Waves and Sound: Study Notes for Phys1310 Exam 2

Simple Harmonic Motion

Definition and Properties

Simple harmonic motion (SHM) describes the oscillatory motion of an object where the restoring force is proportional to its displacement from equilibrium. This type of motion is fundamental in physics and underlies many wave phenomena.

• Equation of Motion: $x(t) = A \\cos(\omega t)$, where A is amplitude and $\omega$ is angular frequency.

• Period (T): The time for one complete cycle, $T = \frac{2\pi}{\omega}$.

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

• Spring-Mass System: $T = 2\pi \sqrt{\frac{m}{k}}$, where m is mass and k is spring constant.

• Pendulum: $T = 2\pi \sqrt{\frac{L}{g}}$, where L is length and g is acceleration due to gravity.

• Mechanical Energy: $E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2$

Example: A mass attached to a spring oscillates back and forth, with its position described by a cosine function.

Waves

Introduction to Waves

A wave is a disturbance that propagates from one place to another, transferring energy without transferring matter. Waves are central to many physical phenomena, including sound, light, and water waves.

• Propagation: Waves can travel through solids, liquids, and gases.

• Connection to SHM: Many waves are generated by oscillatory sources undergoing simple harmonic motion.

Types of Waves

Transverse and Longitudinal Waves

Waves are classified based on the direction of particle displacement relative to the direction of wave propagation.

• Transverse Waves: Displacement is perpendicular to the direction of wave motion. Example: Motion of a wave on a string, ripples on a pond.

• Longitudinal Waves: Displacement is parallel to the direction of wave motion. Example: Sound waves, compression and rarefaction in air.

• Combinations: Some waves can have both transverse and longitudinal components.

Periodic Waves

Characteristics of Periodic Waves

Periodic waves consist of cycles or patterns that repeat over time, similar to simple harmonic motion.

• Amplitude (A): Maximum displacement from equilibrium.

• Wavelength (\lambda): Horizontal length of one cycle.

• Period (T): Time for one complete cycle.

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

Wavelength, Frequency, and Speed

Wave Equation

The speed of a wave is related to its frequency and wavelength by the fundamental wave equation:

• Wave Speed: $v = f\lambda$

• Units:

  • Speed: m/s

  • Frequency: Hz (1/s)

  • Wavelength: m

Determinants of Wave Speed

The speed of a wave depends on the properties of the medium through which it travels.

• On a String: Determined by tension and mass per unit length ($\mu = m/L$).

• General: Wave velocity depends on medium's elasticity and inertia.

Longitudinal Waves and Sound

Nature of Sound Waves

Sound waves are longitudinal waves that propagate through gases, liquids, and solids. The distance between adjacent compressions (condensations) is the wavelength.

• Propagation: Sound travels at different speeds in different media.

• Compressions and Rarefactions: Regions of high and low pressure, respectively.

Pitch and Loudness

Perceptual Qualities of Sound

Pitch and loudness are two key perceptual qualities of sound, related to frequency and amplitude, respectively.

• Pitch: Determined by frequency; higher frequency means higher pitch.

• Loudness: Related to amplitude; greater amplitude means louder sound.

• Pure Tone: Sound with a single frequency.

Sound Intensity

Definition and Calculation

Sound intensity quantifies the power transmitted by a sound wave per unit area perpendicular to the direction of propagation.

• Intensity (I): $I = \frac{P}{A}$, where P is power and A is area.

• Spherical Waves: For a point source, $I = \frac{P}{4\pi r^2}$, where r is distance from the source.

Example Calculation

• If intensity at 4.0 m is 120 W/m2, power $P = 4\pi (4\text{ m})^2 \times 120 \text{ W/m}^2 = 24115 \text{ W}$

• Intensity at 9.0 m: $I = \frac{P}{4\pi (9\text{ m})^2}$

Intensity and Hearing

Decibel Scale and Intensity Level

Loudness is measured by the intensity level, expressed in decibels (dB), which is a logarithmic scale.

• Reference Intensity: $I_0 = 1.00 \times 10^{-12} \text{ W/m}^2$

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

• Logarithmic Nature: Increasing intensity by a factor of 10 increases intensity level by 10 dB.

Example Calculation

• If one person talking produces 60 dB, eight people talking simultaneously: $\beta = 60 \text{ dB} + (10 \text{ dB}) \log_{10}(8) \approx 69 \text{ dB}$

Grading and Exam Information

Exam Structure and Grading Policy

Exams consist of three night exams and a final exam. The lowest night exam score may be replaced by the final exam score if it is higher. Extra credit is available through quizzes and other means.

Letter Grade Table

Letter Grade	Range
A	93.00-100
A-	90.00-92.99
B+	87.00-89.99
B	83.00-86.99
B-	80.00-82.99
C+	77.00-79.99
C	73.00-76.99
C-	70.00-72.99
D	67.00-69.99
D-	63.00-66.99
F	60.00-62.99 and below

Score Replacement Option Table

Original Scores	Replacement
60, 70, 80, 75 (285)	75, 70, 80, 75
80, 70, 75, 90 (315)	80, 90, 75, 90
80, 75, 85, 70 (310)	80, 75, 85, 70

Note: The replacement option cannot be used to replace a zero score for an exam not taken. All students must take four exams (three night exams and the final exam).

Final Exam Information

• Date: Tuesday, December 16

• Time: 4:30 pm - 6:30 pm

Summary Table: Key Equations

Concept	Equation
SHM Position	$x(t) = A \\cos(\omega t)$
Period (Spring)	$T = 2\pi \sqrt{\frac{m}{k}}$
Period (Pendulum)	$T = 2\pi \sqrt{\frac{L}{g}}$
Wave Speed	$v = f\lambda$
Sound Intensity	$I = \frac{P}{A}$
Spherical Wave Intensity	$I = \frac{P}{4\pi r^2}$
Intensity Level (dB)	$\beta = (10 \text{ dB}) \log_{10} \left( \frac{I}{I_0} \right)$

Additional info: Some context and definitions have been expanded for clarity and completeness.