Wave Properties
Wave Properties
Before we can describe a wave using a mathematical function, we need to first define what a wave is. A wave is a disturbance in a medium that carries energy without a net movement of particles. A medium can be something like sound, water, or a string. Any physical thing that can have a disturbance. A disturbance of a medium like water would be something like water waves moving up and down, or the contraction or expansion of a slinky.
Now that we have defined what a wave is, let us describe it mathematically. In the simulation below, you can see how you can change the various properties of the wave. Let us go over each of them:
Amplitude:
Firstly, a wave must have a height to see the disturbance of the medium. The height of a wave is called the amplitude. This is the maximum displacement of the wave from the neutral position, or node of the wave. Displacement is the position of a particular point in the medium as it moves as the wave passes. Maximum displacement is the amplitude of the wave. The node of a wave is a point where there is no displacement. The antinode of a wave on the other hand is the point of maximum displacement of a wave. We will talk more about this in the next section discussing Standing Waves. The amplitude also represents the energy of the wave. The greater the amplitude a wave has, the greater the energy the wave carries. Amplitude is denoted as:
\[ A = \text{amplitude} \]
Wavelength:
Secondly, a wave must have a certain distance between two successive identical parts of the wave. One cycle of a wave has three nodes and two antinodes. We have a node at the beginning of the wave signaling the start of the wave, then we have the first antinode which is the positive maximum displacement of the wave. Then we have the second node of the wave, which is the first half of the wave. This middle node connects the wave with the positive and negative maximum displacement of the wave. Then we have the second antinode which is the negative maximum displacement of the wave. Then we have the third node which closes the wave, before repeating the same pattern to infinity. One of these cycles is the wavelength of the wave. Wavelength is denoted as:
\[ \lambda = \text{wavelength} \]
Each wave ranges from 0 to 2π . If we take this into account, we can calculate the wave number of a wave. The wave number is simply that range of 2π divided by the wavelength. The wave number is denoted as:
\[ k = \frac{2 \pi}{\lambda} = \text{wave number} \]
Frequency:
Thirdly, a wave must have a rate in which the number of wavelengths is repeated per second. This is called the frequency. The frequency is the number of wavelength repetitions per second in Hz. Hertz (Hz) is denoted as:
\[ \frac{1}{s} = \text{frequency} \]
The period,
\[ T = \text{time} \]
on the other hand, is the time for one wavelength to pass a point. The difference between frequency and period is that the frequency measures the number of wavelengths passing through a wave motion per second (Hz). The period on the other hand measures how long in time (s) a single wavelength passes through a wave motion.
Similarly with the wave length of a wave, since each wave ranges from 0 to 2π , we can calculate the angular frequency of the wave. The angular frequency is simply that range of 2π multiplied by the period. The angular frequency is denoted as:
\[ \omega = 2 \pi \cdot T = \text{angular frequency} \]
Phase Constant:
Fourthly, although this is optional, a wave can have a phase constant. A phase constant is the radian change of the starting point of a wave. Waves being sinusoidal by nature, can take the form of a sine or cosine wave. A sine wave starts at a point 0, 2π , 2nπ …, a cosine wave starts at a point pi, 3π , (2n +1)π … A phase constant therefore is a number that changes the initial position of a wave from the standard sine wave starting point of 0. The phase constant is denoted as:
\[ \phi = \text{phase constant} \]
Velocity:
Waves also possess a velocity. Using the total distance a wave travels, the wave in this case refers to all the cycled waves combined together with their distances and using the relation of period and frequency, the velocity measures the traveling speed of the wave. We will talk more about how to calculate the velocity of a wave in the section discussing Calculating the Wave Velocity. The velocity is denoted as:
\[ v = \text{velocity} \]
Together with all of there components, we can create an equation that defines all the properties of waves. That equation is:
\[ y = A \sin(k x - \omega t - \phi) \]
This is the general form of the formula. The positve amplitude can be changed to be negative, and the angular frequency as well as phase constant can be changed to be positve. When you use the simulation above, you will see how the wave will change as a result of those different inputted values. The wave will either have different phases, or move in different directions.
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