The Pulse of the Universe: Mastering Waves

If Oscillations are about a single particle dancing in place, Waves are about that dance spreading through a crowd. A wave is a disturbance that carries energy from one point to another without moving the medium itself. From the silent ripples on a pond to the thunderous roar of a jet engine, waves are the primary way the universe communicates.

The Core Pillars of Wave Motion

1. Longitudinal vs. Transverse

• Transverse Waves: Particles move perpendicular to the direction of the wave (e.g., light, ripples on water, waves in a string).
• Longitudinal Waves: Particles move parallel to the direction of the wave, creating compressions and rarefactions (e.g., sound waves).

2. The Speed of Sound: Newton vs. Laplace

Newton originally calculated the speed of sound assuming an isothermal process. He was wrong by about 15%. Laplace corrected this by realizing that sound travels so fast that heat cannot escape, making it an adiabatic process.

• Formula: v = √(γP/ρ)

3. Standing Waves: The Physics of Music

When two identical waves traveling in opposite directions meet, they interfere to form a Standing Wave. This creates “Nodes” (points of zero motion) and “Antinodes” (points of maximum vibration). This is how every guitar, piano, and flute produces sound.

4. The Doppler Effect

Ever noticed how an ambulance siren sounds higher as it approaches and lower as it moves away? That is the Doppler Effect—the change in frequency due to the relative motion between the source and the observer.

The Gauntlet: 10 Challenging Aptitude Questions

Question 1: The String Tension Shift

A transverse wave travels along a string with a speed of 50 m/s. If the tension in the string is increased by 44% while keeping the linear mass density constant, what is the new speed of the wave?

Question 2: The Open vs. Closed Pipe

An open organ pipe has a fundamental frequency of 300 Hz. If one end of the pipe is closed, what will be the new fundamental frequency?

Question 3: The Beat Frequency

Two tuning forks, A and B, produce 4 beats per second. When fork A is slightly loaded with wax, the beat frequency increases to 6 beats per second. If the frequency of B is 256 Hz, what was the original frequency of A?

Question 4: Speed of Sound and Temperature

At what temperature (in Celsius) will the speed of sound be double its value at 0°C?

Question 5: The Third Harmonic

A string of length 2m is fixed at both ends. If it vibrates in its third harmonic with a frequency of 450 Hz, what is the speed of the transverse wave in the string?

Question 6: Intensity and Distance

A point source of sound emits energy equally in all directions. If you move from a distance of 2m to 10m from the source, by how many decibels (dB) does the sound level decrease?

Question 7: The Doppler “Wall” Reflect

A car blowing a horn of frequency 400 Hz moves at 20 m/s toward a tall vertical wall. What is the frequency of the echo heard by the driver? (Speed of sound = 340 m/s).

Question 8: Phase Difference to Path Difference

Two sound waves from coherent sources reach a point with a path difference of 15 cm. If the wavelength of the sound is 10 cm, is the interference at that point constructive or destructive?

Question 9: Velocity of a Particle vs. Wave

A wave is represented by y = 0.5 sin(10t – 2x). Find the ratio of the maximum particle velocity to the wave velocity.

Question 10: The Resonance Column

In a resonance column experiment, the first resonance occurs at a tube length of 16 cm and the second at 49 cm. What is the end correction of the tube?

Detailed Explanations & Solutions

1. String Tension

Wave speed v = √(T/μ). So, v₂/v₁ = √(T₂/T₁).

Increasing tension by 44% means T₂ = 1.44 T₁.

v₂ = v₁ × √1.44 = 50 × 1.2.

Result: 60 m/s.

2. Organ Pipes

Fundamental of open pipe = v/2L. Fundamental of closed pipe = v/4L.

The closed pipe frequency is exactly half of the open pipe.

Result: 150 Hz.

3. Beat Frequency

Original A was either 252 or 260 (256 ± 4). Loading A with wax decreases its frequency. If 252 is lowered, it gets further from 256 (beats increase). If 260 is lowered, it gets closer to 256 (beats decrease).

Result: 252 Hz.

4. Temperature Effect

v ∝ √T (where T is in Kelvin). To double v, T must quadruple.

Initial T = 273 K. Final T = 273 × 4 = 1092 K.

Result: 1092 – 273 = 819°C.

5. Third Harmonic

In a fixed string, fₙ = nv/2L. For n=3: 450 = 3v / (2 × 2).

450 = 3v / 4 → 1800 = 3v.

Result: 600 m/s.

6. Decibel Decrease

Intensity I ∝ 1/r². Ratio of intensities = (10/2)² = 25.

Change in dB = 10 log₁₀(I₁/I₂) = 10 log₁₀(25) = 20 log₁₀(5).

Result: Approximately 14 dB.

7. Doppler Echo

The wall “hears” frequency f’ = f [v / (v-vₛ)]. The wall then acts as a stationary source reflecting f’ to a moving observer.

f_final = f’ [(v+v₀) / v] = f [(v+v₀) / (v-vₛ)].

f_final = 400 [(340+20) / (340-20)] = 400 [360/320].

Result: 450 Hz.

8. Interference

Path Difference Δx = 1.5 λ.

When path difference is an odd multiple of half-wavelengths (0.5λ, 1.5λ, 2.5λ…), the interference is destructive.

Result: Destructive (the point will be quiet).

9. Particle vs. Wave Velocity

Wave velocity v_w = ω/k = 10/2 = 5.

Max particle velocity v_p = Aω = 0.5 × 10 = 5.

Result: Ratio = 1.

10. End Correction

L₁ + e = λ/4 and L₂ + e = 3λ/4.

Subtracting gives λ/2 = L₂ – L₁ = 49 – 16 = 33 cm. So λ = 66 cm.

e = (λ/4) – L₁ = (66/4) – 16 = 16.5 – 16.

Result: 0.5 cm.

Pro-Tip: The “Wave” Visualization

When dealing with string waves, remember that no matter how fast the wave travels, the particles themselves only move up and down. They never travel with the wave.