PROGRESSIVE WAVES (OR) TRAVELLING WAVES
If a wave that propagates in a medium is continuous then it is known as progressive wave or travelling wave.
2. The phase of every particle ranges from 0 to 2π.
3. No particle remains at rest permanently. During wave propagation, particles come to the rest position only twice at the extreme points.
4. Transverse progressive waves are characterized by crests and troughs whereas longitudinal progressive waves are characterized by compressions and rarefactions.
5. When the particles pass through the mean position they always move with the same maximum velocity.
6. The displacement, velocity and acceleration of particles separated from each other by nλ are the same, where n is an integer, and λ is the wavelength.
Suppose we give a jerk on a stretched string at time t = 0s. Let us assume that the wave pulse created during this disturbance moves along positive x direction with constant speed v as shown in Figure 11.23 (a). We can represent the shape of the wave pulse, mathematically as y = y(x, 0) = f(x) at time t = 0s. Assume that the shape of the wave pulse remains the same during the propagation. After some time t, the pulse moving towards the right and any point on it can be represented by x' (read it as x prime) as shown in Figure 11.23 (b). Then,
Similarly, if the wave pulse moves towards left with constant speed v, then y = f(x + vt). Both waves y = f(x + vt) and y = f(x − vt) will satisfy the following one dimensional differential equation known as the wave equation
where the symbol ∂ represent partial (read ∂y/∂x as partial y by partial x). Not all the solutions satisfying this differential equation can represent waves, because any physical acceptable wave must take finite values for all values of x and t. But if the function represents a wave then it must satisfy the differential equation. Since, in one dimension (one independent variable), the partial derivative with respect to x is the same as total derivative in coordinate x, we write
This can be extended to more than one dimension (two, three, etc.). Here, for simplicity, we focus only on the one dimensional wave equation.
Sketch y = x −a for different values of a.
This implies, when increasing the value of a, the line shifts towards right side. For a = vt, y = x − vt satisfies the differential equation. Though this function satisfies the differential equation, it is not finite for all values of x and t. Hence, it does not represent a wave.
How does the wave y = sin(x − a) for a = 0, a = π/4, a = π/2, a = 3π/2, a = π look like?. Sketch this wave.
From the above picture we observe that y = sin (x−a) for a = 0, a = π/4, a = π/2, a = 3π/2 and a = π, the function y = sin (x−a) shifts towards right. Further, we can take a = vt and v = π/4 , and sketching for different times t = 0s, t = 1 s, t = 2s etc., we once again observe that y = sin(x−vt) moves towards the right. Hence, y = sin(x−vt) is a travelling (or progressive) wave moving towards the right. If y = sin(x+vt) then the travelling (or progressive) wave moves towards the left. Thus, any arbitrary function of type y = f(x−vt) characterising the wave must move towards right and similarly, any arbitrary function of type y = f(x+vt) characterizing the wave must move towards left.
Check the dimensional of the wave y = sin(x−vt). If it is dimensionally wrong, write the above equation in the correct form.
Dimensionally it is not correct. we know that y = sin(x−vt) must be a dimensionless quantity but x−vt has dimension. The correct equation is y = sin (k x−ωt), where k and ω have the dimensions of inverse of length and inverse of time respectively. The sine functions and cosine functions are periodic functions with period 2π. Therefore, the correct expression is y = sin ([2π/λ]x, [2π/T]t) where λ and T are wavelength and time period, respectively.
In general, y(x,t)=A sin(k x−ωt).
Let us graphically represent the two forms of the wave variation
(a) Space (or Spatial) variation graph
(b) Time (or Temporal) 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
where k is called wave number. This measures how many wavelengths are present in 2π radians.
The spatial periodicity of the wave is
At t = 0 s y(x, 0) = y(x + λ, 0)
At any time t, y(x, t) = y(x + λ, t)
The wavelength of two sine waves are λ1 = 1m and λ2 = 6m. Calculate the corresponding wave numbers.
By keeping the position fixed, the change in displacement with respect to time is plotted. Let us consider a sinusoidal graph, y =A sin(ωt) as shown in the Figure 11.25, where ω is angular frequency of the wave which measures how quickly wave oscillates in time or number of cycles per second.
The temporal periodicity or time period is
The angular frequency is related to frequency f by the expression ω = 2 πf, where the frequency f is defined as the number of oscillations made by the medium particle per second. Since inverse of frequency is time period, we have,
This is the time taken by a medium particle to complete one oscillation. Hence, we can define the speed of a wave (wave speed, v) as the distance traversed by the wave per second
which is the same relation as we obtained in equation (11.4).
Particle velocity and wave velocity
In a plane progressive harmonic wave, the constituent particles in the medium oscillate simple harmonically about their equilibrium positions. When a particle is in motion, the rate of change of displacement at any instant of time is defined as velocity of the particle at that instant of time. This is known as particle velocity.
Similarly, we can define velocity (here speed) for the travelling wave (or progressive wave). In order to determine the velocity of a progressive wave, let us consider a progressive wave (shown in Figure 11.23) moving towards right. This can be mathematically represented as a sinusoidal wave. Let P be any point on the phase of the wave and yP be its displacement with respect to the mean position. The displacement of the wave at an instant t is
y = y(x,t) = A sin(k x− ω t)
At the next instant of time tʹ = t + ∆t the position of the point P is xʹ = x + ∆x. Hence, the displacement of the wave at this instant is
Since the shape of the wave remains the same, this means that the phase of the wave remains constant (i.e., the y- displacement of the point is a constant). Therefore, equating equation (11.42) and equation (11.44), we get
y(x',t') = y(x,t), which implies
A sin[k (x + ∆x)− ω (t + ∆t)]= A sin(k x− ω t)
On simplification of equation (11.45), we get
where vp is called wave velocity or phase velocity.
By expressing the angular frequency and wave number in terms of frequency and wave length, we obtain
A mobile phone tower transmits a wave signal of frequency 900MHz. Calculate the length of the waves transmitted from the mobile phone tower.
Frequency, f = 900 MHz = 900 ×106 Hz
The speed of wave is c = 3 × 108m s−1