# Physics Tutorial: Types of Waves. The Simplified Equation of Waves

In this Physics tutorial, you will learn:

• The meaning of waves
• Definition of waves
• How to classify waves?
• Which are elements of a wave?
• What is the simplified equation of waves?

## Introduction to Types of Waves. The Simplified Equation of Waves

What happens to the guitar strings when you play music? Are the string oscillations regular or irregular?

What do you understand with "water waves"? How can you generate them?

What happens when you shake a rope fixed at one end? Do you think the rope's particles move forward? Why? / Why not?

The answers of the above questions will be discussed in this tutorial. Thus, here you will learn some key concepts regarding waves as phenomenon and their behaviour. Also, you will learn how to classify waves based on pre-defined criteria and how to express the waves mathematically through their equation.

## What are Waves?

As we know, everything in universe is in motion. Nothing can be considered as absolutely at rest. As discussed in the tutorial "Motion. Types of Motion", objects can perform linear, projectile, circular, elliptic and other types of motion. One of the most widespread types of motion is oscillatory (or vibrational) motion, which represents a kind of periodic motion, typical for waves. Obviously, this kind of motion needs the suitable conditions to take place such as its own medium, a source that generates the disturbance in the medium, etc., otherwise it is difficult to produce a sustainable oscillatory motion.

By definition, a wave is a regular disturbance on the medium (caused by a source), which travels through space and matter (i.e. through medium), transferring energy from one place to another.

If we split the sentence of the above definition in parts, we will observe three key concepts that need explication. They are:

### 1. Regular disturbance

Regular disturbance represents an alteration or displacement of a region of a medium from its equilibrium state. In waves, his alteration or displacement occurs in regular shapes and it is periodical, i.e. it repeats itself in equal intervals. For example, when you stretch a rope, it is in equilibrium. When you shake it, you cause a periodical disturbance in the rope and the surrounding air and as a result, the rope moves in regular periodical cycles like this ### 2. Medium of propagation

Medium of propagation represents any substance that allows the disturbance to propagate in other regions. In the above example, the medium of wave propagation is the rope, not the air, as the disturbance caused by the up-and-down shake of the hand, propagates along the rope.

### 3. Transferring energy

A wave transfers only energy, not matter. This is obvious, since the rope in the example above does not move from its place, it only shakes but it does not displaces horizontally. If we consider a certain piece of the rope, we can see that it moves only vertically. Only the disturbance shifts horizontally; this causes a wave to carry energy during its propagation. However, no matter is transferred throughout the wave. Indeed, if you throw a ball on seawater, it will move only up and down at the same place due to water waves if no external factors such as wind, any pushing force etc., are present as shown in the figure below. In other words, we can define a wave as "a regular oscillation that shifts in space". In this sense, a wave is an extension of the concept of oscillations in SHM we discussed in the previous section.

## Elements of a Wave

Consider the wave shown below. The equilibrium position shows a horizontal line that shows the rope, water surface, etc., if no wave were present. When the wave is shown in a coordinate system, the equilibrium position is at the horizontal axis.

The highest point of a wave is known as "crest". In the above figure, crest is shown by the point a.

The opposite of crest is known as "trough". It is shown by the point b in the above figure. Trough represents the lowest point a wave can reach.

Waves represent a periodical motion. This means the cycle repeats itself periodically. The distance between two similar points of two successive cycles is known as "wavelength". It is denoted by the Greek letter λ (lambda). Wavelength is similar to the circumference in rotational motion, i.e. it shows the distance the wave travels during an entire cycle.

Like in rotational and simple harmonic motion, the time needed to complete one cycle is called "period", T. Also, the inverse of period is known as frequency, f as well. Frequency represents the number of cycles in one second. It is very important, especially when the vibrations are very fast.

## Classification of Waves

Waves are classified in two groups based on two different criteria. They are:

### I. According to the method of propagation

When waves' motion is perpendicular to particles' motion as shown below, waves are known as "transverse". This means amplitude is perpendicular to wavelength (A ┴ λ). Water waves, rope waves and light waves are some examples of transverse waves in daily life.

