Traveling Waves and Sound: Physics for Life Sciences I (Lecture 24, Ch. 15.1-15.4)

Recap: Simple Harmonic Motion (SHM)

Fundamental Properties of SHM

Simple harmonic motion (SHM) describes oscillatory motion that can be represented by sinusoidal functions. Systems undergoing SHM share common mathematical forms for position, velocity, and acceleration as functions of time.

• Position-versus-time: $x(t) = A \, \cos(2\pi f t)$

• Velocity-versus-time: $v(t) = -[2\pi f A] \, \sin(2\pi f t)$

• Acceleration-versus-time: $a(t) = -[2\pi f]^2 A \, \cos(2\pi f t)$

Example: A mass attached to a spring oscillates back and forth, following the above equations.

Recap: SHM and Uniform Circular Motion

Connection Between SHM and Circular Motion

The x-component of a particle in uniform circular motion exhibits simple harmonic motion. The position as a function of time is:

• Equation: $x(t) = A \, \cos(2\pi f t)$

• Application: The projection of uniform circular motion onto one axis yields SHM.

Example: The shadow of a rotating wheel's point moves in SHM.

Recap: Mass on a Spring and Pendulum

Restoring Forces and Oscillator Properties

SHM occurs when a linear restoring force returns a system to equilibrium. The frequency and period depend on system details, but not on amplitude.

• Mass on Spring: $F_{net} = -kx$

• Pendulum: $F_{net} = -\left(\frac{mg}{L}\right)s$

Frequency and Period Formulas

Oscillator	Frequency $f$	Period $T$
Mass on Spring	$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$	$T = 2\pi \sqrt{\frac{m}{k}}$
Pendulum	$f = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$	$T = 2\pi \sqrt{\frac{L}{g}}$

Key Point: For pendulums, frequency and period depend on length and gravity, not mass. For springs, they depend on mass and spring constant.

Recap: Damping and Resonance

Damping in SHM

Damping (e.g., due to drag) causes the amplitude of SHM to decrease over time. The time constant $\tau$ determines the rate of decay.

• Damped amplitude: $A(t) = A_0 e^{-t/\tau}$

Resonance

A system oscillates at its natural frequency $f_0$. If driven at $f_{ext} = f_0$, resonance occurs, producing large amplitude oscillations.

Chapter 15: The Wave Model, Traveling Waves, and Sound

Introduction to Waves

The wave model describes phenomena from light to earthquakes. A traveling wave is an organized disturbance moving with a well-defined speed.

• Wave model: Emphasizes properties common to all waves.

• Traveling wave: Disturbance propagating through a medium or field.

Section 15.1: The Wave Model

Types of Waves

• Mechanical Waves: Require a material medium (e.g., water, air, string). The medium's particles are displaced from equilibrium.

• Electromagnetic Waves: Oscillations of electromagnetic fields (e.g., light, radio). No medium required; can travel through vacuum.

• Matter Waves: Wave-like properties of particles (e.g., electrons, atoms) at quantum scales.

Energy Transfer

• Key Point: Waves transfer energy, not matter.

Section 15.1: Transverse and Longitudinal Waves

Classification of Waves

• Transverse Waves: Medium particles move perpendicular to wave direction. Example: Waves on a string, S-waves in earthquakes.

• Longitudinal Waves: Medium particles move parallel to wave direction. Example: Sound waves, P-waves in earthquakes.

Earthquake Waves Comparison

Type	Direction of Particle Motion	Speed	Destructiveness
S-wave (Transverse)	Perpendicular	Slower	More destructive (larger amplitude)
P-wave (Longitudinal)	Parallel	Faster	Less destructive

Section 15.2: Traveling Waves

Wave and Medium Motion

It is important to distinguish between the motion of the wave and the motion of the medium. Newton's laws apply to the medium's particles, not the wave itself.

• Example: A pulse on a string moves right, but each string segment moves up and down.

Section 15.2: Waves on a String

Transverse Wave Pulse

• Each point on the string moves perpendicular to the wave's direction.

• Once initiated by an external force, the pulse continues due to the medium's internal dynamics.

Section 15.2: Sound Waves

Nature of Sound Waves

• Sound waves: Longitudinal waves in air (or other media).

• Compression: Region of higher pressure.

