Waves & Acoustics: Types, Properties & Applications

TAKEAWAYS:

• Waves transfer energy without medium displacement; transverse (e.g., light) and longitudinal (e.g., sound) differ in particle motion.
• Sound waves in air follow $v=f\lambda$, where speed depends on medium density/temperature (331 m/s at 0°C).
• Doppler effect shifts frequency for moving sources/observers (e.g., ambulance sirens, radar speed guns).
• Resonance amplifies vibrations (e.g., bridges collapsing, musical instruments tuning).
• Acoustic impedance determines reflection/transmission at boundaries (key in ultrasound and speaker design).
• Standing waves form nodes/antnodes (e.g., guitar strings, organ pipes) with harmonics at $f_n=n{f}_1$.

Core Concepts

1. Wave Basics: Types and Properties

Waves are disturbances propagating through a medium (or vacuum) via energy transfer, not particle movement. Classify them by:

• Direction of oscillation:
  • Transverse: Oscillation perpendicular to propagation (e.g., light, water ripples).
  • Longitudinal: Oscillation parallel to propagation (e.g., sound, seismic P-waves).
• Medium requirement:
  • Mechanical waves: Need a medium (sound, water waves).
  • Electromagnetic waves: Travel in vacuum (light, radio).

Key Parameters:

• Amplitude (A): Maximum displacement (determines intensity/loudness).
• Wavelength (λ): Distance between crests (m).
• Frequency (f): Cycles per second (Hz).
• Speed (v): $v=f\lambda$ (m/s). In air, $v_{\text{sound}}=331+0.6T$ (°C).

graph LR
    A["Wave Types"] --> B["Transverse"]
    A --> C["Longitudinal"]
    B --> D["Light\nEM Wave"]
    B --> E["Water Ripples"]
    C --> F["Sound\nAir"]
    C --> G["Seismic P-Waves"]

Shows displacement vs. propagation for both types. (Image: Guy vandegrift, CC0, via Wikimedia Commons)

2. Sound Waves: Physics and Behavior

Sound is a longitudinal pressure wave in a medium (air, water, solids). Key properties:

• Speed in air: $v=331+0.6T$ (m/s). Faster in denser media (e.g., 1482 m/s in water).
• Intensity (I): Power per unit area ($I\propto A^2$), measured in decibels (dB): $$\beta =10{\log }_{10}\left(\frac{I}{I_0}\right),I_0=10^{-12}{\text{ W/m}}^2.$$
• Pitch vs. Loudness:
  • Pitch = frequency (high $f$ = high pitch).
  • Loudness = amplitude/intensity (dB scale).

Worked Example: Doppler Effect in Kathmandu Traffic A police siren (500 Hz) moves toward you at 30 m/s. Air speed = 340 m/s.

• Frequency heard: $f^{\prime }=f\left(\frac{v}{v-v_s}\right)=500\left(\frac{340}{340-30}\right)=551.4\text{ Hz}$.
• Result: Siren sounds ~10% higher-pitched when approaching.

Shows frequency shift for moving source/observer. (Image: Prokaryotic Caspase Homolog, CC BY-SA 4.0, via Wikimedia Commons)

3. Wave Interference and Superposition

When two waves meet, they superpose (add amplitudes). Outcomes:

• Constructive interference: Crests align → amplified wave.
• Destructive interference: Crest meets trough → cancellation.
• Standing waves: Fixed nodes/antnodes (e.g., guitar strings, organ pipes).

Standing Wave Formula: For a string fixed at both ends: $$f_n=\frac{nv}{2L},n=1,2,3,\dots$$ Example: A 1 m guitar string vibrates at 440 Hz (fundamental). Its 3rd harmonic is $3\times 440=1320$ Hz.

4. Resonance and Applications

Resonance occurs when a system’s natural frequency matches an external force, leading to amplified oscillations.

• Dangers: Tacoma Narrows Bridge (1940) collapsed due to wind-induced resonance.
• Applications:
  • Musical instruments: Tuning forks, violins.
  • Ultrasound imaging: High-frequency sound waves reflect off tissues.
  • Radio tuning: Circuits resonate at specific frequencies.

5. Acoustic Impedance and Reflection

When sound meets a boundary (e.g., air/water), part reflects/transmits based on acoustic impedance (Z): $$Z=\rho v,$$ where $\rho$ = density, $v$ = speed.

• High impedance mismatch → more reflection (e.g., sound bouncing off walls).
• Low mismatch → transmission (e.g., ultrasound through skin).

Example: Ultrasound gel reduces impedance mismatch between transducer and skin, improving signal transmission.

6. Beats and Waveforms

Beats occur when two close frequencies interfere: $$f_{\text{beat}}=|f_1-f_2|.$$ Example: Tuning pianos using a 440 Hz reference tone. If a string produces 444 Hz, beats at 4 Hz help adjust pitch.

In the Real World

1. eSewa/Khalti Payments:

   • Doppler effect is used in RFID-based payment authentication. The system emits a signal and measures the reflected frequency shift to verify card proximity, preventing fraud.

2. Pathao/Ncell Ride-Hailing:

   • Ultrasound sensors in autonomous vehicles (e.g., Tesla) detect obstacles via echo time-of-flight, similar to bats’ echolocation. The system calculates distance using $d=v\times t$, where $t$ is the round-trip time of the reflected sound wave.

3. NTC Power Grid Monitoring:

   • Standing waves are analyzed in high-voltage transmission lines to detect faults. Resonance frequencies help identify cracks or loose connections before they cause blackouts.

4. NEPSE Stock Market Alerts:

   • Beats and interference principles are used in audio-based trading signals. Some algorithms convert market data into sound waves; traders listen for "beats" (frequency differences) to spot trends.

5. Kathmandu Traffic Noise Pollution:

   • Acoustic impedance explains why traffic noise is louder in valleys (sound reflects off buildings/mountains). Engineers use sound-absorbing materials (e.g., foam panels) to reduce impedance mismatches and dampen echoes in concert halls or metro stations.

Exam Tip

1. Memorize formulas:

   • $v=f\lambda$, Doppler shift ($f^{\prime }=f\frac{v\pm v_o}{v\mp v_s}$), standing wave harmonics ($f_n=n{f}_1$).
   • Decibel scale: $\beta =10{\log }_{10}(I/I_0)$.

2. Graphs are critical:

   • Sketch displacement-time graphs for transverse/longitudinal waves.
   • Plot standing wave patterns with nodes/antnodes labeled.
   • Draw Doppler effect frequency shifts for moving sources/observers.

3. Real-world applications:

   • Link resonance to musical instruments or structural failures.
   • Relate Doppler effect to radar guns or medical imaging (e.g., blood flow measurement).
   • Explain acoustic impedance in ultrasound or speaker design.

4. Unit consistency:

   • Always convert temperatures to Celsius for sound speed calculations.
   • Use SI units (Hz, m, s) in all problems.

5. Common pitfalls:

   • Transverse vs. longitudinal: Sound is always longitudinal in fluids.
   • Doppler sign conventions: Observer moving toward source → + in numerator; source moving toward observer → – in denominator.
   • Standing waves: Only odd harmonics exist for fixed-fixed ends (e.g., strings); all harmonics for open pipes.

Visual Summary Table: