7.2 Frequency, Period, Wavelength & Waveform Math

7.2 Frequency, Period, Wavelength & Waveform Math

Quick Answer: f = 1/T. λ = c/f with c ≈ 3 × 10⁸ m/s. Frequency and wavelength are inversely proportional. For sine waves: RMS = 0.707 × peak, peak = 1.414 × RMS, P-P = 2 × peak. Duty cycle = on-time / period (× 100 for percent).

Topic 3-B expects you to move fluently between time, frequency, wavelength, and amplitude measures. These conversions appear in receiver LO math, transmitter channels, antenna length estimates, and scope-to-meter comparisons.

Frequency and period

Frequency (f) is the number of cycles completed per second, measured in hertz (Hz).

Period (T) is the duration of one complete cycle, measured in seconds.

[ f = \frac{1}{T} \qquad T = \frac{1}{f} ]

Unit conversion ladder

Worked example: A square-wave clock has period 250 ns. Frequency?

[ f = \frac{1}{250 \times 10^{-9}} = 4 \times 10^6,\mathrm{Hz} = 4,\mathrm{MHz} ]

Worked example: An RF carrier is 2182 kHz (MF distress-related channel context). Period?

[ T = \frac{1}{2.182 \times 10^6} \approx 458,\mathrm{ns} ]

Convert kHz → Hz before reciprocating, or think “2.182 MHz → about 0.458 µs.”

Wavelength and the speed of light

In free space (and approximately in air):

[ \lambda = \frac{c}{f} \qquad f = \frac{c}{\lambda} \qquad c \approx 3.00 \times 10^8,\mathrm{m/s} ]

Frequency and wavelength are inversely proportional: double f → half λ; half f → double λ. They are not directly proportional and not unrelated.

Convenient RF forms:

[ \lambda_{\mathrm{m}} \approx \frac{300}{f_{\mathrm{MHz}}} \qquad \lambda_{\mathrm{ft}} \approx \frac{984}{f_{\mathrm{MHz}}} ]

Worked example — half-wave dipole estimate

Approximate free-space half-wave length at 150 MHz:

[ \lambda = \frac{300}{150} = 2,\mathrm{m} \Rightarrow \frac{\lambda}{2} = 1,\mathrm{m} ]

Element 3-style practice often asks this 1-meter half-wave result. Real antennas are slightly shorter because of end effects and velocity factor—but the pool math starts from λ = c/f.

Worked example — wavelength from frequency in Hz

At f = 10 MHz = 10⁷ Hz:

[ \lambda = \frac{3 \times 10^8}{10^7} = 30,\mathrm{m} ]

Same answer via 300/f(MHz) = 300/10 = 30 m.

Sine-wave amplitude relationships

For a pure sine wave with peak voltage (V_p):

RMS definition for exam day: the RMS AC value produces the same heating in a resistor as a DC voltage of the same numerical value. That is why power calculations on AC sine waves use RMS unless the stem explicitly says peak or PEP.

Household / reference numbers (continuity from Topic 3-A)

Worked example — scope vertical math

A sine wave spans 4.0 divisions peak-to-peak at 2 V/div:

1. (V_{pp} = 4.0 \times 2 = 8.0,\mathrm{V})
2. (V_p = 4.0,\mathrm{V})
3. (V_{\mathrm{RMS}} = 0.707 \times 4.0 \approx 2.83,\mathrm{V})

If you wrongly treat 8 V as RMS, any (P = V^2/R) calculation is off by a factor of about (8/2.83)² ≈ 8.

Worked example — RMS to peak RF voltage

An RF voltmeter (true-RMS, sine assumption) reads 70.7 V. Peak voltage?

[ V_p = 1.414 \times 70.7 \approx 100,\mathrm{V} ]

Peak-to-peak ≈ 200 V.

Average values and meter factors (sine context)

Element 3 keeps three amplitude ideas separate:

1. Peak — crest of the waveform.
2. RMS — effective heating value (what most AC meters are calibrated to display for sine waves).
3. Average — mean of the absolute or rectified waveform over a cycle (context-dependent wording).

Pool-style factor often tested: multiply a sine-wave AC voltmeter (RMS) reading by 0.9 to obtain the average value in that framing. Going the other way, peak from RMS is ×1.414, not ×0.9 or ×0.707.

Duty cycle basics

For a periodic pulse waveform:

[ \text{Duty cycle} = \frac{t_{\mathrm{on}}}{T} \quad (\text{or} \times 100% ) ]

If a rectangular pulse train switches between 0 V and V_p, the average voltage is approximately (V_p \times) duty cycle (as a fraction). RMS of a 0-to-Vp rectangular pulse train is (V_p \sqrt{D}) where D is the duty fraction—useful when estimating heating or supply loading under pulsed keying.

Worked example — duty cycle

A transmitter pulse is 2 ms wide and the pulse repetition period is 10 ms:

[ D = \frac{2}{10} = 0.20 = 20% ]

Average power for constant peak envelope power during the pulse would be about 0.20 × PEP if the “off” power is zero—tying this section to peak-envelope concepts in §7.3.

Putting frequency and waveform math together

A VHF channel at 156.8 MHz (marine Ch-16 vicinity):

1. Period: (T = 1/(1.568 \times 10^8) \approx 6.4,\mathrm{ns})
2. Wavelength: (\lambda \approx 300/156.8 \approx 1.91,\mathrm{m})
3. Quarter-wave vertical (free-space ideal): ≈ 0.48 m before velocity-factor correction

You do not need marine-channel memorization for pure math stems—but the same arithmetic powers antenna, LO, and filter questions throughout Element 3.

Exam-day checklist for §7.2

1. f ↔ T are reciprocals—watch ms vs µs vs ns.
2. λ = 300/f(MHz) meters free space; inverse relationship with f.
3. Sine: 0.707 peak→RMS, 1.414 RMS→peak, 2× peak→P-P.
4. RMS = heating equivalence, not “half of peak.”
5. Duty cycle = on-time/period; average of pulses scales with D.
6. Convert kHz/MHz/GHz before plugging into λ or T formulas.

With time-frequency-wavelength and amplitude conversions secure, you are ready for power formulas and the decibel language of RF gains and losses.

Instantaneous amplitude at an angle (degrees and radians)

Sub-topic 3-B-012 asks for the instantaneous value of a sine wave at a stated angle, and it states that angle in degrees in some items and radians in others. One formula covers both:

v = V_peak × sin θ

Radians are just a different unit for the same angle: π radians = 180°, so π/3 rad = 60° and π/2 rad = 90°. Convert to degrees first if that is easier, then take the sine.

Two habits prevent almost every wrong answer here:

1. Watch the sign. Angles between 180° and 360° are in the negative half-cycle, so 240° gives −4.3 V, not +4.3 V. The pool offers both.
2. Note the symmetry. sin 150° = sin 30° = 0.5, and sin 240° = −sin 60° = −0.866. The sine of an angle and of its supplement are equal; the sine of an angle and of that angle plus 180° are equal and opposite.

Contrast this with RMS, which is a single number describing the whole waveform (0.707 × peak for a sine), and with average (0.637 × peak). Instantaneous amplitude is a snapshot at one instant; RMS and average summarise the entire cycle. The pool asks all three in this topic, so read the stem carefully for the word instantaneous.