• Rarefaction: Region of lower pressure.

• Sound wave speed depends on properties of the medium (e.g., air temperature, molecular mass).

Section 15.2: Wave Speed as a Property of the Medium

Wave Speed on a String

• Depends on string's tension $T$ and linear density $\mu$ ($\mu = \frac{m}{L}$).

• Formula: $v_{string} = \sqrt{\frac{T}{\mu}}$

Wave Speed in Gases (Sound)

• Formula: $v_{sound} = \sqrt{\frac{\gamma k_B T}{m}} = \sqrt{\frac{\gamma R T}{M}}$

• $M$ = molar mass, $\gamma$ = adiabatic index, $k_B$ = Boltzmann constant, $T$ = temperature in kelvin.

Wave Speed in Different Media

• Speed increases with temperature and decreases with molecular mass.

• Speed is higher in liquids than gases, and highest in solids.

Table: Speed of Sound in Various Media

Medium	Speed (m/s)
Air (0°C)	331
Air (20°C)	343
Helium (0°C)	972
Water	1480
Human tissue (ultrasound)	1640
Lead	1210
Glass	5100
Granite	6000
Diamond	12000

Additional info: Table values inferred from context and standard sources.

Speed of Electromagnetic Waves

• All electromagnetic waves travel at $c = 3.00 \times 10^8$ m/s in vacuum.

Section 15.3: Graphical and Mathematical Descriptions of Waves

Snapshot and History Graphs

• Snapshot graph: Displacement vs. position at a fixed time.

• History graph: Displacement of a fixed point vs. time.

Mathematical Description of Sinusoidal Waves

• General equation: $y(x, t) = A \cos[2\pi (\frac{x}{\lambda} - \frac{t}{T})]$ for a wave traveling right

• For a wave traveling left: $y(x, t) = A \cos[2\pi (\frac{x}{\lambda} + \frac{t}{T})]$

• Amplitude ($A$): Maximum displacement

• Wavelength ($\lambda$): Distance between crests

• Period ($T$): Time for one cycle

• Frequency ($f$): $f = 1/T$

Fundamental Relationship

• Wave speed: $v = \lambda f$

Section 15.4: Sound and Light Waves

Sound Waves

• Pressure oscillates sinusoidally around atmospheric pressure.

• Humans hear frequencies from 20 Hz to 20,000 Hz.

• Ultrasound: Frequencies above human hearing, used in medical imaging and animal echolocation.

Example: Range of Wavelengths of Sound

• At 20 Hz: $\lambda = \frac{343}{20} = 17$ m

• At 500 Hz: $\lambda = \frac{343}{500} = 0.686$ m

• At 20,000 Hz: $\lambda = \frac{343}{20000} = 0.017$ m

Example: Ultrasound in Medicine

• Required wavelength: 0.50 mm

• Speed in tissue: 1540 m/s

• Frequency: $f = \frac{v}{\lambda} = \frac{1540}{0.0005} = 3.1 \times 10^6$ Hz = 3.1 MHz

Light and Electromagnetic Waves

• Electromagnetic waves: Oscillations of electric and magnetic fields.

• Visible light: Wavelengths 400-700 nm (violet to red).

• All electromagnetic waves travel at $c = 3.00 \times 10^8$ m/s in vacuum.

Electromagnetic Spectrum

• Visible spectrum is a small part of the full electromagnetic spectrum (includes radio, microwave, infrared, ultraviolet, x-ray, gamma ray).

Summary: General Principles

Wave Model Key Points

• Traveling waves are organized disturbances moving at a defined speed.

• Transverse waves: Medium moves perpendicular to wave direction.

• Longitudinal waves: Medium moves parallel to wave direction.

• Waves transfer energy, not matter.

• Mechanical waves require a medium; electromagnetic waves do not.

Graphical and Mathematical Representation

• Snapshot graph: Displacement vs. position at a fixed time.

• History graph: Displacement of a fixed point vs. time.

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

• Wave speed: $v = \lambda f$

Applications

• Sound and light waves are examples of traveling waves.

• Ultrasound imaging uses high-frequency sound waves for medical diagnostics.

• Electromagnetic waves enable communication and imaging technologies.

Additional info: All equations and table entries are standard in introductory physics and inferred from context.