WHAT IS A WAVE
A wave is a disturbance that travels or
propagates and transports energy and momentum without
the transport of matter. The ripples on a pond, the sound we hear,
visible light, radio and TV signals are a few examples of waves. Sound, light
and radio waves provide us with an effective means of transmitting and
receiving energy and information.
Waves are of two types : mechanical and
electromagnetic.
Mechanical waves require material medium
for their propagation. Elasticity and density of the medium
play an important role in propagation of mechanical waves. That is why the
mechanical waves sometimes are referred to as elastic waves.
Electromagnetic waves require absolutely no
material medium for their propagation. They can travel through vacuum. Light,
TV signals, radio waves, X-rays, etc. are examples of non mechanical waves.
These are electromagnetic in nature. In an electromagnetic wave,
energy travels in the form of electric and magnetic fields.
Classification of Waves
There are three ways of classifying mechanical waves.
1. Based on Direction of Motion of Particles
Waves differ from one another in the manner the particles of
medium oscillate (or vibrate) with reference to the direction of propagation.
(i) Transverse Waves : In such waves, the oscillatory
motion of the particles of the medium is transverse to the direction of
propagation. Consider the wave travelling along a rope.
The direction of
propagation of the wave is along the rope, but the individual particles of the
rope vibrate up and down. The electromagnetic wave (light, radio waves, X-rays,
etc.) through not mechanical, are said to be transverse, as the electric
and magnetic field vibrate in direction perpendicular to the direction of
propagation.
(ii) Longitudinal Waves : In these waves, the
direction of vibration of the particles of the medium is parallel to the
direction of propagation.
The figure shows a long
and elastic spring. When we repeatedly push and pull on end of the spring, the
compression and rarefaction of the spring travel along the spring. A particle
on the spring moves back and forth, parallel and anti-parallel to the direction
of the wave velocity.
Sound waves in air are also
longitudinal. Some waves (for example,ripples on the surface of a
pond) are neither transverse nor longitudinal but a combination of the two. The
particles of the medium vibrate up and down, and back and forth simultaneously
describing ellipses in a vertical plane.
In strings, mechanical waves are always transverse,
when the string is under a tension.
In gases and liquids,
mechanical waves are always longitudinal, e.g., sound waves in air
or water. This is because fluids cannot sustain shear. They do not posses rigidity.
They posses volume elasticity, because of which the variations of
pressure (i.e., compression and rarefaction) can travel through them. For this
reason, the longitudinal waves are also called pressure waves.
The waves on the surface of water are of two kinds: Capillary
waves and gravity waves. Capillary waves are ripples of fairly short
wavelength - no more than a few centimeters. The restoring force that
produces these waves is the surface tension of water.
Gravity waves have wavelength of several meter and restoring
force is the pull of gravity.
In solids, mechanical waves (may be sound) can be
either transverse or longitudinal depending on the mode of excitation. The
speeds of the two waves in the same solid are different (longitudinal waves
travel faster than transverse waves).
2. Based on Dimensionality of Propagation
One-dimensional wave travels along a
straight line, e.g., waves produced on a string.
Two-dimensional wave propagates
over a surface, e.g, water ripples, vibration of the surface of a drum.
Three-dimensional wave propagates in all
directions, e.g., sound waves.
3. Based on Particle Behaviour in Time
These can be two types of waves – wave pulse and wave
train.
(i) Wave pulse: In this case, the
motion of a particle of the medium has following time sequence. First the
particle is in equilibrium (no motion) state. It then gets some type of motion
or disturbance, and finally it returns to it equilibrium position. We can
generate a transverse wave pulse on a string by once displacing one end of the
string up and down.
As the displacement pulse
travels along the string each particle in the string begins at rest,
experiences a displacement as the pulse passes through it, and then returns to
the equilibrium.
(ii) Wave Train : In a wave train all the particles
of the medium undergo a continuous periodic motion. Any continuous succession
of pulses constitution a wave train. Specially, if the periodic motion of the
particles is simple harmonic motion, the wave is called sinusoidal
wave train.
Wave
Function
The disturbance created by a wave is represented by wave
function. For a string, the wave function is a (Vector) displacement;
whereas for sound waves, it is (scalar) pressure or density fluctuation. In the
case of light or radio waves, the wave function is either an electric or magnetic
field vector.
Mathematical
Representation of Wave Function
Consider a disturbance or a pulse travelling along x-direction
with a velocity v. Let us look at this pulse from two different
frames of reference. The xy-frame is stationary, whereas the other
frame x'y' is moving with velocity v along x-axis,
as shown in the figure. We assume that the origins of the two frames concede
at t = 0.
In the moving frame, the pulse appears to be at rest, since both
the pulse and the x'y'- frame are moving with the same
velocity v. Therefore, at any time the vertical displacement y' at
position x' is given by some function f(x') that
describes the shape of the pulse;
y′=f(x′) ........(i)
In the stationary frame, the pulse has the same shape but it is
moving with a velocity v. It means that the displacement y is
a function of both x and t.
The coordinates of any point on the pulse as measured in the two
frame are related as
y′=y
x′=x−vt
Thus, Eq. (i) may be modified as
y=f(x−vt) ..........(ii)
This equation represents a wave motion along +ve x-direction.
Any given feature (phase) of the pulse, for
example, its peak, has a fixed value of x'. It means that
x′=x−vt=constant ....(iii)
The quantity x−vt is
called the phase of the wave function.
