Decibels, Attenuation, Impedance, Reflection, and Refraction

Physics: Decibels & Attenuation - Chapter 6

Interaction of Sound and Media

Logarithms

• Definition: An alternative method to rank numbers, especially when quantities vary over a large range.

• Base 10 Logarithm: Represents the number of times that $10$ has to be multiplied together to create the original number.

  • Example: The log of $100$ is $2$ (since $10 imes 10 = 100$).

  • Example: The log of $10,000$ is $4$ (just count the zeros: $10 imes 10 imes 10 imes 10 = 10,000$).

  • Example: The log of $1,000,000$ is $6$.

• Relationship: A logarithm may increase by $1$, but its corresponding value increases $10$-fold.

• Real-world Application: The Richter scale for earthquake severity is logarithmic.

  • Earthquake Severity Richter Magnitudes and Effects:

    • < 3.5: Generally not felt, but recorded.

    • $3.5-5.4$: Often felt, but rarely causes damage.

    • < 6.0: At most slight damage to well-designed buildings. Can cause major damage to poorly constructed buildings over small regions.

    • $6.1-6.9$: Can be destructive in areas up to about $100$ kilometers across where people live.

    • $7.0-7.9$: Major earthquake. Can cause serious damage over larger areas (e.g., Haiti $7.0$).

    • $8$ or >: Great earthquake. Can cause serious damage in areas several hundred kilometers across.

  • Richter Scale Unit Interpretation: Each unit increase on the Richter scale represents a $10$-fold increase in shaking amplitude.

    • A $5$ to $6$ magnitude increase means the shaking amplitude is $10$ times greater.

    • A $5$ to $7$ magnitude increase means the shaking amplitude is $100$ times greater (e.g., $10 imes 10$).

Attenuation: Introduction

• Definition: As sound propagates through a medium, it loses magnitude, a process called attenuation.

• Decibels (dB): Used to report the change in signal strength or amplitude.

Decibel Notation

• Description: Decibels (dB) are a logarithmic scale.

• Purpose: Used to describe the difference between two acoustic signals.

• Nature: It is a relative scale, not an absolute measurement.

• Calculation: Represents the ratio of final to initial intensity levels.

• Requirement: Two intensities are needed to calculate a decibel change.

• Units: A ratio of two intensities, powers, or amplitudes is typically expressed in dB.

Positive Decibels (Gain)

• Meaning: A positive dB value indicates that the final intensity exceeds the original (signal is getting bigger).

• Key Relationships for Intensity:

  • $+3$ dB: Means the intensity is $2$ times bigger (doubles).

    • Example: A sound wave with an initial intensity of $2 ext{ mW/cm}^2$ undergoes a $+3$ dB change. The final intensity is $2 imes 2 = 4 ext{ mW/cm}^2$.

  • $+6$ dB: Means the intensity is $4$ times bigger ($2 imes 2$).

    • Example: A $+6$ dB change to a $2 ext{ mW/cm}^2$ wave results in $2 imes 4 = 8 ext{ mW/cm}^2$.

  • $+9$ dB: Means the intensity is $8$ times bigger ($2 imes 2 imes 2$).

    • Example: A $+9$ dB change to a $2 ext{ mW/cm}^2$ wave results in $2 imes 8 = 16 ext{ mW/cm}^2$.

  • $+10$ dB: Means the intensity is $10$ times bigger.

    • Example: A sound beam with $5 ext{ mW/cm}^2$ undergoing a $+10$ dB change will have a final intensity of $5 imes 10 = 50 ext{ mW/cm}^2$.

  • $+20$ dB: Means the intensity is $100$ times bigger ($10 imes 10$).

    • Example: A $35 ext{ mW/cm}^2$ sound wave experiencing a $+20$ dB change will have a final intensity of $35 imes 100 = 3,500 ext{ mW/cm}^2$.

• Important: dB values do not measure absolute numbers; they represent a comparison or ratio between an initial and a final intensity.

Negative Decibels (Attenuation/Loss)

• Meaning: A negative dB value indicates that the intensity is decreasing (signal is getting smaller or attenuating).

• Key Relationships for Intensity:

  • $-3$ dB: Means the intensity is $1/2$ of the original value.

