Math, Units & Decibels You Must Know

Why Math Fluency Comes Before Physics

SPI is a physics exam, and physics on this exam is expressed almost entirely through simple algebra: ratios, reciprocals, and one logarithmic scale, the decibel. You will not need calculus or trigonometry beyond reading a cosine value off a table, but you must be fast and error-free with unit conversions and reciprocal relationships, because nearly every later chapter in this guide — wavelength, pulse-echo timing, resolution, Doppler shift — builds directly on the skills covered in this section. Getting the math automatic now prevents it from becoming the bottleneck once you reach conceptually harder material.

Metric Prefixes and Scientific Notation

Ultrasound physics operates across an enormous range of scale: frequencies measured in millions of cycles per second, time intervals measured in millionths of a second, distances measured in thousandths of a meter. The metric prefixes below are the ones that appear constantly on SPI.

The two conversions you will use most are MHz to Hz (multiply or divide by 1,000,000) and µs to s (divide or multiply by 1,000,000). A frequency given in MHz and a time given in µs are already scaled so they cancel conveniently in most SPI formulas — that pairing of units is not a coincidence; it is why the field standardized on them.

Scientific notation is the safest way to avoid decimal-point errors when a problem mixes scales. For example, a frequency of 5 MHz is 5 × 10⁶ Hz, and a period of 0.2 µs is 2 × 10⁻⁷ s. If a calculation produces an answer with an implausible number of zeros, re-check your exponent before you re-check your algebra.

Reciprocal Relationships

A large share of SPI's formula questions are really just reciprocal pairs — two quantities that multiply together to equal 1 (or a constant), so that knowing one always gives you the other by simple division. You will meet these formally in later chapters, but the pattern to internalize now is:

• Period and frequency are reciprocals: period = 1/frequency, and frequency = 1/period.
• Pulse repetition period (PRP) and pulse repetition frequency (PRF) are reciprocals: PRP = 1/PRF.
• A reciprocal relationship is always an inverse relationship: as one quantity goes up, its reciprocal partner goes down by the same factor. Doubling a frequency halves its period; tripling a PRF cuts its PRP to one-third.

Recognizing "this is a reciprocal-pair question" instantly tells you whether to multiply or divide. Inverting a reciprocal relationship under time pressure is one of the most common SPI errors, so practice converting back and forth until it requires no conscious effort at all.

The Decibel (dB) Scale

Ultrasound intensities span an enormous range, from the transmitted pulse down to the faint returning echo, so instead of a linear scale, SPI and ultrasound instrumentation itself express relative intensity changes in decibels (dB), a logarithmic ratio. The defining formula is:

dB = 10 · log(I₂/I₁)

where I₁ and I₂ are two intensities being compared — for example, output intensity versus received echo intensity, or a gain adjustment's "before" and "after" values. Because the scale is logarithmic, decibels do not add the way linear percentages do: a few dB represents a large multiplicative change in intensity. The relationships below are fixed values you should memorize cold; they appear constantly in gain, TGC, dynamic range, and attenuation questions throughout this guide.

Notice the pattern: every +10 dB multiplies intensity by a power of ten (+10 dB = ×10, +20 dB = ×100), while every +3 dB roughly doubles intensity and every −3 dB roughly halves it. These two anchor points, 10 dB for an order of magnitude and 3 dB for doubling or halving, let you estimate almost any dB question on the exam without a calculator, and they combine directly: −6 dB is two −3 dB steps stacked together, so intensity is halved twice (½ × ½ = ¼), which is exactly the quarter-intensity value shown in the table above.

Hold onto both the reciprocal-relationship habit and the decibel table as you move into Chapter 2: propagation speed, wavelength, and pulse-echo timing all reuse these exact math patterns, just applied to new physical quantities. If a later chapter's formula question feels unfamiliar, it is worth returning here first — the underlying arithmetic pattern, whether reciprocal or logarithmic, is almost always one already introduced in this section, simply relabeled with new physics terms.