Amplitude and Frequency

Amplitude is how far an oscillation moves from equilibrium. Frequency is how many complete cycles it makes each second. In short, amplitude tells you the size of the motion, and frequency tells you how often it repeats.

A wave can have a large amplitude and a low frequency, or a small amplitude and a high frequency. In an ideal linear model, changing one does not automatically change the other.

Amplitude Measures Maximum Displacement

Amplitude is measured from the equilibrium position to a crest or to a trough. If the displacement reaches $+3\ \mathrm{cm}$ at its highest point and $-3\ \mathrm{cm}$ at its lowest point, the amplitude is $3\ \mathrm{cm}$.

This is a common place to slip. The full top-to-bottom distance is the peak-to-peak value, which would be $6\ \mathrm{cm}$ in that example, not the amplitude.

Frequency Counts Cycles Per Second

Frequency tells you how often the motion repeats. Its SI unit is the hertz, where $1\ \mathrm{Hz} = 1$ cycle per second.

If a vibration completes $5$ full cycles in $1$ second, then its frequency is $5\ \mathrm{Hz}$. If you know the period $T$, which is the time for one cycle, then

$$f = \frac{1}{T}$$

So a shorter period means a higher frequency.

Worked Example From A Sine Wave

Consider a wave described by

$$y(t) = 4 \sin(10\pi t)$$

Assume $y$ is measured in centimeters and $t$ in seconds.

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

$$A = 4\ \mathrm{cm}$$

To find the frequency, compare the expression with the standard form

$$y(t) = A \sin(2\pi f t)$$

Here,

$$2\pi f = 10\pi$$

so

$$f = 5\ \mathrm{Hz}$$

This wave reaches a maximum displacement of $4\ \mathrm{cm}$ from equilibrium and completes $5$ full cycles every second.

The example shows the split clearly:

• amplitude answers "how far?"
• frequency answers "how often?"

Common Mistakes With Amplitude And Frequency

Mixing up amplitude with peak-to-peak distance

Amplitude is half of the full vertical range. If the motion goes from $-2$ to $+2$, the amplitude is $2$, not $4$.

Assuming a larger amplitude means a higher frequency

That is not generally true. 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 describes repetition rate. Wave speed describes how fast the disturbance travels through space. They are related in many wave models, but they are not the same quantity.

Where Amplitude And Frequency Matter

Amplitude and frequency show up 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 the motion repeats.

The exact physical effect depends on the system, so it is better to treat amplitude and frequency as general descriptors first and then add context.

Try A Similar Problem

Take

$$y(t) = 2 \cos(6\pi t)$$

Find the amplitude and frequency. Then change the $2$ to $7$ without changing the rest of the equation. You will see that the amplitude changes, but the frequency stays the same.

If you want a useful next step, compare this with simple harmonic motion to see how amplitude and frequency fit into a full oscillation model.