Unit 1: Formulas and Math Concepts in Ultrasound Physics

Importance of Formulas

• Understanding formulas is key to grasping the physical concepts of ultrasound.
• Involves manipulating formulas to understand relationships between variables.

Simple Formula Manipulation

• Formulas often structured as A = B / C.
• Example: If A = 10 and C = 2, B can be found using algebraic principles.
• To solve for B:
  • Multiply both sides by C: C * A = B * C / C.
  • C / C = 1, so C * A = B.
• To solve for C from C * A = B:
  • Divide both sides by A: C * A / A = B / A.
  • So, C = B / A.

Relationships Between Variables

• Direct Relationships: Increase or decrease together.
• Inverse Relationships: One increases while the other decreases.
• Reciprocal Relationships: Product of factors equals to 1.

Ultrasound Physics Examples

• Wavelength formula: λ = C / F.
  • C = propagation speed, F = frequency.
  • Rearrange to solve for C: C = λ * F.
  • Rearrange to solve for F: F = C / λ.
• Frequency and Period:
  • Frequency (F) increases, Period (P) decreases. (Inversely related)
• Power and Intensity:
  • Power (P) increases, Intensity (I) increases. (Directly related)

Key Relationships

• Direct relationship: Both variables increase or decrease together.
• Inverse relationship: One variable increases while the other decreases.
• Reciprocal relationship: Multiplying two factors results in 1.

Identifying Relationships in Formulas

1. Multiplication (Product):

• Factors are directly proportional to the product.
• Example: A * B = C.
  • If A increases, C increases (directly proportional).
  • If A decreases, C decreases.

2. Division (Quotient):

• Numerator is directly related to the quotient.
• Denominator is inversely related to the quotient.
• Example: A = B / C.
  • If B increases, A increases (directly proportional).
  • If C increases, A decreases (inversely proportional).

Solving Real-world Problems

• Practice in rearranging formulas and understanding variable relationships is crucial.
• Create a formula sheet for quick reference:
  • Write formulas.
  • Define each variable & units.
  • Include relationships and transpose for all variables.

Mathematical Basics in Physics

• Algebra: Fundamental for manipulating formulas.
• Factor Relationships: Focus on increasing and decreasing variables.
• Conversions: Practice converting units (e.g., millimeters to centimeters).
• Understanding Exponents: Positive exponents make numbers larger, negative make them smaller.

Scientific Notation and Exponents

• Large numbers: Use positive exponents (e.g., 10^9 for a billion).
• Small numbers: Use negative exponents (e.g., 10^-9 for a billionth).
• Zero exponent: Any number to the power of zero equals one.
• Powers of other numbers (e.g., 2^3 means 2 * 2 * 2 = 8).
• Converting between fractions and decimals using long division.

Graphs and Axes

• X-axis: Horizontal line (time, distance, etc).
• Y-axis: Vertical line (velocity, amplitude, etc).
• Z-axis: Adds depth for a 3D representation.

Final Tips

• Understand relationships within formulas for tests and real-world applications.
• Practice converting units and numbers to scientific notation.
• Keep a practical formula sheet for ultrasound physics.
• Perform practice problems to reinforce understanding.