Chapter 12: Mechanical Waves & Sound – Study Notes

Mechanical Waves & Sound

Introduction to Mechanical Waves

Mechanical waves are collective periodic disturbances that propagate through a medium. Unlike electromagnetic waves, mechanical waves require a material medium for transmission. Their behavior can be described mathematically, especially when they exhibit simple harmonic motion, using sine and cosine functions.

• Wave: A periodic disturbance that transfers energy through a medium.

• Stationary vs. Traveling Waves: Waves can be stationary (standing waves) or travel through the medium.

• Simple Harmonic Motion: If the wave is sinusoidal, it can be described by $y(t) = A \sin(\omega t)$ or $y(x, t) = A \sin\left[2\pi\left(\frac{t}{T} - \frac{x}{\lambda}\right)\right]$.

Types of Mechanical Waves

Mechanical waves are classified based on the direction of disturbance relative to the direction of propagation.

• Transverse Waves: Disturbance is perpendicular to the direction of propagation (e.g., waves on a rope).

• Longitudinal Waves: Disturbance is parallel to the direction of propagation (e.g., sound waves in air).

Examples:

• Rope wiggles – Transverse

• Sound – Longitudinal

• Water – Both transverse (surface) and longitudinal (depth)

• Earthquake – Both types (P-waves: longitudinal, S-waves: transverse)

Wave Properties and Equations

Key properties of waves include amplitude, period, frequency, and wave speed.

• Amplitude (A): Maximum displacement from equilibrium.

• Period (T): Time to complete one cycle (units: seconds).

• Frequency (f): Number of cycles per second (units: hertz, Hz). $f = \frac{1}{T}$

• Wave Speed (v): $v = \lambda f = \frac{\lambda}{T}$

Wave Speed on a String

The speed of a wave on a stretched string depends on the tension and the linear mass density of the string.

• Wave speed: $v = \sqrt{\frac{F_T}{\mu}}$

• Linear mass density: $\mu = \frac{m}{L}$

Example: For a guitar string with tension $F_T$ and mass per unit length $\mu$, use the above formula to calculate wave speed.

Standing Waves and Normal Modes

Standing waves are formed by the superposition of two waves traveling in opposite directions. They exhibit nodes (points of zero displacement) and antinodes (points of maximum displacement).

• Standing wave equation: $y(t) = A \sin(\omega t)$

• Traveling wave equation: $y(x, t) = A \sin\left[2\pi\left(\frac{t}{T} - \frac{x}{\lambda}\right)\right]$

Normal modes occur at specific frequencies and wavelengths determined by the boundary conditions of the system.

Normal/Allowed Modes for Strings and Pipes

For a string fixed at both ends, the allowed wavelengths and frequencies are:

• $L = n \frac{\lambda}{2}$, $n = 1, 2, 3, ...$

• $\lambda_n = \frac{2L}{n}$

• $f_n = \frac{v}{2L} n = n f_1$

• Fundamental frequency: $f_1 = \frac{v}{2L}$

• Using tension: $f_1 = \frac{1}{2L} \sqrt{\frac{F_T}{\mu}}$

Example: For a bass string of length 0.86 m, mass per unit length 0.015 kg/m, and fundamental frequency 41 Hz, calculate tension and higher harmonics.

Resonators: Open and Closed Pipes

Resonant frequencies in pipes depend on whether the ends are open or closed.

• Open pipe (both ends open): $L = n \frac{\lambda}{2}$, $f_n = \frac{n v}{2L}$, $n = 1, 2, 3, ...$

• Closed pipe (one end closed): $L = n \frac{\lambda}{4}$, $f_n = \frac{n v}{4L}$, $n = 1, 3, 5, ...$

Example: For a tube of hydrogen gas, given frequency and node distance, calculate wave speed.

Wave Interference

When two waves overlap, they interfere constructively or destructively depending on their phase relationship.

• Constructive interference: Path difference is an integer multiple of wavelength: $|d_1 - d_2| = n \lambda$

• Destructive interference: Path difference is a half-integer multiple: $|d_1 - d_2| = (n + \frac{1}{2}) \lambda$

Sound Intensity and Decibel Scale

Sound waves carry energy, and their intensity is the power delivered per unit area. The decibel scale is used to quantify sound intensity relative to a reference level.

• Power: $P_{av} = \frac{W}{\Delta t}$

• Intensity: $I = \frac{P}{A}$

• Reference intensity: $I_0 = 10^{-12} \text{ W/m}^2$

• Loudness (Bels): $\text{loudness} = \log_{10} \frac{I}{I_0}$

• Decibel (dB): $\beta = (10 \text{ dB}) \log_{10} \left( \frac{I}{I_0} \right)$

Type of Noise	Sound Level (dB)	Intensity (W/m2)
Rock concert	140	100
Threshold of pain	120	1
Riveter	95	3.2 × 10-3
Elevated trains	90	10-3
Busy street traffic	70	10-5
Ordinary conversation	65	3.2 × 10-6
Quiet automobile	50	10-7
Quiet radio in home	40	10-8
Average whisper	20	10-10
Rustle of leaves	10	10-11
Threshold of hearing	0	10-12

Wave Beats

When two waves of similar frequency are superimposed, the result is a beat pattern where the amplitude varies periodically.

• Beat frequency: $f_{\text{beat}} = |f_1 - f_2|$

• Trigonometric identity: $\sin(A) + \sin(B) = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)$

Example: Two sound waves of frequencies 440 Hz and 444 Hz produce a beat frequency of 4 Hz.

Doppler Effect

The Doppler effect describes the change in frequency of a wave as observed by a listener moving relative to the source.

• Doppler formula: $f_L = \frac{v + v_L}{v + v_S} f_S$

• $f_L$ = frequency heard by listener

• $f_S$ = frequency emitted by source

• $v$ = speed of sound

• $v_L$ = velocity of listener (positive if moving toward source)

• $v_S$ = velocity of source (positive if moving away from listener)

Example: A train horn at 320 Hz, train speed 40 m/s, speed of sound 340 m/s. Frequency heard as train approaches: 363 Hz; as train recedes: 286 Hz.

Additional info:

• Wave reflection and superposition are essential for understanding phenomena such as echoes and standing waves.

• Wave simulations (e.g., PhET) can help visualize wave behavior and interference.