One way to produce waves and study their behavior is to observe the vibration of a string that has a fixed end. When producing a disturbance at the other end, each particle of the string oscillates and with it the energy of the disturbance is transmitted in the form of a succession of pulses throughout.
Waves manifest in various ways in nature. Source: Pixabay
As the energy spreads out, the string that should be perfectly elastic adopts the typical sinusoidal shape with grooves and troughs shown in the figure below in the next section.
Wavelength characteristics and meaning
Amplitude A is the distance between the crest and the reference axis or level 0. If preferred, between a valley and the reference axis. If the disturbance in the string is small, the amplitude A is small. If, on the contrary, the disturbance is intense, the amplitude will be greater.
A model for describing the wave consists of a sinusoidal curve. The wavelength is the distance between a crest or valley and the reference axis. Source: PACO [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0/)]
The amplitude value is also a measure of the energy the wave carries. It is intuitive that greater amplitude is associated with greater energies.
In fact, energy is proportional to the square of the amplitude, which mathematically expresses is:
I ∝A 2
Where i is the wave intensity, in turn related to energy.
The type of wave produced in the example sequence belongs to the category of mechanical waves. An important feature is that each particle in the string is always kept very close to its equilibrium position.
Particles do not move or move through the string. They swing up and down. This is indicated in the diagram above with the green arrow; however, the wave, along with its energy, travels from left to right (blue arrow).
The waves that propagate through the water provide the evidence needed to convince yourself of this. Observing the movement of a leaf that has fallen into a pond, you can see that it simply oscillates along with the movement of the water. It doesn’t go very far unless, of course, there are other forces that provide other movements.
The waveform shown in the figure consists of a repeating pattern in which the distance between two crests is the wavelength λ . If desired, the wavelength also separates two identical points on the wave, even if they are not on the crest.
The mathematical description of a wave
Of course, the wave can be described by a mathematical function. Periodic functions like sine and cosine are ideal for the task whether you want to represent the wave in space and time.
If we call the vertical axis in the figure “y” and the horizontal axis we call “t”, then the behavior of the wave in time is expressed by:
y = A cos (ωt + δ)
For this ideal movement, each particle of the string oscillates with a simple harmonic movement, which originates thanks to a force directly proportional to the displacement effected by the particle.
In the proposed equation, A, ω and δ are parameters that describe the movement, where A is the amplitude defined above as the maximum displacement experienced by the particle in relation to the reference axis.
The cosine argument is called the phase of motion and δ is the phase constant , which is the phase where t = 0. Both the cosine function and the sine function are suitable for describing a wave, as they differ only one of the wave. another π / 2)
You can usually choose t = 0 with δ = 0 to simplify the expression, getting:
y = A cos(ωt)
As motion is repetitive in space and time, there is a characteristic time which is the period T , defined as the time it takes for the particle to perform a complete oscillation.
Description of the wave over time: characteristic parameters
Now, sine and cosine repeat their value when the phase increases by 2π, so:
ωT = 2π → ω = 2π / T
Ω is called the angular frequency of motion and has dimensions inverse to time, with its units being in the international system radian / second or second -1 .
Finally, the frequency of movement f can be defined as inverse or reciprocal of the period. Represents the number of grooves per unit of time, in this case:
f = 1 / T
ω = 2πf
F and ω have the same dimensions and units. In addition to the -1 second , called Hertz or hertz, it is common to hear about revolutions per second or revolutions per minute .
The velocity of the wave v , which must be emphasized that it is not the same as that experienced by particles, can be easily calculated if the wavelength λ and the frequency f are known:
v = λf
If the oscillation experienced by the particles is of the simple harmonic type, the angular frequency and frequency depend only on the nature of the oscillating particles and the characteristics of the system. Wave amplitude does not affect these parameters.
For example, when playing a musical note on an acoustic guitar, the note will always have the same pitch, even if it is played more or less intensely; that way, a do will always sound like a do, even if it’s heard louder or softer in a composition, piano, or guitar.
In nature, waves that are transported in a material medium in all directions are attenuated because the energy dissipates. For this reason, the amplitude decreases with the inverse of the distance r to the source, and it is possible to state that:
A∝1 / r
The figure shows the y(t) function for two waves, where y is in meters and t is in seconds. For each, find:
d) The equation of each wave in terms of sines or cosines.
a) It is measured directly on the graph, with the help of the grid: blue wave: A = 3.5 m; fuchsia wave: A = 1.25 m
b) It is also read on the graph, determining the separation between two consecutive peaks or valleys: blue wave: T = 3.3 seconds; fuchsia wave T = 9.7 seconds
c) It is calculated remembering that the frequency is reciprocal of the period: blue wave: f = 0.302 Hz; Fuchsia wave: f = 0.103 Hz.
d) Blue wave: y (t) = 3.5 cos (ωt) = 3.5 cos (2πf.t) = 3.5 cos (1.9 t) m; Fuchsia wave: y (t) = 1.25 sin (0.65 t) = 1.25 cos (0.65 t + 1.57)
Note that the fuchsia wave is out of phase π / 2 with respect to the blue one, and it is possible to represent it with a sine function. Or cosine offset π / 2.