Sound waves are fascinating phenomena that shape our auditory world. From the soothing melody of a guitar to the blaring siren of an ambulance, these mechanical waves travel through various media, exhibiting unique properties like frequency, wavelength, and amplitude.

Understanding sound's behavior is crucial in many fields. We'll explore how sound intensity decreases with distance, the Doppler effect's impact on perceived frequency, and how resonance creates standing waves in musical instruments and acoustic spaces.

Wave Properties and Sound Characteristics

Properties of sound waves

• Mechanical waves require medium for propagation cannot travel through vacuum
• Longitudinal waves particles oscillate parallel to wave direction create compressions and rarefactions
• Wave properties include frequency (oscillations per second), wavelength (distance between compressions), speed (frequency × wavelength), amplitude (max displacement)
• Speed of sound depends on medium properties varies with temperature (343 m/s in air at 20℃)
• Harmonic content consists of fundamental frequency and overtones
• Sound waves exhibit reflection, refraction, and diffraction

Sound intensity and decibels

• Sound intensity measures power per unit area $I = \frac{P}{A}$ in W/m²
• Inverse square law intensity decreases with distance $I \propto \frac{1}{r^2}$ (concert speakers)
• Decibel scale logarithmic measure of sound intensity level uses reference $I_0 = 10^{-12}$ W/m²
• Decibel formula $\beta = 10 \log(\frac{I}{I_0})$ relates intensity to perceived loudness
• Hearing threshold 0 dB pain threshold 120 dB (jet engine)
• Sound levels add logarithmically not linearly (multiple sources)

Acoustic Phenomena

Doppler effect for sound

• Observed frequency changes due to relative motion between source and observer
• Approaching source or observer increases frequency receding decreases (ambulance siren)
• Doppler formula $f' = f(\frac{v \pm v_o}{v \mp v_s})$ calculates observed frequency
• Applications include weather radar blood flow measurement speed detection (police radar)

Resonance in acoustic systems

• Resonance occurs at natural vibration frequencies increases amplitude
• Standing waves form from superposition create nodes (zero displacement) and antinodes (max displacement)
• String resonance fundamental frequency $f_1 = \frac{v}{2L}$ harmonics $f_n = n\frac{v}{2L}$ (guitar strings)
• Air column resonance open-ended $f_n = n\frac{v}{2L}$ closed-ended $f_n = (2n-1)\frac{v}{4L}$ (flutes, organ pipes)
• Applications include musical instruments acoustic design noise reduction (soundproofing)