  • $-6$ dB: Means the intensity is $1/4$ of the original value ($1/2 imes 1/2$).

  • $-9$ dB: Means the intensity is $1/8$ of the original value ($1/2 imes 1/2 imes 1/2$).

  • $-10$ dB: Means the intensity is $1/10$ of the original value.

    • Example: A $100 ext{ mW/cm}^2$ sound wave undergoing a $-10$ dB change will have a final intensity of $100 imes (1/10) = 10 ext{ mW/cm}^2$.

• General Rule: If the final intensity is greater than the initial intensity, the gain in dB is positive ($+$). If it's smaller, it's negative ($-$).

Attenuation: Detailed

• Definition: The decrease in intensity, power, and amplitude of a sound wave as it travels through a medium.

  • The further sound travels, the more it attenuates.

• Units: Attenuation is measured in decibels (dB), often reported as a negative value (e.g., $-3$ dB), though sometimes simply given as a magnitude (e.g., $3$ dB attenuation).

• Factors Affecting Attenuation in Soft Tissue:

  • Frequency ($f$): Greater frequency leads to greater attenuation.

    • Practical Implication: Lower frequency transducers are needed to image deep structures because they attenuate less.

  • Path Length: Greater path length leads to greater attenuation.

• Components of Attenuation: Attenuation results from three primary processes:

  1. Absorption

  2. Reflection

  3. Scattering

1. Absorption

• Definition: The conversion of sound energy into heat.

• Significance: In soft tissue, absorption accounts for approximately $80$% of total attenuation.

• Material Properties: Bone is a particularly strong absorber of ultrasound energy.

2. Reflection

• Definition: Sound changes direction at an interface (boundary) between two tissues or media and returns to the transducer.

• Importance: Reflection is the fundamental principle behind creating an ultrasound image.

• Effect on Wave: It weakens the portion of the sound wave that continues to propagate forward.

Specular Reflection

• Description: Reflections that occur off smooth, large reflectors (relative to the wavelength).

• Characteristics: Act like a mirror and are highly angle-dependent.

  • If sound waves strike a specular reflector at an angle other than $90^ ext{o}$ (normal incidence), the reflections do not return to the transducer.

• Examples: Diaphragm, capsules of organs (e.g., liver capsule), pleura, wall of the aorta.

Diffuse Reflection (Backscatter)

• Description: Reflections that occur off an irregular or rough surface.

• Characteristics: The reflected sound will not be organized and will scatter in more than one direction.

• Disadvantage: Typically produces lower strength reflections compared to specular reflection.

• Advantage: Can produce reflections even when the sound beam strikes a surface at a suboptimal angle (i.e., not $90^ ext{o}$), which is beneficial for imaging complex anatomy.

3. Scattering

• Definition: Occurs when a boundary between two media is uneven or its size is equal to or less than the wavelength of the sound (e.g., reflector size approx $0.15 - 0.8$ mm).

• Mechanism: Sound is reflected in many directions.

• Material Properties: Lung tissue scatters sound significantly due to its many air-filled alveoli.

• Frequency Relationship: High frequency sound waves scatter more than low frequency waves.

Rayleigh Scattering

• Description: A specific type of scattering where the reflector is much smaller than the wavelength of the sound.

• Mechanism: Ultrasonic energy is uniformly diverted in all directions, similar to how a pebble dropped in a pond creates ripples in all directions.

• Example: Red blood cells (RBCs) produce Rayleigh scattering (e.g., RBCs are approximately $8$ microns, while typical ultrasound wavelengths are much larger, e.g., $800$ microns).

• Frequency Relationship: There is a direct and strong relationship between frequency and Rayleigh scattering.

  • Rayleigh scattering is proportional to the frequency to the fourth power ($ext{Rayleigh scattering} imes ext{frequency}^4$).

  • Implication: If you double the frequency, Rayleigh scattering increases by a factor of $2^4 = 16$.

Attenuation Coefficient

• Definition: The amount of attenuation that occurs per centimeter of tissue traversed by the sound wave.

• Units: dB/cm.

• Determinants of Total Attenuation: Total attenuation depends on:

  • The specific tissue through which the sound travels.

  • The distance (path length) the sound beam travels.

  • The frequency ($f$) of the sound wave.

