Frequency, Period, Wavelength, and Velocity Revision Notes for HSC SSCE Physics

Frequency, Period, Wavelength, and Velocity

Introduction

Wave motion can be described using four fundamental characteristics that are connected through mathematical relationships. These characteristics are frequency, period, wavelength, and velocity. Understanding how these properties relate to each other is essential for analysing any type of wave, whether it's a wave on a string, a sound wave, or an electromagnetic wave. The relationships apply universally to all wave phenomena.

How frequency and period are related

Frequency and period describe different aspects of wave motion, but they are intimately connected through an inverse relationship. This means that when one quantity increases, the other decreases by the same proportion.

Frequency ( $f$ ) is defined as the number of complete waves that pass a fixed point in one second. It is measured in Hertz (Hz), where $1 \text{ Hz} = 1 \text{ wave per second}$ .

Period ( $T$ ) is defined as the time interval between successive waves passing a point. In other words, it's how long it takes for one complete wave to pass. Period is measured in seconds.

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The inverse relationship between frequency and period means they are mathematical reciprocals of each other. When one value doubles, the other is cut in half. When one increases by a factor of ten, the other decreases by a factor of ten. This reciprocal relationship is constant for all waves.

To understand their inverse relationship, consider these examples:

If two wave crests pass a point in one second, the frequency is $f = 2 \text{ Hz}$ . Since two waves occur in one second, each wave takes $0.5 \text{ s}$ , so the period is $T = 0.5 \text{ s}$ .

If ten wave crests pass a point in one second, the frequency is $f = 10 \text{ Hz}$ . Each wave takes $0.1 \text{ s}$ to pass, giving a period of $T = 0.1 \text{ s}$ .

Notice the pattern: when frequency increases five-fold (from $2 \text{ Hz}$ to $10 \text{ Hz}$ ), the period decreases by the same factor (from $0.5 \text{ s}$ to $0.1 \text{ s}$ ).

chatImportant

The inverse relationship between frequency and period can be expressed mathematically as:

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

Alternatively, rearranging this equation gives:

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

These two forms are equally valid - use whichever is more convenient for the problem you're solving.

Worked example: Calculating frequency from period

lightbulbExample

Worked Example: Calculating frequency from period

Question: A string is being moved up and down with a continuous vibration, taking $0.02 \text{ s}$ to complete one full oscillation. What is the frequency of this vibration?

Solution approach:

Answer	Logic
$T = 0.02 \text{ s}$	• Identify the relevant data in the question.
$f = \frac{1}{T}$	• Identify the appropriate formula.
$= \frac{1}{0.02 \text{ s}}$	• Substitute the known values, with units, into the formula.
$= 50 \text{ Hz}$	• Calculate the answer with the correct significant figures and units.

The vibration has a frequency of 50 Hz, meaning fifty complete oscillations occur each second.

How velocity, frequency, and wavelength are related

The velocity of a wave is determined by the properties of the medium through which it travels. However, velocity is also connected to the wave's frequency and wavelength through a fundamental equation.

To derive this relationship, we start with the basic equation for velocity from kinematics:

$v = \frac{d}{t}$

where $v$ is velocity, $d$ is distance travelled, and $t$ is time taken.

Consider a wave with period $T$ . During one complete period, the wave pattern moves forward by exactly one wavelength, $\lambda$ . This is because the period is the time for one complete oscillation, and each oscillation corresponds to one complete wave pattern.

Therefore, we can substitute $d = \lambda$ (the distance) and $t = T$ (the time) into the velocity equation:

$v = \frac{\lambda}{T}$

Since we know that $f = \frac{1}{T}$ , we can substitute $T = \frac{1}{f}$ into this equation:

$v = \frac{\lambda}{\frac{1}{f}}$

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Simplifying this expression gives us the fundamental wave equation:

$v = f\lambda$

This equation tells us that wave velocity equals the product of frequency and wavelength. This is one of the most important equations in wave physics and applies to all types of waves.

Practical example: Imagine you're creating pulses in a slinky spring at a rate of one pulse every $0.5 \text{ s}$ . This is the period, $T = 0.5 \text{ s}$ . You measure the distance between the peaks of two adjacent pulses and find it to be $0.6 \text{ m}$ . This is the wavelength, $\lambda = 0.6 \text{ m}$ .

In one period ( $0.5 \text{ s}$ ), a wave crest travels one wavelength ( $0.6 \text{ m}$ ). Therefore:

$v = \frac{\text{distance}}{\text{time}} = \frac{0.6 \text{ m}}{0.5 \text{ s}} = 1.2 \text{ m s}^{-1}$

Worked example: Calculating wavelength

lightbulbExample

Worked Example: Calculating wavelength

Question: A sound wave with a frequency of $200 \text{ Hz}$ is travelling through air at $340 \text{ m s}^{-1}$ . What is its wavelength, $\lambda$ ?

Solution approach:

Answer	Logic
$f = 200 \text{ Hz}$ ; $v = 340 \text{ m s}^{-1}$	• Identify the relevant data in the question.
$v = f\lambda$	• Identify the appropriate formula.
$\lambda = \frac{v}{f}$	• Rearrange formula.
$= \frac{340 \text{ m s}^{-1}}{200 \text{ Hz}}$	• Substitute the known values, with units, into the formula.
$= 1.70 \text{ m}$	• Calculate the answer and express with the correct significant figures and units.

