how to represent progressive waves in graph?
Answers
Progressive Wave
A progressive wave is defined as the onward transmission of the vibratory motion of a body in an elastic medium from one particle to the successive particle.
Wave
Equation of a plane progressive wave
An equation can be formed to represent generally the displacement of a vibrating particle in a medium through which a wave passes. Thus each particle of a progressive wave executes simple harmonic motion of the same period and amplitude differing in phase from each other.
Let us assume that a progressive wave travels from the origin O along the positive direction of X axis, from left to right (Fig. 7.6). The displacement of a particle at a given instant is
y = a sin ωt …... (1)
where a is the amplitude of the vibration of the particle and ω = 2πn.
The displacement of the particle P at a distance x from O at a given instant is given by,
y = a sin (ωt - φ) …... (2)
If the two particles are separated by a distance λ, they will differ by a phase of 2π. Therefore, the phase φ of the particle P at a distance
x is φ = (2π/λ) x
y = a sin (ωt - 2πx/λ) …... (3)
Since ω = 2πn = 2π (v/λ), the equation is given by,
y = a sin [(2πvt/λ) - (2πx/λ)]
y = a sin 2π/λ (vt – x) …... (4)
Since, ω = 2π/T, the equation (3) can also be written as,
y = a sin 2π (t/T – x/λ) …... (5)
If the wave travels in opposite direction, the equation becomes,
y = a sin 2π (t/T + x/λ) …... (6)
(i) Variation of phase with time
Plane Progressive Wave
The phase changes continuously with time at a constant distance.
At a given distance x from O let φ1 and φ2 be the phase of a particle at time t1 and t2 respectively.
φ1 = 2π (t1/T - x/λ)
φ2 = 2π (t2/T - x/λ)
φ2 – φ1 = 2π (t2/T – t1/T) = 2π/T (t2 – t1)
?φ = (2π/T) ?t
This is the phase change ?φ of a particle in time interval ?t. If ?t = T, ?φ = 2π. This shows that after a time period T, the phase of a particle becomes the same.
(ii) Variation of phase with distance
At a given time t phase changes periodically with distance x. Let φ1 and φ 2 be the phase of two particles at distance x1 and x2 respectively from the origin at a time t.
Then, φ1 = 2π (t/T - x1/λ)
φ2 = 2π (t/T - x2/λ)
So, φ2 – φ1 = – 2π/λ (x2 – x1)
Thus, ?φ = – 2π/λ (?x)
The negative sign indicates that the forward points lag in phase when the wave travels from left to right.
When ?x = λ, ?φ = 2π, the phase difference between two particles having a path difference λ is 2π.
Let us graphically represent the two forms of the wave variation
(a) Space (or Spatial) variation graph
(b) Time (or Temporal) variation graph
(a) Space variation graph
By keeping the time fixed, the change in displacement with respect to x is plotted. Let us consider a sinusoidal graph, y = A sin(kx) as shown in the Figure 11.24, where k is a constant. Since the wavelength λ denotes the distance between any two points in the same state of motion, the displacement y is the same at both the ends y = x and y = x + λ, i.e.,
The sine function is a periodic function with period 2π. Hence,
Comparing equation (11.38) and equation (11.39), we get
kx + k λ = kx + 2π
This implies
where k is called wave number. This measures how many wavelengths are present in 2π radians.
The spatial periodicity of the wave is
Then,
At t = 0 s y(x, 0) = y(x + λ, 0)
and
At any time t, y(x, t) = y(x + λ, t)