To understand what a wave is, a simple description isn’t enough. We should also learn the properties of waves. Now, let’s delve deeper into waves. 


A waveform is the shape of a wave. We use it to describe the properties of waves. Usually, we draw it on two axes. The horizontal axis is to define time or distance whereas, the vertical axis is to define the extent of the disturbance of a wave. The most common waveforms are sine, square, triangle, and sawtooth.

Sine Wave

The most basic wave is sine wave. It is a continuous wave that takes its name from a smooth curve called sinusoid. They retain their shape when we add another sine wave of the same frequency. This is why we use sine waves in many fields such has mathematics and physics. 

Sine Wave
Sine Wave

Here is a sine wave. Now let’s learn how to make sense of this graph. 

Properties of Waves

Direction of Propagation 

Propagation is the spreading of waves. So when we talk about the direction of propagation, we refer to the direction of spreading.

Direction of propagation of a wave
Direction of Propagation

In this graph, this wave travels from left to right. So its direction of propagation is from left to right.

Rest Position

Rest position is the point where there is no disturbance. 

Rest Positions
Rest Positions

You can see the rest positions in this figure.

Crest and Trough

A crest is the point on a wave with the maximum amount of upward displacement. In other words, a crest is a point where the displacement of the medium is at a maximum. A trough is the opposite of a crest. It is the minimum or lowest point in a cycle. We can say that both crest and trough are the peak points of disturbance.

Crests and Troughs
Crests and Troughs

You can see the crests and troughs in this figure.


The cycle of a wave is a single change from up to down to up. We can also say that it is a change from positive to negative to positive. 

A Cycle
A Cycle

You can see a cycle in this figure.

Wavelength (λ)

The wavelength is the distance between two identical points on back-to-back cycles of a wave. So the wavelength is the length of one complete cycle of a wave. We represent the wavelength by the Greek letter lambda (λ). We use meters (m) to measure wavelengths. 


As you can see from this figure, we can measure the wavelength from a crest to another crest, or from a trough to another trough. We can also measure it from one rest position to the corresponding one on the next cycle. Once we go past one wavelength the pattern starts to repeat.  

Frequency (f) 

The frequency of a wave is the number of times per second that the wave cycles. So it is how often the particles of the medium vibrate when a wave passes through it. 

We use the term frequency as a part of our everyday language. For example, we can hear a question like “How often (or frequently) do you go to cinema?” So just as the name implies, it’s something that happens over and over again. 


As you can see from this figure, high frequency means there are more cycles per second. 

We represent the frequency by the lowercase f. We use Hertz (Hz) or cycles per second to measure the frequency of a wave. 1 Hz is equivalent to 1 cycles per second.

Period (T)

The period of a wave is the time it takes to complete one cycle. When something happens over and over again, we say that it is periodic.


We use the uppercase T to represent the period. We use seconds to measure the period of a wave.

Frequency and period are inverses of each other. If we flip cycles per second (f), it becomes seconds for each cycle (T). Their formulas are like this:

T= 1/f  and f=1/T 

Amplitude (A)

Amplitude is the height, force or power of the wave. It is the maximum amount of the disturbance of the wave from its rest position. So it is related to the energy which a wave transports. We represent the amplitude by the uppercase A. Amplitude is a generic term. So, the unit we use to measure the amplitude depends on the physical quantity that we measure. For waves on a string, the amplitude is a displacement. So we use meters to measure it. For sound waves, there are several units of measurement. 

There is one thing to remember here. Frequency, cycle, and wavelength remain constant. However, the height of the waveform is dynamic. It is based on the power of the wave. The higher the power or amplitude of a wave, the higher the waveform peaks. 

There are also various definitions of amplitude. 

Peak-to-peak Amplitude 

Peak-to-peak Amplitude
Peak-to-peak Amplitude

It is the change between crest and trough. This is one of the most common way of defining amplitude. 

Peak Amplitude 

Peak Amplitude
Peak Amplitude

It is the change between crest and the rest position. In other words, it is the height of the wave from its rest position. 

There are also terms such as Semi-amplitude and Root mean square amplitude. But the most common used terms are the ones above. 

Velocity (v)

One of the important properties of a wave is its velocity or speed. This is how fast the disturbance of the wave is moving. The velocity of a wave depends on the medium that the wave is traveling through. For example, sound will travel at a different speed in air than in water. We represent the speed of a wave by the lowercase v. We can calculate the velocity by multiplying the frequency by the wavelength. So the formula of velocity is like this:

velocity= frequency * wavelength

These are the main properties of waves. Although it is not a property of a wave, I would like to write about phase too. 


Phase is not one of the properties of waves. It is the position of a point in time. We use degrees or radians to measure the phase. Phase also refers to the relative displacement between two corresponding features of two waves having the same frequency.

Phase Difference

We also use phase to describe the relationship between two waves having the same frequency.  The phase difference is the difference in phase angle between two waves. 

For example, if the crests of two waves with the same frequency are in exact alignment at the same time, we call them in phase. If they are not in exact alignment at the same time, we call them out of phase. 

Phase Difference
Phase Difference

In this figure, you can see two waveforms which are out of phase.

In Phase Points

A cycle of a wave is 360 degrees. To be in phase, two points need to be at the same location on the wave in two consecutive cycles. Let’s find cycles first. In this example, A to E  and E to I are two cycles. So we can say that there are two complete cycles in this figure. Now, let’s have a look at another example and try to find the points that are in phase and out of phase.  


In this figure, the following points are in phase: 

  • A and E
  • B and F
  • C and G
  • D and H
  • E and I

Here is the important part. If you notice, the points A, C, E, G, and I are all at the rest positions. Then why do the points A and C are out of phase? Because they are not the same. At the point A, the wave is going up. However, at the point C, the wave is going down. So although they are both at the rest positions, they are not the same points on consecutive cycles. This is why they are out of phase. 

Remember that the distance between two consecutive locations that are in phase also gives us the wavelength. So we can measure the wavelength from the pairs of points that are in phase.

90 Degrees Out Of Phase

The following points are 90 degrees out of phase: 

  • A and B
  • B and C
  • C and D
  • D and E
  • E and F
  • F and G
  • G and H
  • H and I

Why? Remember when I mentioned that a cycle of a wave is 360 degrees? If A to E is a cycle, then the phase difference between A to C and C to E are 180 degrees. Because they are two halves of a cycle. Likewise, the phase difference E to G and G to I are 180 degrees too. If we divide this halves into two, we’ll have quarters of a cycle. For example, A to B is a quarter of a cycle. This is why the phase difference between these points is 90 degrees.

180 Degrees out of Phase

The following points are 180 degrees out of phase:

  • A and C
  • B and D
  • C and E
  • D and F
  • E and G
  • F and H
  • G and I

The same logic applies here. 

270 Degrees out of Phase

The following points are 270 degrees out of phase:

  • A to D
  • B to E
  • C to F
  • D to G
  • E to H
  • F to I

The same logic applies here. 

Anti-Phase and Interference

If the phase difference between two waves is 180 degrees, then we call them to be in anti-phase. When two waves meet at a point where they are in anti-phase, then interference will occur. 

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Gökhan Damgacı

Gökhan Damgacı

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