Some other waves propagate in the same direction in which the oscillations occur. In these cases, there are many successive compressions and rarefactions of medium as shown in the figure below. In the first figure, no waves are produced in the spring. As a result, all turns have the same distance from each other.

In the second figure, we see compressions and rarefactions occurring in the spring because of a disturbance. As a result, linear waves known as "longitudinal" are produced in the spring. The spring here acts as medium of waves propagation.

Sound waves, spring waves etc., are examples of longitudinal waves. In such waves, amplitude is in the same direction of waves' propagation (they are parallel). Thus, A || λ.

### II. According to the medium of propagation

Some waves need a material medium to propagate. Otherwise, they cannot exist. Such waves are known as "mechanical waves". For example, sound is a mechanical wave as it needs a material medium such as air, metals, wood etc. to propagate. No sound waves exist in vacuum. Also rope waves, water waves, etc. are all examples of mechanical waves.

On the other hand, there are other waves which do not need any material medium to propagate. These waves can also propagate in vacuum and are known as "electromagnetic waves" (in short, EM). Light waves are the most common example of EM waves.

Thus, for example water waves are mechanical and transverse because they need a material medium (water) to propagate and the direction of oscillations is perpendicular to the direction of propagation. On the other hand, sound waves are mechanical and longitudinal waves as they also need a material medium to propagate but their amplitude is parallel to the direction of oscillations.

## The Simplified Equation of Waves

If waves travel through the same medium, they propagate at constant speed. Therefore, we can use the standard equation of uniform motion

v = s/t

to calculate the waves speed in a certain medium, where s is the distance travelled by the wave and t is the time of motion.

However, since waves move at the same speed throughout the entire process, often it is more appropriate to limit the study in a single cycle only. This eases the study of waves' motion as we can extend the outcome obtained in a single cycle in the entire process as well without the need for further data. We have discussed about a similar approach in circular motion, where we used circumference and period to calculate the speed of rotation instead of total distance and total time.

Thus, we can replace the distance s with wavelength t with period v = λ/T

for the speed of waves. Yet, to eliminate fractions in the formula, we can replacing period with its inverse, i.e. with frequency, as

f = 1/T

Therefore, the above equation becomes

v = λ × f

The last equation is known as the simplified equation of waves.

Obviously,

s = N × λ

and

t = N × T

where

1. What kind of wave is it?
2. What is the amplitude of this wave?
3. What is the maximum displacement a point of the wave can have?
4. What is the wavelength?
5. What is the period of this wave?
6. What is its frequency?
7. How fast does the wave move?

### Solution 1

a) This is a transverse wave as the direction of particles oscillations (up-down) is perpendicular to the direction of wave's propagation (right).

b) Amplitude is the maximum displacement from equilibrium position to the highest point. From the figure we can see that the wave displaces 3 m (from y = 0 to y = +3 m) at maximum, so its amplitude is A = 3 m.

c) The maximum displacement a point of the wave can have is calculated from crest to trough, i.e. from the highest to the lowest position it can reach. Therefore, the maximum displacement is

Δy = (+3 m) - (-3 m)
= 3 m + 3 m
= 6 m

d) From the figure, it is easy to see that there are 3.5 cycles in total, (N = 3.5). During this process, the wave has travelled by s = 28 m. Therefore, wavelength is

λ = s/N
= 28 m/3.5
= 8 m

e) Period is calculated in the same way as wavelength. Thus, since the total time t = 7 s and N = 3.5, we obtain

T = t/N
= 7 s/3.5
= 2 s

f) Frequency f is the inverse of period. Thus,

f = 1/T
= 1/2 s
= 0.5 s-1
= 0.5 Hz

g) "Calculate how fast does the wave move" means, "calculate the wave's speed". Thus, using the equation of waves.

v = λ × f

we find for the wave speed after substitutions,

v = 8 m × 0.5 s-1
= 4 m/s

## Physics Revision: Types of Waves. The Simplified Equation of Waves Summary

A wave is a regular disturbance on the medium (caused by a source), which travels through space and matter (i.e. through medium), transferring energy from one place to another.