Differentiating Eq. (iii) w.r.t. time, we get
dxdt=v
where v is the wave velocity or phase
velocity. It is the velocity at which a particular phase
f the disturbance travels through space. If the wave is travelling
along the negative x-axis, the wave function is given by Eq. (ii)
modified as
y=f(x+vt)
In general, the wave motion in one dimension is given by
y=f(x±vt)
The
Wave Equation
A travelling wave satisfies a differential equation, called
the linear wave equation,
∂2y∂x2=1v2∂2ydt2
Any function of space and time which satisfies above
differential equation is a wave.
Functions such as y = A sin wt
or y = A sin kx do
not satisfy above equation, hence do not represent waves. On the other hand;
functions such as
Asin(kx−wt)
Asinkxsinwt
[Asin(kx−wt)+Bcos(kx+wt)]
(ax−bt)−−−−−−−√
(ax−bt)2
Ae−B(x−vt)2
or Acos2(kx−wt) satisfy the wave equation, and hence these are wave
functions.
Note that for a function to be wave function, the three
quantities x, t and v must appear
in the combinations (x−vt) or (x+vt). Thus, (x−vt)2 is acceptable but (x2−v2t2) is
not.
Negative sign between t and x implies
that the wave is travelling along positive x-axis and vice-versa.
Example 1
The wave function of a pulse is given by y=32+(x−4t)2, where y is in metres and t is seconds.
Determine the wave velocity of the pulse and indicate the direction of
propagation of the wave.
Solution
On comparing the given expression with
y=f(x−vt)
we get the velocity of the wave as
v=4m/s
Since these occurs negative sign between x and t in
the given expression, the wave propagates along the +ve x-axis.
Harmonic Wave Train
If the source of the wave is a simple harmonic oscillator, the
function f(x±vt) is sinusoidal and it represents a harmonic wave
train or simply, a plane progressive wave.
When such a wave passes through a given region, the particles of the medium
execute simple harmonic motion.
A 1-D (one dimensional) plane progressive wave in
its most general form is given by
y=Asin(kx∓wt+ϕ0)
Clearly a set of four parameters A, ϕ0w and k completely describes a plane
progressive wave.
(1) Amplitude (A)
It represents the maximum value of the wave function from its
equilibrium value.
(2) Phase Constant (ϕ0)
The phase constant or initial phase ϕ0 enables
us to find the position from where time is considered. If all t =
0, x = 0 and y is also zero, then ϕ0 will be
zero which is usually the case with a wave. Henceforth, we shall assume ϕ0 =
0 and the wave is travelling along positive x-axis unless stated
otherwise.
(3) Angular Frequency (w)
At a given position, the wave function at time t' is
given as
y,=Asin(kx−wt,)
The wave will repeat itself, if y' = y
or t′=t+(2πw) [as sin(0+2π)=sinθ]
The time after which a wave repeats itself is called time period (T),
given by
T=t′−t=2πw
It is exactly the same time that it takes for one wavelength to
pass the point.
Further, the rate at which the wave repeats itself is
called its frequency (f),
f=1T=w2π
The SI unit of f is Hz (hertz).
It is same as the number of complete vibrations of a point that
occur in one second.
The angular frequency (w) is related to
the frequency as
w=2πf
w is measured in rad/s.
Note that w, f or T are the
characteristics of the source producing the wave and are independent of the
nature of the medium in which the wave propagates.
(4) Wave Number (k)
The wave number or propagation constant (k)
of a wave train is defined as
k=2πλ
where λ is
the wavelength. The wavelength (λ) is
the distance between two consecutive points vibrating with the same phase (for
example, two creasts).
The constant k or wavelength λ depends on the nature of the medium.
Example 2
The equation of a transverse wave is a stretched string is given
as
y=2sin2π{x30−t0.01}
where y and x are cm and t is in s. Find
(a) the amplitude
(b) the
frequency
(c) the wavelength, and
(d) the wave velocity
Solution
Comparing the given equation with the standard equation,
y=Asin(kx−wt)=Asin{2πλx−2πTt} =Asin2π{xλ−tT}
we get,
(a) amplitude, A = 2 cm
(b) frequency, f = 100 Hz
(c) wave length, λ = 30
cm
(d) wave velocity, v=fλ=100×30=3000 cm/s = 30 m/s
Example 3
Calculate the velocity of sound in a gas, in which the
difference in frequencies of two waves of wavelength 1.0 m and 1.01 m is 4 Hz.
Solution
Let the frequencies of the two waves be f1 and f2. Then
f1−f2=4
Since, v=fλ, we
can write
vλ1−vλ2=4
or v(1λ1−1λ2)=4 or v(11.0−11.01)=4
v=4×1.010.01=404ms−1
Phase Difference and Path Difference
The argument of the harmonic function,
y=Asin(kx−wt+ϕ0)
is called phase of the wave, ϕ. Thus,
ϕ=kx−wt+ϕ0
.................(i)
The phase ϕ changes
both with distance x and time t.
The change in phase Δϕ with change in time Δt for fixed value of x is found by
partially differentiating. Eqn (i) w.r.t. t, as
Δϕ=wΔt=2πTΔt
Δφ=2πTΔt .................(ii)
Similarly, the change in phase Δf with
change in distance Δx for fixed value of time t is given as
Δφ=kΔx=2πλΔx
Δφ=2πλΔx .......................(iii)
The change in x is also called path
difference. If Δx=λ = l, we get Δϕ=2π. That is, a path difference λ corresponds to a phase difference of 2π rad.
Example 4
A progressive wave of frequency 500 Hz is travelling with a
velocity of 360 m/s. How far apart are two points 60° out of phase ?
Solution
We know that for a wave v=fλ.
λ=vf=360500=0.73m
Given,
Δφ=60∘=π180∘×60∘=π3rad
We know that
phase difference,Δφ=2πλ (path
diffrence,Δx)
Δx=λ2πΔφ=0.722π×π3 = 0.12m