The wavelength of this sound wave is 1.70 m.

Worked example: Calculating wave velocity

lightbulbExample

Worked Example: Calculating wave velocity

Question: A wave with a frequency of $440 \text{ Hz}$ has a wavelength of $0.75 \text{ m}$ . What is its speed?

Solution approach:

Answer	Logic
$f = 440 \text{ Hz}$ ; $\lambda = 0.75 \text{ m}$	• Identify the relevant data in the question.
$v = f\lambda$	• Identify appropriate formula.
$= 440 \text{ Hz} \times 0.75 \text{ m}$	• Substitute the known values, with units, into the formula.
$= 330 \text{ m s}^{-1}$	• Calculate the answer and express with the correct significant figures and units.

The wave is travelling at 330 m s⁻¹.

Investigation: Modelling wave equations using technology

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This investigation uses spreadsheet software to create visual models of the mathematical relationships between wave characteristics. By generating graphs from the wave equations, you can see how changing one variable affects the others. This helps develop a deeper understanding of the inverse and direct relationships governing wave behaviour.

Aim

To model the wave equations $v = f\lambda$ and $f = \frac{1}{T}$ using technology to visualise the relationships between the variables.

Materials

Laptop or tablet with suitable spreadsheet software (Note: the instructions are for Microsoft Excel and may need to be modified for other spreadsheet software)

Method

Part 1: Modelling the relationship between period and frequency

Open a new spreadsheet and save it as 'Wave relationships'.
Enter data into worksheet 1 as shown in the table below:

ROW	COLUMN A: Period, $T$ (s)	COLUMN B: Frequency, $f$ (Hz)
1	Period, $T$ (s)	Frequency, $f$ (Hz)
2	0.1	=1/A2
3	0.2
4	0.5
5	1.0
6	2.0
7	5.0
8	10.0

Into cell B2, type the formula =1/A2, and then copy and fill column B with this formula to cell B8.
Next, to produce a graph, highlight cells A1 to B8, and click on 'Insert' (on the ribbon or select from the menu).
Select the scatter graph with data points joined by a line. You may wish to add axis labels and a chart title.
Analyse the graph and describe it.

Part 2: Modelling the relationship between frequency and wavelength

Copy the data into worksheet 2 as shown below:

ROW	COLUMN A: Frequency, $f$ (Hz)	COLUMN B: Wavelength, $\lambda$ (m)
1	Frequency, $f$ (Hz)	Wavelength, $\lambda$ (m)
2	100	=B9/A2
3	200
4	400
5	800
6	1600
7	3200
8	6400

Type a value for the speed of sound into cell B9. Use '340' to begin with, as this is the approximate speed of sound in air.
Copy and fill column B with the formula entered into cell B2.
Repeat steps 4 to 6 to insert a scatter graph as before.

Results

Record your analysis and description beneath each graph on the spreadsheet.

Analysis of results

Analyse the graph and describe the relationship between $f$ and $\lambda$ for waves with the same speed. Identify the speed of the wave you have graphed.

Discussion

Explain how the investigation can show how characteristics of waves are related.
Enter a new value in cell B9 (try '2000' to begin with). Describe the change observed in the graph produced from the data in the table. Is the shape of the graph similar? Why?

Conclusion

With reference to the data obtained and its analysis, write a conclusion based on the aim of this investigation.

Safety considerations

This investigation uses computer equipment only. Ensure cables are positioned safely to avoid trip hazards, and take regular breaks from screen time.

What happens when waves change velocity

Wave velocity is a property determined by the medium through which the wave travels. Different media support different wave speeds. The frequency of a wave is determined by how quickly the source is oscillating or vibrating. The wavelength depends on both the velocity and the frequency through the equation $v = f\lambda$ .

When a wave enters a new medium or when the properties of the medium change, the wave's velocity will change. The key question is: what happens to the wave's other characteristics?

chatImportant

When wave velocity changes:

The wavelength changes to accommodate the new velocity
The frequency remains constant

This makes physical sense because the frequency is set by the source of the wave (how fast it's vibrating), which doesn't change just because the wave enters a different medium. However, for the wave equation $v = f\lambda$ to remain valid with a constant frequency $f$ , the wavelength $\lambda$ must adjust when velocity $v$ changes.

For example, when a wave slows down (velocity decreases), its wavelength must decrease proportionally to maintain the relationship $v = f\lambda$ . Conversely, when a wave speeds up, its wavelength increases.

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Important distinction between graphs:

A displacement versus time graph shows the period, $T$ , of a wave
A displacement versus distance graph shows the wavelength, $\lambda$ , of a wave

Understanding these graphs helps you identify which wave characteristic is being displayed and extract the correct information for calculations.

Remember!

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Key Points to Remember:

Frequency and period are inversely related - when one doubles, the other halves. The relationship is $f = \frac{1}{T}$ .
The wave equation $v = f\lambda$ connects three fundamental properties - velocity equals frequency multiplied by wavelength. This applies to all types of waves.
When using the wave equation, ensure units are consistent - frequency in Hertz, wavelength in metres, velocity in metres per second.
Wave velocity is determined by the medium - different materials allow waves to travel at different speeds. Frequency is determined by the source.
When a wave changes medium and its velocity changes, only the wavelength adjusts - the frequency remains constant because it's controlled by the source, not the medium.

Frequency, Period, Wavelength, and Velocity (HSC SSCE Physics): Revision Notes