Types of Waves. The Simplified Equation of Waves

In these revision notes for Types of Waves. The Simplified Equation of Waves, we cover the following key points:

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

Types of Waves. The Simplified Equation of Waves Revision Notes

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

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

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

Therefore, the above equation becomes

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

Obviously,

and

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