Chapter 14: Waves and Sound – Structured Study Notes

Chapter 14: Waves and Sound

Sections Covered

• 14-1 Types of Waves

• 14-2 Waves on a String

• 14-3 Harmonic Wave Functions

• 14-4 Sound Waves

• 14-5 Sound Intensity

• 14-6 The Doppler Effect

• 14-7 Superposition and Interference

• 14-8 Standing Waves

• 14-9 Beats

14-1 Types of Waves

Definition and Classification of Waves

• Wave: A disturbance that propagates from one place to another, carrying energy and characterized by a well-defined speed determined by the properties of the medium.

• Waves are classified as transverse, longitudinal, or a combination of both.

• Transverse wave: The displacement of particles is perpendicular to the direction of wave propagation (e.g., waves on a string).

• Longitudinal wave: The displacement of particles is parallel to the direction of wave propagation (e.g., sound waves in air).

• Some waves, such as water surface waves, exhibit both transverse and longitudinal motion.

Wave Properties

• Crest: The highest point of a wave.

• Trough: The lowest point of a wave.

• Wavelength (λ): The distance between two consecutive crests or troughs. Definition: λ = distance over which a wave repeats.

• Period (T): The time it takes for a wave to travel one wavelength.

• Frequency (f): The number of wave cycles per second, $f = \frac{1}{T}$ (unit: Hz).

• Wave speed (v): $v = \frac{\lambda}{T} = \lambda f$ (unit: m/s).

Example

• Sound waves travel in air at 343 m/s. If middle C on a piano produces sound waves with a wavelength of 1.31 m, the frequency is $f = \frac{343}{1.31} \approx 262$ Hz.

• Dogs can hear up to 45,000 Hz. The wavelength of such a sound is $\lambda = \frac{343}{45,000} \approx 0.0076$ m.

14-2 Waves on a String

Wave Generation and Propagation

• Vibrating a string fixed at one end creates a periodic wave characterized by wavelength (λ), frequency (f), and speed (v).

• A wave pulse is a single disturbance that travels along the string.

Wave Speed on a String

• The speed depends on the tension (F) in the string and the linear density (μ) ($\mu = \frac{m}{L}$, where m is mass and L is length).

• Formula: $v = \sqrt{\frac{F}{\mu}}$

• Greater tension increases speed; greater density decreases speed.

Reflection of Waves

• At a fixed end, a wave pulse reflects and is inverted.

• At a free end, a wave pulse reflects without inversion.

Example

• A guitar string with μ = $0.401 \times 10^{-3}$ kg/m and F = 71.4 N: $v = \sqrt{\frac{71.4}{0.401 \times 10^{-3}}} \approx 422$ m/s.

14-3 Harmonic Wave Functions

Mathematical Description of Waves

• A harmonic wave is described by a sine or cosine function.

• The general form: $y(x, t) = A \cos\left(2\pi \left(\frac{x}{\lambda} - \frac{t}{T}\right)\right)$

• Amplitude (A): Maximum displacement from equilibrium.

• As time progresses, the wave moves along the x-axis, and the phase changes accordingly.

Example

• Given $y(x, t) = (0.12\,\text{m}) \cos(\pi x/4 - 8\pi t)$, amplitude = 0.12 m, wavelength = 8 m, period = 0.25 s.

14-4 Sound Waves

Nature and Properties of Sound Waves

• Sound waves are longitudinal waves consisting of compressions and rarefactions in air.

• Speed of sound in air at room temperature: approximately 343 m/s.

• Speed in solids depends on material stiffness (e.g., Aluminum: 6420 m/s, Steel: 5960 m/s).

• Pitch is determined by frequency; humans hear 20 Hz to 20,000 Hz.

• For a constant speed, increasing frequency decreases wavelength: $v = f\lambda$.

14-5 Sound Intensity

Definition and Calculation

• Intensity (I): Power per unit area, $I = \frac{P}{A}$ (unit: W/m2).

• Intensity decreases with distance from the source: $I = \frac{P}{4\pi r^2}$.

• Examples: Jet engine at 50 m: $10^2$ W/m2, Conversation at 1 m: $10^{-6}$ W/m2.

Intensity Level (Decibels)

• Intensity level (β): $\beta = (10\,\text{dB}) \log\left(\frac{I}{I_0}\right)$, where $I_0 = 10^{-12}$ W/m2 is the threshold of hearing.

• Each 10 dB increase corresponds to a tenfold increase in intensity.

Sound Source	Intensity (W/m2)	Intensity Level (dB)
Loudest sound (lab)	109	190
Jet engine (50 m)	102	140
Conversation (1 m)	10-6	60
Threshold of hearing	10-12	0

14-6 The Doppler Effect

Frequency Shift Due to Relative Motion

• The Doppler Effect is the change in frequency or wavelength of a wave in relation to an observer moving relative to the source.

• For an observer moving towards a stationary source: $f' = f\left(1 + \frac{u}{v}\right)$

• For a source moving towards a stationary observer: $f' = \frac{f}{1 - \frac{u}{v}}$

• General case (both moving): $f' = \left(\frac{1 \pm u_o/v}{1 \mp u_s/v}\right)f$

• Where $u$ is speed of observer/source, $v$ is speed of sound, and signs depend on direction.

	Stationary Observer	Observer Moving Towards Source	Observer Moving Away from Source
Stationary Source	$f_s$	$f_s(1 + \frac{u_o}{v})$	$f_s(1 - \frac{u_o}{v})$
Source Moving Towards Observer	$\frac{f_s}{1 - \frac{u_s}{v}}$
Source Moving Away from Observer	$\frac{f_s}{1 + \frac{u_s}{v}}$

14-7 Superposition and Interference

Principle of Superposition

• When two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements.

• Constructive interference: Waves in phase add to produce larger amplitude.

• Destructive interference: Waves out of phase subtract, reducing amplitude.

• Interference patterns result from the superposition of waves from different sources.

14-8 Standing Waves

Formation and Properties

• A standing wave is formed by the interference of two waves traveling in opposite directions with the same frequency and amplitude.

• Nodes: Points of zero amplitude.

• Antinodes: Points of maximum amplitude.

• For a string fixed at both ends, the allowed wavelengths are $\lambda_n = \frac{2L}{n}$, $n = 1, 2, 3, ...$

• Frequencies: $f_n = \frac{nv}{2L}$

14-9 Beats

Beat Frequency

• When two waves of slightly different frequencies interfere, they produce beats—periodic variations in amplitude.

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

• Beats are heard as fluctuations in loudness at the beat frequency.

Additional info: The above notes are structured to provide a comprehensive yet concise overview of the main concepts, equations, and examples relevant to Chapter 14: Waves and Sound, suitable for college-level physics students.