Amplitude and Frequency

Strike a tuning fork softly, then strike it hard. The pitch you hear does not change, but the loudness does. That everyday split is exactly the difference between frequency and amplitude: how often the motion repeats versus how big each swing is. In an ideal linear model, you can change one without touching the other.

The Two Formulas And Their Symbols

Two short relations carry most amplitude-and-frequency problems.

Amplitude $A$ is the maximum displacement from the equilibrium position to a crest or a trough. It is a distance, so it carries a length unit such as centimeters.

Frequency $f$ counts complete cycles per second. Its SI unit is the hertz, where $1\ \mathrm{Hz} = 1$ cycle per second. Frequency is tied to the period $T$, the time for one cycle, by

So a shorter period means a higher frequency.

Why $f = 1/T$ Holds

The link between frequency and period is just a counting statement. If one cycle takes time $T$, then in one second you fit $1/T$ of those cycles. That ratio is the frequency. So if a vibration completes $5$ full cycles in $1$ second, each cycle takes $1/5$ of a second, and the frequency is $5\ \mathrm{Hz}$. Amplitude does not enter this reasoning at all, which is the first hint that the two quantities are independent.

Worked Example: Reading A Sine Wave

Consider a wave described by

with $y$ in centimeters and $t$ in seconds.

In the standard form, the number in front of the sine function gives the amplitude, so

To find the frequency, compare with the standard form

Matching the time coefficient,

so

This wave reaches a maximum displacement of $4\ \mathrm{cm}$ from equilibrium and completes $5$ full cycles every second. The split is clean: amplitude answers "how far?" and frequency answers "how often?"

Practice And Check Your Answer

Take

Read off the amplitude, then match $6\pi$ against $2\pi f$ to get the frequency. You should find $A = 2$ and $f = 3\ \mathrm{Hz}$. Now change the leading $2$ to $7$ and recheck: the amplitude jumps to $7$, but the frequency is still $3\ \mathrm{Hz}$, because the time coefficient never moved. That single substitution is the fastest proof that amplitude and frequency are independent in this model.

Traps That Cost Points

Mixing up amplitude with peak-to-peak distance. Amplitude is half the full vertical range. If the motion goes from $-2$ to $+2$, the amplitude is $2$, not $4$. The full top-to-bottom value is the peak-to-peak distance.

Assuming a larger amplitude means a higher frequency. In an ideal linear system, you can change amplitude without changing frequency. Some real systems behave differently, but that depends on the system.

Counting half-cycles as full cycles. Frequency counts complete repeats. Going from the middle to the top and back to the middle is only half a cycle.

Confusing frequency with wave speed. Frequency is a repetition rate. Wave speed describes how fast the disturbance travels through space. They are related in many wave models but are not the same quantity.

Where The Two Quantities Show Up

Amplitude and frequency appear in sound, light, springs, circuits, and water waves. In sound, frequency is related to pitch, while a larger amplitude usually means a stronger signal and, in the same setup, a louder sound. In simple harmonic motion, amplitude sets the size of the oscillation and frequency sets how quickly it repeats. The exact physical effect depends on the system, so treat amplitude and frequency as general descriptors first, then add context.

For the full oscillation model, see how these fit together in simple harmonic motion.