16.1 Hooke’s Law: Stress and Strain Revisited

• An oscillation is a back and forth motion of an object between two points of deformation.
• An oscillation may create a wave, which is a disturbance that propagates from where it was created.
• The simplest type of oscillations and waves are related to systems that can be described by Hooke’s law:
  $$F=-\text{kx},$$

  where $F$ is the restoring force, $x$ is the displacement from equilibrium or deformation, and $k$ is the force constant of the system.

• Elastic potential energy ${\text{PE}}_{\text{el}}$ stored in the deformation of a system that can be described by Hooke’s law is given by
  $${\text{PE}}_{\text{el}}=(1/2){\mathit{\text{kx}}}^2.$$

16.2 Period and Frequency in Oscillations

• Periodic motion is a repetitious oscillation.
• The time for one oscillation is the period $\mathrm{T.}$
• The number of oscillations per unit time is the frequency $\mathrm{f.}$
• These quantities are related by
  $$f=\frac{1}{T}.$$

16.3 Simple Harmonic Motion: A Special Periodic Motion

• Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke’s law. Such a system is also called a simple harmonic oscillator.
• Maximum displacement is the amplitude $X$. The period $T$ and frequency $f$ of a simple harmonic oscillator are given by

  $T=\mathrm{2\pi }\sqrt{\frac{m}{k}}$ and $f=\frac{1}{\mathrm{2\pi }}\sqrt{\frac{k}{m}}$, where $m$ is the mass of the system.

• Displacement in simple harmonic motion as a function of time is given by $x(t)=X\text{cos}\frac{\mathrm{2\pi }t}{T}.$
• The velocity is given by $v(t)=-v_{\text{max}}\text{sin}\frac{\mathrm{2\pi }\text{t}}{T}$, where $v_{\text{max}}=\sqrt{k/m}X$.
• The acceleration is found to be $a(t)=-\frac{\mathrm{kX}}{m}\text{cos}\frac{\mathrm{2\pi }t}{T}.$

16.4 The Simple Pendulum

• A mass $m$ suspended by a wire of length $L$ is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about $\text{15º}.$

  The period of a simple pendulum is

  $$T=\mathrm{2\pi }\sqrt{\frac{L}{g}},$$

  where $L$ is the length of the string and $g$ is the acceleration due to gravity.

16.5 Energy and the Simple Harmonic Oscillator

• Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant:
  $$\frac{1}{2}{\text{mv}}^2+\frac{1}{2}{\text{kx}}^2=\text{constant.}$$
• Maximum velocity depends on three factors: it is directly proportional to amplitude, it is greater for stiffer systems, and it is smaller for objects that have larger masses.
  $$v_{\text{max}}=\sqrt{\frac{k}{m}}X$$

16.6 Uniform Circular Motion and Simple Harmonic Motion

A projection of uniform circular motion undergoes simple harmonic oscillation.

16.7 Damped Harmonic Motion

• Damped harmonic oscillators have non-conservative forces that dissipate their energy.
• Critical damping returns the system to equilibrium as fast as possible without overshooting.
• An underdamped system will oscillate through the equilibrium position.
• An overdamped system moves more slowly toward equilibrium than one that is critically damped.

16.8 Forced Oscillations and Resonance

• A system’s natural frequency is the frequency at which the system will oscillate if not affected by driving or damping forces.
• A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
• The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.

16.9 Waves

• A wave is a disturbance that moves from the point of creation with a wave velocity $v_{\text{w}}$.
• A wave has a wavelength $\lambda$, which is the distance between adjacent identical parts of the wave.
• Wave velocity and wavelength are related to the wave’s frequency and period by $v_{\text{w}}=\frac{\lambda }{T}$ or $v_{\text{w}}=\mathrm{f\lambda }.$
• A transverse wave has a disturbance perpendicular to its direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.

16.10 Superposition and Interference

• Superposition is the combination of two waves at the same location.
• Constructive interference occurs when two identical waves are superimposed in phase.
• Destructive interference occurs when two identical waves are superimposed exactly out of phase.
• A standing wave is one in which two waves superimpose to produce a wave that varies in amplitude but does not propagate.
• Nodes are points of no motion in standing waves.
• An antinode is the location of maximum amplitude of a standing wave.
• Waves on a string are resonant standing waves with a fundamental frequency and can occur at higher multiples of the fundamental, called overtones or harmonics.
• Beats occur when waves of similar frequencies $f_1$ and $f_2$ are superimposed. The resulting amplitude oscillates with a beat frequency given by
  $$f_{\text{B}}=\mid f_1-f_2\mid .$$

16.11 Energy in Waves: Intensity

Intensity is defined to be the power per unit area:

$I=\frac{P}{A}$ and has units of ${\text{W/m}}^2$.