• Formula (for Soft Tissue):
  $ext{Attenuation Coefficient (dB/cm)} = rac{ ext{frequency (MHz)}}{2}$

• Interpretation: This formula gives the number of dB of attenuation that occurs when sound travels $1$ cm in soft tissue.

• Property: The attenuation coefficient value remains constant for a given medium and frequency; it does not change as the sound travels.

  • Example: For a $4$ MHz transducer in soft tissue:
    $ext{Attenuation Coefficient} = rac{4 ext{ MHz}}{2} = 2 ext{ dB/cm}$

  • This shows that higher frequencies lead to higher attenuation coefficients, illustrating the direct relationship between frequency and attenuation.

Total Attenuation

• Formula to Calculate Total Attenuation:
  $ext{Total Attenuation (dB)} = ext{Path length (cm)} imes ext{Attenuation Coefficient (dB/cm)}$

• Example: If the path length is $5$ cm and the frequency is $10$ MHz (in soft tissue):

  1. Calculate Attenuation Coefficient: $ext{Atten. Coeff.} = rac{10 ext{ MHz}}{2} = 5 ext{ dB/cm}$

  2. Calculate Total Attenuation: $ext{Total Attenuation} = 5 ext{ cm} imes 5 ext{ dB/cm} = 25 ext{ dB}$

Attenuation of Ultrasound in Various Media

• Ranking (from Highest to Lowest Attenuation Rate):

  1. Air (Highest rate)

  2. Bone & Lung (High rate)

  3. Soft tissue (Intermediate rate)

  4. Fluids (Low rate)

  5. Water (Lowest attenuation)

• Analogy: A sponge would cause greater attenuation than Jell-O.

Half Value Layer Thickness

• Definition: The thickness of tissue required to reduce the sound beam intensity by one half (i.e., by $-3$ dB).

• Units: Any unit of distance, typically centimeters (cm).

• Factors Influencing Half Value Layer Thickness:

  1. Medium: Different media have different attenuation rates (e.g., Air vs. Water).

  2. Attenuation Rate: A higher attenuation rate results in a thinner half value layer (sound loses intensity faster).

  3. Frequency: A higher frequency sound beam also results in a thinner half value layer (higher frequencies attenuate more rapidly).

     • Example: A $2$ MHz beam will have a thicker half value layer (e.g., $3$ cm) than a $6$ MHz beam (e.g., $1$ cm) or a $12$ MHz beam.

• Other Names: Penetration depth, Depth of penetration, Half-boundary layer.

• Formula:
  $ext{Half-value Layer Thickness (cm)} = rac{3 ext{ dB}}{ ext{Attenuation Coefficient (dB/cm)}}$

• Example: For a $6$ MHz sound wave in soft tissue:

  1. Attenuation Coefficient: $ext{Atten. Coeff.} = rac{6 ext{ MHz}}{2} = 3 ext{ dB/cm}$

  2. Half-value Layer Thickness: $ext{HLT} = rac{3 ext{ dB}}{3 ext{ dB/cm}} = 1 ext{ cm}$

• Example: For a $10$ MHz sound wave in soft tissue:

  1. Attenuation Coefficient: $ext{Atten. Coeff.} = rac{10 ext{ MHz}}{2} = 5 ext{ dB/cm}$

  2. Half-value Layer Thickness: $ext{HLT} = rac{3 ext{ dB}}{5 ext{ dB/cm}} = 0.6 ext{ cm}$

• Relationship to Penetration: As frequency decreases, the depth of penetration (half value layer thickness) increases because attenuation decreases.

Impedance, Reflection & Transmission

Impedance

• Definition: The acoustic resistance to sound traveling through a medium.

• Units: Rayls, represented by the symbol $Z$.

• Measurement: Impedance is calculated, not directly measured.

• Characteristic: It is a characteristic solely of the medium; it does not change with the sound wave's properties.

• Importance: Plays a critical role in the physics of reflection.

• Formula:
  $ext{Impedance (Z) = Density (kg/m}^3) imes ext{Propagation Speed (m/sec)}$

• Relationships: High impedance (Z) corresponds to increased density and/or fast propagation speed.

• Typical Values: In biological tissues, typical impedance values range from $1.25$ to $1.75$ Mrayls.

