WAVE AMPLITUDE: CHARACTERISTICS, FORMULAS AND EXERCISE - PHYSICAL - 2024

The wave amplitude is the maximum displacement that a point of a wave experiences with respect to the equilibrium position. Waves manifest themselves everywhere and in many ways in the world around us: in the ocean, in the sound and on the string of an instrument that produces it, in the light, on the earth's surface and much more.

One way to produce waves and study their behavior is by observing the vibration of a string that has a fixed end. By producing a disturbance at the other end, each particle of the string oscillates and thus the energy of the disturbance is transmitted in the form of a succession of pulses along its entire length.

Waves manifest themselves in many ways in nature. Source: Pixabay.

As the energy propagates, the string that is supposed to be perfectly elastic assumes the typical sinusoidal shape with crests and valleys shown in the figure below in the next section.

Characteristics and meaning of wave amplitude

The 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 slight, the amplitude A is small. If, on the other hand, the disturbance is intense, the amplitude will be greater.

A model to describe the wave consists of a sinusoidal curve. Wave amplitude is the distance between a crest or valley and the reference axis. Source: PACO

The amplitude value is also a measure of the energy carried by the wave. It is intuitive that a great amplitude is associated with higher energies.

In fact the energy is proportional to the square of the amplitude, which mathematically expressed is:

I ∝A ²

Where I is the intensity of the wave, in turn related to energy.

The type of wave produced in the string in the example belongs to the category of mechanical waves. An important characteristic is that each particle in the string is always kept very close to its equilibrium position.

The particles do not move or travel 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 in the water provide the necessary evidence to convince yourself of this. Observing the movement of a leaf that has fallen into a pond, it is appreciated that it simply oscillates accompanying the movement of the water. It does not go very far, unless of course, there are other forces that provide it with other movements.

The wave pattern shown in the figure consists of a repeating pattern in which the distance between two crests is the wavelength λ. If you like, the wavelength also separates two identical points on the wave, even when they are not on the crest.

The mathematical description of a wave

Naturally, the wave can be described by a mathematical function. Periodic functions such as sine and cosine are ideal for the task, whether you want to represent the wave in both 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 simple harmonic movement, which originates thanks to a force that is directly proportional to the displacement made by the particle.

In the proposed equation, A, ω and δ are parameters that describe the movement, A being the amplitude defined above as the maximum displacement experienced by the particle with respect to the reference axis.

The argument of the cosine is called the phase of the movement and δ is the phase constant, which is the phase when t = 0. Both the cosine function and the sine function are appropriate to describe a wave, since they only differ from each other π / two.

In general, it is possible to choose t = 0 with δ = 0 to simplify the expression, obtaining:

y = A cos (ωt)

As the movement is repetitive both in space and in time, there is a characteristic time that is the period T, defined as the time it takes for the particle to execute a complete oscillation.

Description of the wave in time: characteristic parameters

This figure shows the description of the wave in time. the distance between peaks (or valleys) now corresponds to the period of the wave. Source: PACO

Now, both the sine and the cosine repeat their value when the phase increases by the value 2π, so that:

ωT = 2π → ω = 2π / T

A ω is called the angular frequency of the movement and has dimensions of the inverse of time, its units being radian / second or ^-1 second in the international system.

Finally, the frequency of the movement f can be defined as the inverse or reciprocal of the period. Represents in the number of peaks per unit of time, in which case:

f = 1 / T

ω = 2πf

Both f and ω have the same dimensions and units. In addition to the second ^-1, which is called Hertz or hertz, it is common to hear about revolutions per second or revolutions per minute.

The velocity of wave v, which it must be emphasized 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 the frequency depend solely on the nature of the oscillating particles and the characteristics of the system. The amplitude of the wave does not affect these parameters.

For example, when playing a musical note on a guitar, the note will always have the same tone even if it is played with greater or less intensity, in this way a C will always sound like a C, even though it is heard louder or softer in a composition, either on a piano or on a guitar.

In nature, the waves that are transported in a material medium in all directions are attenuated because the energy is dissipated. For this reason, the amplitude decreases with the inverse of the distance r from the source, being possible to affirm that:

A∝1 / r

Exercise resolved

The figure shows the function y (t) for two waves, where y is in meters and t in seconds. For each find:

a) Amplitude

b) Period

c) Frequency

d) The equation of each wave in terms of sines or cosines.

Answers

a) It is measured directly from the graph, using the grid: blue wave: A = 3.5 m; fuchsia wave: A = 1.25 m

b) It is also read from 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 the 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 (1.9t) m; Fuchsia wave: y (t) = 1.25 sin (0.65t) = 1.25 cos (0.65t + 1.57)

Note that the fuchsia wave is out of phase π / 2 with respect to the blue one, being possible to represent it with a sine function. Or cosine shifted π / 2.