) ) (source moving toward observer)
4. Plug in: [ f' = 500 \left( \frac{340}{340 - 20} \right) = 500 \left( \frac{340}{320} \right) = 500 \times 1.0625 = 531.25 \text{ Hz} ]
5. Answer: 531.25 Hz

What we did and why: We applied the Doppler formula, used the correct sign (source moving toward observer → denominator is ( v - v_s )), and calculated the observed frequency. The answer is higher than the actual frequency, which makes sense (approaching source compresses waves).

Example 3 – Exam-Style (Beat Frequency)

Question: Two tuning forks produce frequencies of 256 Hz and 260 Hz. What is the beat frequency heard?

Solution:
1. Identify: Beat frequency → difference between two frequencies.
2. Given: - f₁ = 256 Hz - f₂ = 260 Hz
3. Formula: ( f_b = |f_1 - f_2| )
4. Plug in: [ f_b = |256 - 260| = 4 \text{ Hz} ]
5. Answer: 4 Hz

What we did and why: We directly applied the beat frequency formula. The answer is the absolute difference, so even if f₂ were smaller, the result would be the same.

COMMON MISTAKES

1. MISTAKE: Using wrong units for tension (e.g., kg instead of N). WHY IT HAPPENS: Confusing mass and force. CORRECT APPROACH: Always convert tension to Newtons (1 kg·m/s² = 1 N).

2. MISTAKE: Mixing up numerator/denominator in Doppler effect. WHY IT HAPPENS: Not remembering sign rules. CORRECT APPROACH: Draw a diagram: observer moving → numerator changes; source moving → denominator changes.

3. MISTAKE: Forgetting to take the absolute value in beat frequency. WHY IT HAPPENS: Overlooking the modulus sign. CORRECT APPROACH: Beat frequency is always positive.

4. MISTAKE: Using speed of sound as 340 m/s without checking temperature. WHY IT HAPPENS: Assuming standard conditions. CORRECT APPROACH: Use ( v = 331 + 0.6T ) if temperature is given.

5. MISTAKE: Calculating linear mass density as mass × length. WHY IT HAPPENS: Confusing density with mass. CORRECT APPROACH: μ = mass / length (kg/m).

EXAM TRAPS

1. TRAP: Giving speed of sound in km/h instead of m/s. HOW TO SPOT IT: Units in the question (e.g., "340 km/h"). HOW TO AVOID IT: Convert to m/s (340 km/h = 340 × 1000/3600 ≈ 94.4 m/s).

2. TRAP: Doppler effect with both source and observer moving. HOW TO SPOT IT: Question mentions "car moving toward observer who is also moving." HOW TO AVOID IT: Apply both numerator and denominator changes carefully.

3. TRAP: Beat frequency question where one frequency is unknown. HOW TO SPOT IT: "A tuning fork of 512 Hz produces 4 beats with an unknown fork." HOW TO AVOID IT: The unknown frequency could be 512 ± 4 Hz (two possible answers).

1-MINUTE RECAP (Night Before Exam)

"Okay, listen up—this is your waves cheat sheet for NEET.

String Waves: - Wave speed = √(T/μ). Tension over mass per unit length. - Fundamental frequency = 1/(2L) × √(T/μ). Memorise this!

Sound Waves: - Speed of sound = 331 + 0.6T (T in °C). - Intensity level in dB = 10 log(I/I₀). I₀ is 10⁻¹² W/m².

Doppler Effect: - Observer moving? Change numerator. - Source moving? Change denominator. - Toward? Use + for observer, - for source. - Away? Use - for observer, + for source.

Beat Frequency: - Just subtract the two frequencies. Absolute value!

Common Mistakes: - Units! Tension in N, length in m. - Signs in Doppler—draw a diagram. - Beat frequency is always positive.

You’ve got this. Now go crush those waves questions!