WAVES
Production of waves
Propagation of waves
WAVES
A wave is a disturbance which travels through a medium transferring energy
from one point to another without causing any permanent displacement of the
medium
A wave motion is process of transferring a disturbance from one point to
another without any transfer of particles of the medium.
Types of waves
Waves are broadly classified into two types
a. Based on the medium of propagation: mechanical wave and
electromagnetic wave
b. Based on the comparison of the wave direction with the direction of
vibration of the particle: transverse wave and longitudinal wave
Production and Propagation of waves: Based on the medium of
propagation
1. Production and propagation of mechanical waves
A mechanical wave is the wave that requires material medium for its mode of
propagation (or for it to transfer energy away from the source). Examples are
waves travelling through springs, water waves, and sound waves
2. Production and propagation of electromagnetic waves
Electromagnetic waves are waves that do not need material medium for its
mode of propagation (or for it to transfer energy away from the source).
Examples are radio waves, visible light, ultra-violet rays, x-rays, gamma rays.
Electromagnetic waves travels at the speed of light (3.0×108m).
A wave which travels along a medium transferring energy from one part of the
medium to another is called a progressive wave. The progressive wave can be
divided into transverse and longitudinal waves
y
Direction of wave motion
Progressive or travelling wave
A standing or stationary wave: this is formed when two waves travelling in the
opposite direction meets or by superimposition of incident wave and its
reflection. The amplitude of the standing wave varies along the wave.
Incident wave reflected wave
Standing or stationary wave
Production and Propagation of waves: Based on the comparison of the
wave direction with the direction of vibration of the particle
1. Transverse waves
A transverse wave is a wave in which travel perpendicularly to the direction of
the vibrations producing the waves.
2. Longitudinal wave
Longitudinal waves are waves which travel in a direction parallel to the
vibrations of the medium.
TERMS USED IN DESCRIBING WAVES
1. Phase - particles which are at the same vertical direction from their
positions of rest and are moving in the same direction are said to be in
phase.
2. Cycle - is a complete to-and-fro movement or oscillation of a vibrating
particle
3. The amplitude (A) - is the maximum displacement of a particle from its
rest or mean position. It is measured in meter (m).
4. The period (T) - is the time required for a particle to perform one
complete cycle or oscillation
1
f= ---1
T
1
T= ---2
f
5. Frequency (f) - is the number of complete cycles made in one seconds. It
is measured in Hertz (Hz)
6. Wavelength (λ) - is the distance covered by the waves after one complete
oscillation. For transverse waves, it is the distance between successive
crests or troughs while for longitudinal wave, it is the distance between
successive compressions or rarefactions. It is measured in meter (m).
7. Wave-velocity (v) is the distance traveled by the waves in one second.
The S.I unit is m/s
Displacement
Crest complete oscillation or one cycle
Amplitude
Distance(x)
Trough
One wavelength
MATHEMATICAL RELATIONSHIP
v=wave – velocity
f =frequency(Hz)
λ=wavelength(m)
−1
T = period(S )
Velocity=frequency × wavelength
v=fλ ---3
distance travelled by wave
v=
corresponding time taken
λ
v= ---
T
4
1
From equation 1, f = T
We have:
λ
v= ---
T
5
v=fλ ---6
λ=vT ---7
Worked example
A radio station broadcasts at frequency of 300 KHz. If the speed of the wave is
3 x 108 ms-1, calculate the period and wavelength of the wave?
1
T=
f
1
T=
300000
−6
T =3.3 ×10 S
λ
v=
T
8
3 × 10
v= 5
3× 10
λ=1000 m
Mathematical representation of wave motion – Progressive wave
The general equation for stationary wave is given by:
y= A sin ( 2 πxλ ) ---8
Where
A=amplitude of the wave
λ=wavelength of the wave
y=vertical displacement of the wave
x=horizontal coordinate of the vibrating particle
A
P
O Φ π 2π t
x
Considering O and P that are out of phase by Φ, then we have
y= A sin ( 2 πxλ −Φ) ---8
Where:
Φ x
= ---
2π λ
9
2 πx
Φ= ---
λ
10
But x=vt
2 πvt
Φ= ---
λ
11
Substituting equation 11 into equation 8 gives:
y= A sin ( 2 πxλ − 2 πvtλ ) ---
12
2π
y= A sin ( x−vt ) ---13
λ
Also from equation 12, putting v=fλ we can have
y= A sin ( 2 πxλ − 2 πfλt
λ )
y= A sin ( −2 πft )
2 πx
---14
λ
Recall that ω=2 πf
Thus, equation 12 can be re-written as:
y= A sin ( 2 πxλ −ωt ) ---15
Example:
A plane progressive wave is given by the equation y= Asin(2000 πt−0.5 x )
Calculate: (i) The wavelength of the wave (ii) The speed (iii) The frequency (iv)
The period
Solution:
By comparing the given equation y= Asin(2000 πt−0.5 x ) with the standard
2 πx
equation y= A sin λ −2 πft ( )
We have for:
(i) The wavelength of the wave
2 πft =200 πt
2ft=2000t
f=1000Hz
(ii) The speed
2 πx
=0.5 x
λ
2π
=0.5
λ
λ=2π×2
λ=12.57m
(iii) The frequency
v=f λ
v=1000 × 12.57
v=12570m/s
(iv) The period
1
t=
f
1
t=
1000
t=10-3s-1
CLASSWORK 8
1. What is wave?
2. Elias radio station broadcasts at a frequency of 21MHz. If the speed of
light in the air 3×108ms-1, calculate the wavelength of the broadcast.
3. Define stationary wave
ASSIGNMENT 8
SECTION A
1. An electromagnetic radiation has a speed of 3×108ms-1 and a frequency of
106Hz, calculate its wavelength (a) 3.3×103m (b) 3.0×102m (c) 3.0×10-2m
(d) 3.3×108m (e) 3.3×10-3m
2. A body oscillates in simple harmonic motion according to the equation
π
x=0.05 cos (3 π + ) where x is expressed in meters. What does 0.05
3
represents? (a) velocity (b) frequency (c) period (d) amplitude (e) none of
the above
3. Which of the following is not a mechanical wave (a) wave propagated in
stretched string (b) waves in a closed pipe (c) radio waves (d) water
waves (e) sound waves
4. The maximum displacement of particles of wave from their equilibrium
positions is called (a) wave velocity (b) period (c) amplitude (d)
wavelength (e) frequency
5. D(cm)
0 0.05 0.10 0.15 0.20 0.25 t(s)
The diagram above represents the displacement D versus t graph of a
progressive wave. Deduce the frequency of the wave
(a) 20Hz (b) 10 Hz (c) 5 Hz (d) 4 Hz (e) 50 Hz
SECTION B
1. (a) What is wave motion?
2π
(b) The equation y= A sin λ ( vt−x ) represents a wave train in which y is
the vertical displacement of a particle at a distance x from the origin in
the medium through which the wave travelling. Explain, with the aid of a
diagram, what A and λ represents.
2. A radio waves transmitted from a certain radio station is represented by
the wave equation: y=0.75 sin(0.67 πx−2 ×108 πt)
Calculate the (i) wavelength of the wave (ii) frequency of the wave (ii)
velocity of the wave. Where x, y are in meters while t is in seconds
