# Trig Modelling - Sound

*'Discover the underlying mathematics behind music' *

Do you like music? If you play a musical instrument you’ll know that some combinations of notes sound great, and others sound dreadful. In this investigation you are going to discover that it is mathematics that lies behind these notes, chords and harmonies. Tuning forks are used to tune instruments. Mathematically they are interesting since they produce pure tone sound – there is no echo like with a piano or guitar body, for example. You are going to listen to some different tuning fork sounds and look at the visual representation of these sounds. By reproducing the graphs of these sounds for yourself you should understand how they are related to the musical scale.

### Resources & Description

### Exploring y = **asinbx**

Use the applet below to explore the effect of changing **a** and **b** in the equation **y = asinbx**

For each graph make a note of the equation of the graph, the amplitude and the period of the graph.

### Quiz

Check your understanding: try this QUIZ to see if you can apply what you have learned.

Choose the correct answers to the multiple choice questions below.

Question 1

What is the **amplitude** of this graph?

Question 2

What is the period of this graph?

Question 3

Which of the following graphs has the equation y = 2 sin2x

Question 4

Which of the following graphs has a period of 10°

The graph y = sinax has a period 360/a

So, y = sin36x has a period 360/36 = 10°

Question 5

What is the period of the graph y = sin 360x

The graph y = sinax has a period 360/a

So y = sin 360x has a period of 360/360 = 1°

### Understanding Sound

- The following applet should help you understand the dynamics of sound Sound Wave
- The following short video shows a recording of a pitching fork (pitch A) from http://www.onlinetuningfork.com/and then a visual representation of the sound recorded with Audacity, a sound recording piece of software.

### Analysing Sound

The following video gives an introduction to the activity and instructions of what to do next. Watch it to see the shape of the graph of pitch A4

### Main Activity

Here you will explore the equations of graphs of different notes. You will need access to graphing software. **Click on the image below to download the main activity **The Mathematics of Sound **.**

- Can you find the equation of A
_{4}_{ }? - Can you find the equation of C
_{5}? - What will the graph look like when we play A
_{4}_{ }and C_{5}together?