3.2.1.4. Frequency

Table of Contents

What Frequency Means

Frequency tells us how often a repeating motion or pattern occurs. In waves, it describes how many complete cycles pass a point in one second. If a wave repeats itself many times each second, it has a high frequency. If it repeats more slowly, it has a low frequency.

A complete cycle means one full repetition of the wave pattern. For a transverse wave, this could be from one crest to the next crest. For a longitudinal wave, it could be from one compression to the next compression.

The SI unit of frequency is the hertz, written as $\text{Hz}$. One hertz means one cycle per second.

Frequency is defined as the number of complete cycles per unit time.
$$
f = \frac{N}{t}
$$
where $f$ is frequency, $N$ is the number of cycles, and $t$ is the time.
The SI unit is
$$
1\ \text{Hz} = 1\ \text{s}^{-1}
$$

Frequency and Period

Frequency is closely related to the period. The period is the time required for one complete cycle. If one cycle takes a long time, the frequency is small. If one cycle takes a short time, the frequency is large.

The two quantities are reciprocals of each other.

The relationship between frequency $f$ and period $T$ is
$$
f = \frac{1}{T}
\qquad \text{and} \qquad
T = \frac{1}{f}
$$

If a wave has a period of $0.5\ \text{s}$, then its frequency is

$$
f = \frac{1}{0.5} = 2\ \text{Hz}
$$

So the wave completes 2 cycles every second.

Physical Meaning of Frequency

Frequency tells us how rapidly a source vibrates or oscillates. A vibrating guitar string, a speaker cone, or the surface of water can all create waves. If the source vibrates faster, the wave frequency is higher.

In many cases, the frequency of the wave is determined by the source that produces it. For example, if a string is shaken 10 times each second, the wave frequency is $10\ \text{Hz}$.

For sound, frequency is especially important because it is related to pitch. Higher frequency sounds are heard as higher pitch, and lower frequency sounds are heard as lower pitch.

Counting Frequency

A simple way to find frequency is to count how many cycles occur during a measured time interval.

Suppose 15 complete wave cycles pass a point in 3 seconds. Then

$$
f = \frac{15}{3} = 5\ \text{Hz}
$$

This means the wave repeats 5 times each second.

The period is then

$$
T = \frac{1}{5} = 0.2\ \text{s}
$$

Frequency in Different Kinds of Waves

Frequency applies to all periodic waves, whether they are mechanical, sound, or water waves. The meaning stays the same, even though the wave itself may look different.

Type of wave	One cycle can be measured from	Meaning of frequency
Water wave	crest to crest	number of crests passing each second
Wave on a string	crest to crest	number of oscillations each second
Sound wave	compression to compression	number of pressure cycles each second

What Changes and What Stays the Same

When a wave moves from one place to another in the same medium, its frequency stays fixed by the source. Other properties may change in different situations, but the frequency is tied to how the source oscillates.

This is an important idea because students often confuse frequency with how large the wave looks. A wave can have a large amplitude or a small amplitude and still have the same frequency.

Frequency measures how often the wave repeats, not how large the wave is.
A higher amplitude does not mean a higher frequency.

Visualizing Frequency

A high-frequency wave has cycles packed more closely in time. A low-frequency wave has cycles spread farther apart in time.

Rendering scientific sketch…

Low and high frequency shown over the same time interval

Over the same time interval, the high-frequency wave completes more cycles.

Simple Examples

If a source makes 60 vibrations in 20 seconds, then the frequency is

$$
f = \frac{60}{20} = 3\ \text{Hz}
$$

This means 3 cycles occur each second.

If a wave has frequency $50\ \text{Hz}$, then its period is

$$
T = \frac{1}{50} = 0.02\ \text{s}
$$

So each cycle lasts $0.02$ seconds.

Common Values

Different waves can have very different frequencies.

Example	Approximate frequency
Slow water wave	$0.5\ \text{Hz}$
Hand-shaken rope	$2$ to $5\ \text{Hz}$
Musical note A	$440\ \text{Hz}$

These examples show that frequency can range from less than one cycle per second to hundreds or many more cycles per second.

A Final Practical Rule

When you are given information about repeated motion, ask two questions. How many cycles occur, and over what time? That immediately leads to frequency. If instead you know the time for one cycle, use the reciprocal relationship with the period.

To find frequency, use either
$$
f = \frac{N}{t}
$$
or, if the period is known,
$$
f = \frac{1}{T}
$$
Always express frequency in hertz, $\text{Hz}$.

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