• Example: If media A and media B have the same propagation speed, but media A has $10$% greater density, then media A will have $10$% greater impedance than media B.

Angle of Incidence

• Principle: A sound pulse strikes many tissue interfaces as it courses through soft tissue.

• Effect: The angle at which the sound beam strikes an interface significantly affects the behavior of the pulse (how much is reflected, transmitted, or refracted).

• Normal Incidence: Also known as right angle, perpendicular, orthogonal, or ninety degrees ($90^ ext{o}$).

• Oblique Incidence: Any angle other than $90^ ext{o}$. This includes acute angles (< 90^ ext{o}) and obtuse angles (> 90^ ext{o}).

Intensities and Coefficients

• Incident Intensity: The intensity of the sound beam at the instant prior to striking the boundary between two media. Units are Watts/cm$^2$.

• Reflected Intensity: The intensity of the beam after striking a boundary, changing direction, and returning towards the source.

• Transmitted Intensity: The intensity of the beam after striking a boundary and continuing onward through the second medium.

• Law of Conservation of Energy: Incident Intensity = Reflected Intensity + Transmitted Intensity.

Intensity Reflection Coefficient (IRC)

• Definition: The percentage of ultrasound intensity that is reflected when a sound beam passes from one medium to another.

Intensity Transmission Coefficient (ITC)

• Definition: The percentage of ultrasound intensity that continues onward through an interface or boundary between two media.

• Clinical Significance: In clinical imaging, $99$% or more of a sound wave's intensity is typically transmitted at most soft tissue boundaries.

• Exception: The ITC is smaller (and therefore IRC is larger) when sound strikes a boundary between bone and soft tissue interface, leading to more reflections.

• Relationship: ext{IRC} + ext{ITC} = 100 ext{%}.

  • Example: If a sound wave with an intensity of $50 ext{ W/cm}^2$ strikes a boundary and $25 ext{ W/cm}^2$ reflects, then $25 ext{ W/cm}^2$ was transmitted. The ITC would be (25/50) imes 100 ext{%} = 50 ext{%}.

  • Example: If a sound wave with an intensity of $75 ext{ W/cm}^2$ strikes a boundary and $25 ext{ W/cm}^2$ is transmitted, then the reflected intensity is $75 - 25 = 50 ext{ W/cm}^2$.

Reflection with Normal Incidence

• Condition for Reflection: Reflection occurs only if the acoustic impedance ($Z$) is different between the two media.

• No Reflection: If the impedances of the two media are identical, $100$% of the intensity is transmitted, and no reflection occurs.

• Magnitude of Reflection: The greater the difference in impedance between the two media, the greater the amount of reflection.

Reflection & Transmission with Oblique Incidence

• Complexity: There are no simple rules for predicting reflection and transmission with oblique incidence.

• Impedance Independence: Reflection may occur even if the media impedances are identical.

• Angle of Reflection: When reflection does occur with oblique incidence, the incident angle is always equal to the reflection angle.

• Conservation of Energy: The law of conservation still applies: Incident intensity = Reflected intensity + Transmitted intensity (or 100 ext{%} = ext{reflection coefficient} + ext{transmission coefficient}).

Refraction

• Definition: The change in direction (deflection) of a transmitted sound wave when it crosses an interface.

• Conditions for Refraction: Refraction occurs only under two specific conditions:

  1. The sound must strike the boundary with oblique incidence (i.e., not normal incidence).

  2. There must be different propagation speeds between the two media.

• Direction of Refraction:

  • If the propagation speed of medium 2 is less than the propagation speed of medium 1, the transmission angle will be less than the incident angle (the beam bends towards the normal).

  • If the propagation speed of medium 2 is greater than the propagation speed of medium 1, the transmission angle will be greater than the incident angle (the beam bends away from the normal).

• Snell's Law: This law mathematically defines the physics of refraction. Look at Notes

  • Interpretation: Snell's Law states that the ratio of the sine of the transmission angle to the sine of the incident angle is equal to the ratio of the propagation speed in the second medium to the propagation speed in the first medium. This relationship describes how much a sound wave will bend (refract) when it passes from one medium to another with a different speed, provided it strikes the boundary obliquely.