Harmonic and Circular Motion

Simple Harmonic Motion

• An object's motion around an equilibrium position.
• A force proportional to the displacement causes the motion.
• The force acts in the opposite direction to the displacement.
• Formula for the force is F = -kx, where:
  • F is the force.
  • k is the spring constant.
  • x is the displacement.

Circular Movement

• Angular velocity, ω = Δα/Δt, where α is the angle and t is time.
• Linear velocity is denoted as v.
• Relationship between linear and angular velocity: v = ωR, where R is the radius.

Periodic Motion Description

• Period (T) is the time needed for one full cycle.
• Frequency (f) is the number of cycles per unit time, measured in Hertz (Hz), where 1 Hz = 1/s.
• Relationship between frequency and period: f = 1/T.
• Amplitude (A) is the maximum displacement from equilibrium.

Characteristics of Harmonic Oscillation

• Displacement x(t) is equal to x(t + T).
• Maximum displacement (xmax) is equal to A (amplitude).
• Displacement as a function of time: x = A cos(ωt + φ), where:
  • A is the amplitude.
  • ω is angular frequency.
  • φ is the initial phase.
• Angular frequency: ω = 2π/T.

Velocity and Acceleration

• Velocity (V) is the change in displacement (Δx) over time (Δt): V = Δx/Δt = -Aωsin(ωt)
• Maximum velocity: Vmax = Aω
• Acceleration (a) is the change in velocity (ΔV) over time (Δt): a = ΔV/Δt = -Aω²cos(ωt)
• Maximum acceleration: amax = -Aω²

Total Energy

• Total energy (ET) is the sum of kinetic energy (EK) and potential energy (EP): ET = EK + EP = constant.

Mechanical Waves Overview

• A mechanical wave is a displacement of oscillations in space.
• These waves propagate only through elastic media.
• In an elastic medium, displaced particles experience a restoring force proportional to the displacement, bringing them back to their original positions.
• Mechanical waves propagate energy (information) but do not transport mass.

Wave Description

• Wavelength (λ) is the distance between two successive points with the same phase.
• Velocity of propagation (v) is the speed at which the wave moves through the medium.
• Wave velocity is expressed as: v = λf.

Types of Waves

• Transverse waves: Oscillations are perpendicular to the direction of wave propagation.
• Longitudinal waves: Oscillations are parallel to the direction of wave propagation.
• Flat waves.
• Spherical waves.

Mechanical Wave Velocity

• The velocity depends on the properties of the surrounding medium.
• Velocity changes at the boundary between two media with different acoustic impedances.
• Solids' velocity: v = √(E/ρ), where E is Young's modulus and ρ is density.
• Liquids' velocity: v = √(K/ρ), where K is the bulk modulus.
• Gases' velocity: v = √(κp/ρ), where κ is the bulk modulus and p is gas pressure.

Wave Behavior

• Reflection: The angle of incidence (α) equals the angle of reflection (α').
• Refraction: sin α / sin β = v1 / v2 = n, where n is the refraction index.

Acoustic Pressure

• Acoustic pressure (p) is the difference between atmospheric pressure (P₀) and instantaneous pressure (P) when sound is present: p = P - P₀.

Intensity of Wave

• Formula: I = E/St (measured in J/m²s or W/m²), where E is energy, S is the surface perpendicular to wave propagation, and t is time.
• Intensity depends on the amplitude of pressure changes.
• Intensity: I = (1/2) * (p² / ρv), where p is pressure, v is wave velocity, and ρ is density.

Acoustic Wave Energy

• Acoustic wave energy is partially absorbed by the medium.
• Sound wave is attenuated.
• Intensity with depth: I(x) = I₀e^(-μx), where:
  • I₀ is the initial sound intensity.
  • x is the thickness of the layer.
  • μ is the attenuation coefficient.

Sound Wave Properties

• Speed of sound is constant in a homogenous medium.
• Speed: v = √(B/ρ) where B is the elastic modulus, ρ is density.
• Acoustic impedance (z): z = vρ

Acoustic Impedance and Transmission

• Ir/Ii = (z1 - z2)² / (z1 + z2)², where Ir is an intensity of reflection, and Ii is the intensity of incident wave.
• It / Ii = 1 - Ir / Ii = 4z1z2 / (z1 + z2)².
• The smaller the difference in acoustic impedances between media, the better the wave transmission.

Sound Types

• Infrasound: Below 20 Hz.
• Audible sound: 20 Hz to 20 kHz.
• Ultrasound: Above 20 kHz.

Ultrasound Sources

• Natural sources: Bats and dolphins.
• Artificial sources: Oscillating surfaces of solids.

