Ultrasound Physics Textbook Notes (Chapter 1)

Understanding Ultrasound Physics

Definitions

• Unrelated: Two items that are not associated.

  • Examples:
    • Hair color is unrelated to shoe size.
    • Weight is unrelated to day of birth.
    • Temperature is unrelated to the day of the week.

• Related (or Proportional): Two items that are associated or affiliated. The relationship between the items does not have to be specified.

  • Examples:
    • Weight is related to dieting.
    • Santa is related to Christmas.
    • Exam score is related to studying.
    • Dental health is related to flossing.

• Directly Related (or Directly Proportional): Two items associated such that when one item increases, the other increases.

  • The graph of two directly related items extends from lower left to upper right.

  • Examples:

    • Clothing size is directly proportional to one's weight.
    • Age is directly related to experience.
    • Skill is directly proportional to practice.
    • Quality of wine is directly related or directly proportional to age.

  • Visualization:

    • A graph is shown with age on the x-axis and height on the y-axis, illustrating a direct relationship where height increases with age.

• Inversely Related (or Inversely Proportional): Two items are associated such that when one item increases, the other decreases.

  • The graph of two inversely related items extends from upper left to lower right.

  • Examples:

    • Golf score is inversely related to skill.
    • A car's gas mileage is inversely proportional to its engine size.
    • Grades in school are inversely proportional to partying time.

  • Visualization:

    • A graph is shown with temperature on the x-axis and clothing on the y-axis, illustrating an inverse relationship where clothing decreases as temperature increases.

• Reciprocal Relationship: When two numbers with a reciprocal relationship are multiplied together, the result is one. This is a special form of inverse relationship.

  • Reciprocal numbers are inverse because when one increases, the other decreases.
  • Examples:
    • 2 and 1/2 are reciprocals.
    • 10 and 1/10 are reciprocals.
    • Period and frequency are reciprocals.

Units

• All numerical values must have corresponding units to avoid ambiguity.

  • Example: Asking "how old is Jenny?" requires a numerical response with units (e.g., 6 years).

• Units of length/distance/circumference: cm, feet

• Units of area: cm², ft²

• Units of volume: cm³, ft³

• Any fundamentally correct unit is acceptable.

• "Increase by a factor" means to multiply by that number. Increase by a factor of 6 means six times larger.

• "Decrease by a factor" means to divide by that number. Decrease by a factor of three means one third.

• A number followed by the word "percent" is unitless.

Unit Conversion

• It is important to know how to convert from one unit to another.
• Examples of unit conversion:
  • How many quarters are in 1 dollar?
  • How many days are in 1 month?
• When units change, the "total picture" does not change; only the manner of presentation changes.
  • 12 inches is the same as 1 foot, or 1/3 yard.
• Treat conversion units as fractions with a value of 1 and carry along the units.
• Examples:
  • Convert $5.00 into dimes (hint: $1 = 10 dimes).
  • Convert 12 inches into centimeters (hint: 2.54 cm = 1 inch).

Powers of Ten

• Scientific or engineering notation is a shorthand manner to represent very large or very small numbers.
• A number in scientific notation form with a positive exponent has a value greater than 10.
• A number in scientific notation form with a negative exponent has a value less than 1.
• A number in scientific notation form with an exponent of zero has a value between 1 and 10.
• To correctly calculate the number:
  1. Shift the decimal point so the resulting number is between one and ten.
  2. Multiply by the appropriate power of 10.
• Examples:
  • 1,000,000 = 1.0 \times 10^6
  • 0.000000124 = 1.24 \times 10^{-7}
  • 1742 = 1.742 \times 10^3

Metric System

• Table of Metric System - Powers of Ten:
  • 10^9: giga (G), billion
  • 10^6: mega (M), million
  • 10^3: kilo (k), thousand
  • 10^2: hecto (h), hundred
  • 10^1: deca (da), ten
  • 10^{-1}: deci (d), tenth
  • 10^{-2}: centi (c), hundredth
  • 10^{-3}: milli (m), thousandth
  • 10^{-6}: micro ($\mu$), millionth
  • 10^{-9}: nano (n), billionth

Complementary Metric Units

• Think of the pairs as belonging together.
• For example, if frequency is in megahertz, then period is in microseconds (millions of cycles and millionths of seconds).
• Table of Complementary Metric Units:
  • giga & nano (G & n): billions and billionths
  • mega & micro (M & $\mu$): millions and millionths
  • kilo & milli (k & m): thousands and thousandths
  • hecto & centi (h & c): hundreds and hundredths
  • deca & deci (da & d): tens and tenths

Graphs

• In diagnostic ultrasound, information is often displayed in graphical form.
• The two axes used with all graphs have special names:
  • The horizontal axis, or x-axis, runs side to side.
  • The vertical axis, or y-axis, runs up and down.
• Visualization:
  • A standard graph labeling diagram is shown with the x-axis labeled as horizontal and the y-axis labeled as vertical.