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# Longitudinal waves: Introduction and key concepts

## Introduction

We have already studied transverse pulses and waves. In this chapter we look at another type of wave called longitudinal waves. In transverse waves, the motion of the particles in the medium were perpendicular to the direction of the wave. In longitudinal waves, the particles in the medium move parallel (in the same direction as) to the motion of the wave. Examples of transverse waves are water waves or light waves. An example of a longitudinal wave is a sound wave.

## What is a longitudinal wave?

Definition 1: Longitudinal waves
A longitudinal wave is a wave where the particles in the medium move parallel to the direction of propagation of the wave.

When we studied transverse waves we looked at two different motions: the motion of the particles of the medium and the motion of the wave itself. We will do the same for longitudinal waves.

The question is how do we construct such a wave?

To create a transverse wave, we flick the end of for example a rope up and down. The particles move up and down and return to their equilibrium position. The wave moves from left to right and will be displaced.

A longitudinal wave is seen best in a spring that is hung from a ceiling. Do the following investigation to find out more about longitudinal waves.

### Investigation : Investigating longitudinal waves

1. Take a spring and hang it from the ceiling. Pull the free end of the spring and release it. Observe what happens.
2. In which direction does the disturbance move?
3. What happens when the disturbance reaches the ceiling?
4. Tie a ribbon to the middle of the spring. Watch carefully what happens to the ribbon when the free end of the spring is pulled and released. Describe the motion of the ribbon.

From the investigation you will have noticed that the disturbance moves parallel to the direction in which the spring was pulled. The spring was pulled down and the wave moved up and down. The ribbon in the investigation represents one particle in the medium. The particles in the medium move in the same direction as the wave. The ribbon moves from rest upwards, then back to its original position, then down and then back to its original position.

## Characteristics of Longitudinal Waves

As in the case of transverse waves the following properties can be defined for longitudinal waves: wavelength, amplitude, period, frequency and wave speed. However instead of peaks and troughs, longitudinal waves have compressions and rarefactions.

Definition 2: Compression
A compression is a region in a longitudinal wave where the particles are closest together.
Definition 3: Rarefaction
A rarefaction is a region in a longitudinal wave where the particles are furthest apart.

### Compression and Rarefaction

As seen in Figure 4, there are regions where the medium is compressed and other regions where the medium is spread out in a longitudinal wave.

The region where the medium is compressed is known as a compression and the region where the medium is spread out is known as a rarefaction.

### Wavelength and Amplitude

Definition 4: Wavelength
The wavelength in a longitudinal wave is the distance between two consecutive points that are in phase.

The wavelength in a longitudinal wave refers to the distance between two consecutive compressions or between two consecutive rarefactions.

Definition 5: Amplitude
The amplitude is the maximum displacement from equilibrium. For a longitudinal wave which is a pressure wave this would be the maximum increase (or decrease) in pressure from the equilibrium pressure that is cause when a peak (or trough) passes a point.

The amplitude is the distance from the equilibrium position of the medium to a compression or a rarefaction.

### Period and Frequency

Definition 6: Period
The period of a wave is the time taken by the wave to move one wavelength.
Definition 7: Frequency
The frequency of a wave is the number of wavelengths per second.

The period of a longitudinal wave is the time taken by the wave to move one wavelength. As for transverse waves, the symbol TT is used to represent period and period is measured in seconds (s).

The frequencyff of a wave is the number of wavelengths per second. Using this definition and the fact that the period is the time taken for 1 wavelength, we can define:

f = 1 T f = 1 T
(1)

or alternately,

T = 1 f T = 1 f
(2)

### Speed of a Longitudinal Wave

The speed of a longitudinal wave is defined as:

v = f · λ v = f · λ
(3)

where

• v=speed in m·s-1v=speed in m·s-1
• f=frequency in Hzf=frequency in Hz
• λ=wavelength in mλ=wavelength in m

#### Exercise 1: Speed of longitudinal waves

The musical note “A” is a sound wave. The note has a frequency of 440 Hz and a wavelength of 0,784 m. Calculate the speed of the musical note.

##### Solution
1. Step 1. Determine what is given and what is required :
f = 440 Hz λ = 0 , 784 m f = 440 Hz λ = 0 , 784 m
(4)

We need to calculate the speed of the musical note “A”.

2. Step 2. Determine how to approach based on what is given :

We are given the frequency and wavelength of the note. We can therefore use:

v = f · λ v = f · λ
(5)
3. Step 3. Calculate the wave speed :
v = f · λ = ( 440 Hz ) ( 0 , 784 m ) = 345 m · s - 1 v = f · λ = ( 440 Hz ) ( 0 , 784 m ) = 345 m · s - 1
(6)
4. Step 4. Write the final answer :

The musical note “A” travels at 345m·s-1345m·s-1.

#### Exercise 2: Speed of longitudinal waves

A longitudinal wave travels into a medium in which its speed increases. How does this affect its... (write only increases, decreases, stays the same).

1. period?
2. wavelength?
##### Solution
1. Step 1. Determine what is required :

We need to determine how the period and wavelength of a longitudinal wave change when its speed increases.

2. Step 2. Determine how to approach based on what is given :

We need to find the link between period, wavelength and wave speed.

3. Step 3. Discuss how the period changes :

We know that the frequency of a longitudinal wave is dependent on the frequency of the vibrations that lead to the creation of the longitudinal wave. Therefore, the frequency is always unchanged, irrespective of any changes in speed. Since the period is the inverse of the frequency, the period remains the same.

4. Step 4. Discuss how the wavelength changes :

The frequency remains unchanged. According to the wave equation

v = f λ v = f λ
(7)

if ff remains the same and vv increases, then λλ, the wavelength, must also increase.

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