The definition of waves contains three key concepts that need explication. They are:

1. Regular disturbance. It represents an alteration or displacement of a region of a medium from its equilibrium state. In waves, his alteration or displacement occurs in regular shapes and it is periodical, i.e. it repeats itself in equal intervals.
2. Medium of propagation. It represents any substance that allows the disturbance to propagate in other regions.
3. Transferring energy. A wave transfers only energy, not matter.

Thus, a wave as "a regular oscillation that shifts in space". In this sense, a wave is an extension of the concept of oscillations and SHM.

The main elements of a wave are:

1. Equilibrium position. It shows a horizontal line that shows the rope, water surface, etc., if no wave were present. When the wave is shown in a coordinate system, the equilibrium position is at the horizontal axis.
2. Crest. It represents the highest point of a wave.
3. Trough. It is the opposite of crest, i.e. it represents the lowest point a wave can reach.
4. Wavelength. It represents the distance between two similar points of two successive cycles. Wavelength is denoted by the Greek letter λ (lambda).
5. Period. Like in SHM and rotational motion, it represents the time needed to complete one cycle.
6. Frequency. It is the inverse of period. Frequency is very useful when dealing with fast vibrations.

Waves are classified in two groups based on two different criteria. They are:

I. According to the method of propagation

1. Transverse waves. These waves exist when waves' motion is perpendicular to particles' motion. This means amplitude is perpendicular to wavelength (A ┴ λ). Water waves, rope waves and light waves are some examples of transverse waves in daily life.
2. Longitudinal waves. These waves propagate in the same direction in which the oscillations occur. This means in longitudinal waves, amplitude is in the same direction of waves' propagation (they are parallel). Thus, A || λ. Sound waves, spring waves etc., are examples of longitudinal waves.

II. According to the medium of propagation

1. Mechanical waves. Some waves need a material medium to propagate. Otherwise, they cannot exist. Such waves are known as "mechanical waves". For example, sound is a mechanical wave as it needs a material medium such as air, metals, wood etc. to propagate. No sound waves exist in vacuum. Also rope waves, water waves, etc. are all examples of mechanical waves.
2. Electromagnetic (EM) waves. They are waves that do not need any material medium to propagate; they can propagate in vacuum as well. Light waves are the most common example of EM waves.

If waves travel through the same medium, they propagate at constant speed. Therefore, we can use the standard equation of uniform motion

v = s/t

to calculate the waves speed in a certain medium, where s is the distance travelled by the wave and t is the time of motion.

Often it is more appropriate to limit the study in a single cycle only. This eases the study of waves' motion as we can extend the outcome obtained in a single cycle, in the entire process as well, without the need for further data. Thus, we can replace the distance s with wavelength λ (the distance travelled by the wave in one cycle) and the total time t with period T (the time needed to complete one cycle) in the equation of uniform motion. Hence, we obtain

v = λ/T

for the speed of waves. Yet, to eliminate fractions in the formula, we can replacing period with its inverse, i.e. with frequency, as

f = 1/T

Therefore, the above equation becomes

v = λ × f

The last equation is known as the simplified equation of waves.

Obviously,

s = N × λ

and

t = N × T

where N is the number of wave cycles during the entire motion.

## Physics Revision Questions for Types of Waves. The Simplified Equation of Waves

1. What wave below is both transverse and longitudinal? 1. Wave (a)
2. Wave (b)
3. Wave (c)
4. Wave (d)

2. Which wave below has the lowest frequency? 3. The speed of the wave shown in the figure below is 20 cm/s. What it the wave's period?

1. 10 s
2. 0.1 s
3. 2 s
4. 11.5 s