10.1 Waves and Energy
A wave arises from the to and fro motions of the individual particles of a medium. In a wave, energy is
transferred from one point to another without the transfer of particles.
Definitions of terms
· Displacement (x or y) of an oscillating particle is its distance from its mean or rest position.
· Amplitude (a) is the maximum displacement.
· Period or periodic time (T) is the time for one complete oscillation.
· Frequency (f) is the number of oscillations (or cycle) in one second. f =
· Wavelength (l) is the distance between two successive vibrating particles which are in the same phase.
· Wavefront is a line or surface where all particles are vibrating in the same phase.
· Wave velocity (v) is the velocity of advance of the wavefronts. v = fl
· Particle velocity
is the actual velocity of a particle in the wave at any instant.
· The phase of a particle is a measure of the position of the particle in its vibration. Two particles are said
to have the same phase if they move in the same direction and have the same displacement.
· Phase lag (j): Particle at a distance l from the source is said to have a phase lag of 2p. Thus, particle at
a distance x from the source is said to have a phase lag of j where
j = 2p
= kx
Graphical representation of wave
10.2 Progressive Wave
> A progressive or travelling wave is the movement of a disturbance from a source to surrounding
places as a result of which energy is transferred.
> Progressive wave equation
The displacement y of a progressive wave moving in the +x direction is given by:
y = a sin (wt – kx)
where angular frequency w = 2pf
=
wave number k =
The displacement y of a progressive wave moving in the -x direction is given by:
y = a sin (wt + kx)
Example 1
A progressive wave moving along a string under tension is described by the equation
y =0.020 sin (120t-8x)
where y is the displacement of the particles and x is the distance of particle from the source
Find:
(a) The amplitude of the wave.
(b) The wavelength of the travelling wave.
(c) The frequency of oscillation.
(d) The speed of propagation of the wave.
(e) The speed of the particle at a point x = 0.20 m from the source and at time t = 0.30
solution
10.3 wave Intensity
• Wave intensity is the rate of flow of energy per unit area perpendicular to the direction of travel of the
wave.
• Wave intensity is directly proportional to the square of the amplitude of vibration of particles.
I a a2
• For a point source, the wave intensity I at distance r from the source is given by :
I=
(where P = power radiated by the source A = area of the spherical wavefront )
=
Ia
Example 2
Figure above shows water ripple originated from source Q. The amplitude of the wave at P is 24mm and
the amplitude of the wave at Q is 8 mm. What is the ratio of the wave intensity at P to the wave intensity
at Q?
Solution
Example 3
A desk is illuminated by a low energy 18W bulb from a distance of 1.2 m. Assuming that the bulb acts as
a point source and it is 100% efficient. What is the intensity at the desk?
Solution
Example 4
The power per unit length of the wavefront of a circular water ripple is 54 mWm -1 when it is 0.30 m from
the point source.
(a) What is the power of the point source?
(b) What is the power per unit length of the wavefront when it is 0.50 m away from the source?
Solution
10.4 Principle of Superposition
• Principle of superposition states that when two waves travel through a medium their resultant
displacement at any point is the sum of the separate displacements due to the two wave
10.5 Standing Waves
• If two waves of the same frequency are travelling in opposite directions, they create a standing wave
pattern of displacements.
Derive and interpret the standing wave equation
•Suppose y1 = a sin (wt – kx) is a progressive wave travelling in the +x direction
then y2 = a sin (wt + kx) is a progressive wave travelling in the -x direction
• The resultant displacement, y, is hence given by:
y = y2 + y2
= a sin (wt – kx) + a sin (wt + kx)
= 2a sin wt kos kx
where A = 2a kos kx
• From the equation, it is noticed that:
(a) At various x position, the particles vibrate with amplitude
A = 2a kos kx
= 2a kos
x
(b) When x = 0,
,2l………………..
,l,
A = 2a because kos
x = ±1
These are the antinodes position.
(c) When x =
,
A = 0 because kos
,
x=0
These are the nodes position.
Differences between progressive wave and standing wave
• Progressive wave
(a) The waveform appears to be propagated steadily forward at a velocity v.
(b) There are no nodes and antinodes.
(c) All particles vibrate with the same amplitude.
(d) The neighbouring particle vibrates with different phase.
• Standing wave
(a) The waveform is not propagating.
(b) There are nodes and antinodes.
(c) The particles between two nodes vibrate with different amplitudes.
(d) The particles between two nodes vibrate with the same phase.
10.6 longitudinal Waves and Transverse Waves
> Longitudinal waves are waves which travel in a direction parallel to the direction of vibrations of the
particles. e.g. sound waves.
> Transverse waves are waves which travel in a direction perpendicular to the direction of vibrations of
the particles. e.g. water waves, light waves and other electromagnetic waves.
“Merintis Singularity”