A sigle wave
Superposition of waves
© The scientific sentence. 2010
The concept of pulse was introduced to get the features of a wave.
the more interesting situation is that when the disturbance lasts in
time in order to get energy transported by a wave itself from a point to another.
The useful and simple disturbance wave function is a kind of sine function.
Wiggling an object periodically in time constitutes an harmonic oscillator. Thus
this particular disturbance can be conveyed in a medium to generate an
A harmonic oscillation, at certain point (x = 0) is written as:
y(t) = y(x = 0, t)= A sin(ωt); (1)
where ω is the angular frequency, t the time variable and
A is the amplitude of the oscillation. It is plotted on the figure 2-a:
For a given time (t = 0), the related harmonic wave function is written as:
y(x, t = 0) = A sin[(2π/λ)x] (2)
Where λ is the wavelength that is the repeated distance of the wave.
It is represented in the figure 2-b; which shows a "snapshot" of the wave at the
instant t = 0.
Since the shape of the function have to remain the same (wave function at x
is equal the wave function at the time t before; that is at x - vt. Then
the x-t dependance of the wave function is with x - vt
The equation (2) becomes:
y(x,t) = A sin[(2π/λ)(x - vt)] (3)
The period T of oscillations is equal to the one of the wave.
We know that T = 2π/ω = 1/ν = λ/v. Where ν is the
frequency of the wave. Introducing the wave number k = 2π/λ,
the equation (4) can be written as:
y(x,t) = A sin(kx - ωt) (4)
What's more, introducing the phase constant φ the equation (5)
y(x,t) = A sin(kx - ωt + φ) (3.5)
Often we choose x = 0 at t = 0 such that φ = 0.
The angular frequency is the number of times, per unit time as a second, we complete
a cycle, which is completed in the time T called period. In other word, It
is the number of cycles we get per unit time.
As a moving particle over a circle, we can
express the angular frequency by ω = dα/dt, where α is the angle,
from the origin to the current point at the time t corresponding to the position of the moving particle. Integrating
the former relation ∫dα = ∫ ω dt, we find 2π = ω T, so we have the
expression of the period T = 2π/ω.
In the period T, we define a distance called the wavelength λ. It is the
dispalacement of the disturbance (or the pulse) over the period of time T; hence
λ = vT, where v is the speed of the pulse. Therefore v = λ/T = λ ω/2π.
We define a new number called the wave number k = 2π/λ. Thus kv = ω. With
x = vt, we have ωt = ωx/v = ω xT/λ = 2πx/λ = kx . Finally,
the wave function becomes: y = f(x,t) = Asint[k(x - vt] = Asint[kx - ωt].
©: The scientificsentence.net. 2007.