Unit 1 Formulas and Mathy Stuff - Ultrasound Physics

Key Concepts

• Formulas in Ultrasound Physics: Understanding and manipulating formulas is crucial.
• Variable Relationships: Formulas often have one variable to be isolated.
• Algebraic Manipulation: Keep relationships between variables constant.

Example Formula Breakdown

• Formula: a = b / c
• Transposing Formula: Solve for different variables.
  • To solve for b: Multiply both sides by c a*c = b
  • To solve for c: Divide both sides by a c = b / a

Practice Problem Example

• Wavelength Formula: λ = c / f
  • c is propagation speed, f is frequency
  • Solve for c: c = λ * f
  • Solve for f: f = c / λ*

Types of Relationships Among Variables

1. Unrelated: No association.
2. Related: Connected, but without specified relationship.
3. Directly Related/Proportional: Both increase or decrease together.
4. Inversely Related/Proportional: One increases while the other decreases.
5. Reciprocal: Multiplying two factors equals 1. (a * b = 1)*

Important Relationships in Ultrasound Physics

• Inverse Relationships: Frequency ↑ → Period ↓
• Direct Relationships: Power ↑ → Intensity ↑
• Shorthand Notation: Use arrows for increase (↑) / decrease (↓)

Identifying Relationships in Formulas

• Multiplicative Setups: a * b = c
  • Factors (a, b) are directly related to product (c)
• Division Setups: a = b / c
  • Numerator (b) is directly related to quotient (a)
  • Denominator (c) is inversely related to quotient (a)
• Example:
  • Increased b → Increased a
  • Increased c → Decreased a*

Practical Examples for Relationships

• Nyquist Limit: Nyquist Limit = 1/2 * PRF
  • PRF ↑ → Nyquist Limit ↑ (Direct Relationship)
  • PRF ↓ → Nyquist Limit ↓ (Direct Relationship)
• Intensity Formula: Intensity = Power / Area²
  • Power ↑ → Intensity ↑ (Direct Relationship)
  • Area ↑ → Intensity ↓ (Inverse Relationship)*

Study Tips and Practice

• Create a formula sheet with definitions and units.
• Practice manipulating formulas and recognizing their relationships.

Basic Mathematical Skills

• Operations: Addition, subtraction, multiplication, division.
• Calculator Usage: Understand without a calculator.

Converting Units

• Metric System (Base 10): Meter, Liter, Gram, etc.
• Prefixes and Values:
  • Large: Kilo (10³), Mega (10⁶)
  • Small: Milli (10⁻³), Micro (10⁻⁶)
• Complementary Units: Use corresponding units for related measures (e.g., MHz and μs).

Key Conversion Tips

1. Write out the metric staircase.
2. Large to small units: Move decimal right.
3. Small to large units: Move decimal left.

Example Problems

1. Convert 3534 hL to mL: Move decimal 5 places right.
2. Convert 1230 ms to ks: Move decimal 6 places left.

Scientific Notation

• Positive Exponents: Large numbers (e.g., 10⁹ for billion).
• Negative Exponents: Small numbers (e.g., 10⁻⁹ for billionth).
• Conversion: Move decimal based on exponent direction.
• Special Cases: 10⁰ = 1.

Practice with Scientific Notation

1. Convert to notation: Large number - move decimal to the left.
2. Expand notation: Small number - move decimal to the right.

Powers of Numbers

• Positive Exponents: Multiply base.
• Negative Exponents: Divide by base exponent.

Fractions to Decimals

• Use long division for conversion.
• Numerator inside division symbol and denominator outside.

Practice Examples

1. Convert fraction: ⅙ = 0.166 repeating

Reciprocals in Formulas

• Definition: When two factors multiplied equal 1.
• Physics Example: Frequency × Period = 1

Understanding the Axes

• Axes in Graphs: x (horizontal), y (vertical), z (depth/3D).
• Used for mapping data in ultrasound images.

End of Unit

• Review and practice using workbook questions and quizzes.