Unit 1: Formulas and Mathematical Concepts in Ultrasound Physics

Introduction

• Understanding formulas in ultrasound physics is crucial for grasping physical concepts.
• Focus on manipulating and breaking down formulas to understand relationships.

Simple Algebraic Principles

• Formulas usually have a variable (answer) on one side, everything else on the other side.
• Example: A = B / C
  • Given numerical values for A and C, solve for B using algebra (transpose the formula).
  • Keep relationships consistent between variables by performing the same operation on both sides.
• Steps to isolate a variable:
  1. Identify the operation affecting the variable (e.g., division by C).
  2. Perform the inverse operation (e.g., multiply by C) on both sides.
  3. Simplify (e.g., C * A = B * C / C becomes C * A = B).*

Practice Problems

• Rearrange formulas for wavelength, frequency, and propagation speed.
  • Formula: λ = C / f
  • Solve for C and f using similar steps as above.

Describing Relationships within Formulas

• Five Relationship Types:
  1. Unrelated
  2. Related
  3. Directly related/proportional (both factors increase or decrease together)
  4. Inversely related/proportional (one factor increases while the other decreases)
  5. Reciprocal (product equals one)
• Ultrasound Physics Examples:
  • Frequency and period are inversely related.
  • Power and intensity are directly related.

Practice: Recognizing Relationships

• Determine if scenarios are directly related, unrelated, or inversely related.
• Example: Time spent on Instagram and data usage are directly related.

Recognizing Relationships in Formulas

• Rule 1: Factors are directly related to the product.
• Rule 2:
  • Quotient is directly related to the numerator.
  • Quotient is inversely related to the denominator.

Conversion Practice

• Convert numbers between different metric units using the metric staircase.
• Understand the metric system and prefixes (e.g., kilo, mega, giga).

Scientific Notation and Powers of Ten

• Convert large and small numbers to/from scientific notation.
• Positive powers of ten increase the number; negative powers decrease it.

Unit Conversion

• Convert fractions (e.g., 1/4) to decimals (e.g., 0.25) using long division.
• Understand reciprocals in formulas (e.g., frequency x period = 1).

Understanding the Metric System

• Metric system is based on powers of ten.
• Memorize metric prefixes, values, and exponents for the test.
• Practice converting between units using the metric staircase method.

Basic Math Skills

• You need to add, subtract, multiply, divide, and convert units without a calculator.
• Focus on understanding relationships in formulas for exams.

Additional Mathematical Concepts

• Raising numbers to powers (e.g., 2^2 = 4).
• Powers of numbers and their rules (e.g., negative exponents).

Graphing: X, Y, and Z Axes

• Understand 2D and 3D graphing concepts.
• Ultrasound data often uses graphs for display (e.g., time vs. velocity).

• Final Note: Prepare a formula sheet, include units, relationships, and transposed formulas. Practice to understand and visualize parameter relationships.