Piezoelectric Effect

• Natural and artificial sources produce ultrasound.
• Converse piezoelectric effect (piezoelectric transducers) acts as the main source of ultrasounds used in medicine.

Transducer Fields

• Near field.
• Far field.
• L = D² / 4λ, where D is the transducer diameter.

Echoes

• Echoes can be used to define the distance X = (1/2)vt.

Ultrasonography

• It is defined as seeing using an echo.
• Echo return time helps determine the distance between the reflecting interface and the ultrasound probe.
• Echo intensity measured depends on damping by the medium and the degree of reflection at tissue boundaries, with acoustic impedance related to tissue density.
• Impulse technique involves signal emission and waiting for echo.
• Emission to recording time ratio is approximately 1:100.
• Gel is used to enhance skin contact (acoustic impedance similar to soft tissue), reducing reflections.

Body Tissue Characteristics Using Ultrasound

• Water: velocity 1496 m/s, acoustic impedance 1.49 g/cm²s.
• Fat: velocity 1476 m/s, acoustic impedance 1.37 g/cm²s
• Muscle: velocity 1568 m/s, acoustic impedance 1.66 g/cm²s
• Kidney: velocity 1560 m/s, acoustic impedance 1.62 g/cm²s
• Liver: velocity 1570 m/s, acoustic impedance 1.66 g/cm²s
• Bone: velocity 3350 m/s, acoustic impedance 6.2 g/cm²s
• Air: velocity 331 m/s, acoustic impedance 0.0004 g/cm²s

Ultrasound Modes

• A-mode: One beam; amplitude depends on echo intensity.
• B-mode: One or many beams; brightness of a pixel depends on echo intensity.
• M-mode: One beam; image moves in time.

Correction Types

• Attenuation: Accounts for progressive ultrasound weakening in deeper structures.
• Beam geometry: Corrects echo return times in cone-shaped beams.

Ultrasound Probes

• Linear probes: Many transducers in a line = rectangular image.
• Rotational probes: Rotating transducers = cone-shaped image.

Resolution of Waves

• Axial resolution: resolution of the wave along the beam.
• Lateral resolution: resolution of the wave perpendicular to the beam.
• Axial resolution is always better than lateral.

Image Distortions

• Acoustic shadow: reduced echo intensity behind a highly reflective or absorptive structure.
• Reverberation: multiple reflections between two highly reflective surfaces appear as repeating echoes.
• Mirror-image artifact: strong reflective surface causes a duplicate image to appear on the opposite side.

Doppler Effect and Measurement

• Formula: f = f₀ (c ± v) / (c ± u)
  • f is the apparent frequency.
  • f₀ is the frequency of the source.
  • c is the wave velocity.
  • v is the receiver's speed.
  • u is the source's speed.
• In frequency changes; when there are increased distance the reduced frequency is observed, when decreased frequencies increase within observed wave.
• Measurement of moving object speed: v = (cΔf) / (2f₀ cos φ).
  • Δf is the frequency change
• Blood flow measurement: Due to impedance differences between blood cells and plasma, blood flow can be measures.
• Doppler angiography:
  • Continuous wave method.
  • Pulsed wave method.
• Spectral method (spectral Doppler):
  • Positive displacement = flows toward the probe; is negative from the probe.
• Colour-coding (colour Doppler):
  • Red = Towards probe.
  • Blue = From probe.
• Aliasing can occur within periodic methods.

Enhancing Ultra Sound waves

• Contrast-enhanced ultrasonography (CEUS): contrast media such as micro bubbles, applied intravenously, enhance USG signal.

Ultrasound Interaction with Matter

• Thermal: Predominant in high absorption coefficient tissues; temperature typically increases by 2-3 °C.
• Non-thermal (mechanical): Includes cavitation and stress.
  • Cavitation: Rapid pressure leads to small gas-filled cavities that can collapse and create shock waves.

Ultrasound Application

• Intensity is the main factor for diagnostics and possible techniques.
• Diagnostic uses – impulse techniques.
• Physiotherapy: Tissue warming, micro-massage, and phonophoresis.
• Surgery: tissue cutting.
• Otolaryngology: Treatment for cases of chronic rhinosinusitis.
• Ultrasound sterilization.
• Phacoemulsification: Process of treating cateracts
• Lithotripsy: A non-invasive treatment for kidney and biliary stones.
• Lithoplasty: A non-invasive procedure where the body's vessels deposits, and is cleaned of calcium.
• Stomatology: Removing dental formation with cleaning.
• Sonodynamic therapy of cancer (SDT): using cytotoxic substance with ultrasound use, where there the reactive generation of such species creates tissue volumes.
• Cosmetology: peeling, liposuction, skin correction.