Waves

1. Types of Waves

Disturbance that propagates energy and momentum without transport of matter.

Transverse Waves: Particle vibration ⊥ Wave propagation. (e.g., Light, String waves).
Longitudinal Waves: Particle vibration || Wave propagation. (e.g., Sound, Spring waves).
Progressive Wave Equation: y(x,t) = A sin(kx - ωt + φ).
Where k = 2π/λ (Wave Number), ω = 2πf.

Speed of Wave

On Stretched String: v = √(T/μ). (μ = mass per unit length).
Sound in Gas (Newton-Laplace): v = √(γP/ρ).
(γ = C_p/C_v). Speed increases with Temp: v ∝ √T.

(Asked in NEET 2013, 2017, 2019)

2. Superposition & Standing Waves

Principle of Superposition: y_net = y₁ + y₂.
Standing Waves: Formed by superposition of two identical waves traveling in opposite directions.
- Nodes: Points of zero amplitude.
- Antinodes: Points of max amplitude.

Organ Pipes

Closed Pipe (One end closed):
- Fundamental: f₁ = v/4L.
- Harmonics: Odd only (f₁, 3f₁, 5f₁...).
Open Pipe (Both ends open):
- Fundamental: f₁ = v/2L.
- Harmonics: All (f₁, 2f₁, 3f₁...).

(Asked in NEET 2016, 2018, 2020, 2021, 2022)

3. Beats & Doppler Effect

Beats: Periodic variation in intensity due to superposition of two waves of slightly different frequencies.
Beat Frequency f_beat = |f₁ - f₂|.

Doppler Effect

Apparent change in frequency due to relative motion.

f' = f [(v ± v_o) / (v ∓ v_s)]
v_o: Observer velocity, v_s: Source velocity.
Sign convention: Direction S → O is positive.

(Asked in NEET 2016, 2017, 2020)

Numericals: Waves

Q1. An open pipe has fundamental frequency 300Hz. Speed of sound is 330 m/s. Find length of pipe.

Solution:
For open pipe, f₁ = v / 2L.
L = v / 2f₁ = 330 / (2 × 300) = 330 / 600.
L = 0.55 m.

Q2. Two tuning forks produce 4 beats/sec. Fork A has freq 256Hz. Loading A decreases beats. Find freq of B.

Solution:
f_A = 256 Hz. f_B = 256 ± 4 = 260 or 252.
Loading A decreses f_A.
If f_A was 260, decreasing it (e.g., to 258) would increase beats with 260 (No).
If f_B is 252, and f_A decreases (256 → 254), difference (254-252 = 2) decreases.
So f_B = 252 Hz.

Q3. Tension in a string is increased by 44%. Determine percentage change in frequency.

Solution:
f ∝ v ∝ √T.
T' = 1.44 T.
f' ∝ √(1.44 T) = 1.2 √T = 1.2 f.
% Increase = (1.2 - 1) × 100 = 20%.

Q4. A train moves towards stationary observer at 33 m/s. Whistle freq 1000 Hz. Speed of sound 330 m/s. Find apparent freq.

Solution:
Observer stationary (v_o=0). Source approaching (v_s is +ve in denominator logic, effectively reducing denom).
f' = f [ v / (v - v_s) ].
f' = 1000 [ 330 / (330 - 33) ] = 1000 [ 330 / 297 ].
f' = 1000 (1.11) = 1111 Hz.

Q5. Path difference between two waves is 5 cm. Wavelength is 20 cm. Find Phase Difference.

Solution:
Δφ = (2π / λ) Δx.
Δφ = (2π / 20) × 5 = 10π / 20.
Δφ = π/2 rad (90°).

Q6. In a closed organ pipe, frequency of 3rd harmonic is 450 Hz. Find fundamental frequency.

Solution:
For closed pipe, harmonics are odd multiples: f₁, 3f₁, 5f₁...
3rd Harmonic corresponds to 1st Overtone? No, 3rd Harmonic means 3f₁.
Wait! Closed pipe harmonics are 1st, 3rd, 5th...
So 3rd harmonic freq = 3 f₁.
3 f₁ = 450 → f₁ = 150 Hz.

Q7. Wave eq: y = 5 sin(100πt - 0.5πx). Find wave velocity.

Solution:
Compare with y = A sin(ωt - kx).
ω = 100π. k = 0.5π.
v = ω / k = 100π / 0.5π = 100 / 0.5.
v = 200 m/s.

Q8. Ratio of amplitudes is 3:1. Find ratio of Max to Min Intensity.

Solution:
I_max / I_min = (A₁+A₂)² / (A₁-A₂)².
Ratio = (3+1)² / (3-1)² = 4² / 2² = 16 / 4.
Ratio = 4:1.

Q9. First resonance length is 15cm. Second resonance frequency same tuning fork. Find length. (Neglect end correction).

Solution:
Closed pipe resonance lengths: L₁ = λ/4, L₂ = 3λ/4.
L₂ = 3 L₁.
L₂ = 3 × 15 = 45 cm.

Q10. Man fires gun before a cliff. Hears echo after 2s. Speed of sound 340 m/s. Distance to cliff?

Solution:
Total distance traveled by sound = 2d.
2d = v × t.
d = vt / 2 = 340 × 2 / 2 = 340 m.

Important Formulae

1. General Wave Equations

Displacement:

y = A sin(kx ± ωt + φ)

Wave Speed (v):

v = ω/k = λf

Particle Velocity:

v_p = -v (Slope) = -v (dy/dx)

2. Sound & Standing Waves

Laplace Correction:

v = √(γP / ρ)

Closed Pipe Frequency:

f_n = (2n-1)v / 4L (Odd Harmonics)

Open Pipe Frequency:

f_n = nv / 2L (All Harmonics)

3. Beats & Doppler

Beat Frequency:

f_b = |f₁ - f₂|

Doppler Shift:

f' = f (v ± v_o) / (v ∓ v_s)

20 NEET Golden Facts

1. Sound Waves: Cannot travel in vacuum. Require medium. Longitudinal in nature.
2. Light Waves: Can travel in vacuum. Transverse electromagnetic waves.
3. Phase Difference: Path Diff Δx of λ corresponds to Phase Diff Δφ of 2π.
4. Speed of Sound: Solids > Liquids > Gases. Max in Solids due to high elasticity.
5. Effect of Pressure: No effect on speed of sound (if Temp constant). ρ changes with P.
6. Effect of Humidity: Speed increases with humidity (Density of moist air < Dry air).
7. Intensity (I): I ∝ A² f². Energy per unit area per unit time.
8. Constructive Interference: Path diff = nλ. Intensity max = (A₁+A₂)².
9. Destructive Interference: Path diff = (2n+1)λ/2. Intensity min = (A₁-A₂)².
10. Fundamental Tone: Frequency of 1st harmonic. Simplest mode of vibration.
11. End Correction (e): Antinode forms slightly outside open end. e = 0.6r.
12. Beats limit: Audible only if diff < 10 Hz (Persistence of hearing).
13. Doppler source only: Source moves towards observer → wavelength decreases, freq increases.
14. Doppler observer only: Observer moves towards source → relative velocity increases, freq increases.
15. Node in String: At fixed ends.
16. Open pipe preferred: Produces richer quality sound (all harmonics present) vs closed pipe.
17. Distance between Node/Antinode: λ/4.
18. Echo: Reflection of sound. Min distance 17.2 m.
19. Wave Pulse reflection: From fixed end phase change π. From free end no phase change.
20. Red Shift (Light): Source moving away, freq decreases (shift towards red).

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