CHAPTER 2 MECHANICAL WAVES

CHAPTER 2
MECHANICAL WAVES

(a) Define the mechanical wave and its formation.
(b) State the properties of transverse wave and longitudinal wave.
(c) Define amplitude, A, frequency, f, period, T, wavelength, λ and wave
number k.
(d) Interpret and use equation for progressive wave,

(f) State particle vibrational velocity as v =



(g) Use the wave propagation velocity,

(Experiment 2 : Sound wave)

2 (A) MECHANICAL WAVE

WAVES

•DEFINITION :
•THE PROPAGATION OF A DISTURBANCE

THAT CARRIES THE ENERGY AND
MOMENTUM AWAY FROM THE SOURCES

OF DISTURBANCE

2 (A) MECHANICAL

MECHANICAL WAVES

A disturbance that The particles Examples: water waves,
travels through oscillate around sound waves, waves on
particles of the their equilibrium a string (rope), waves in
position but do not a spring, seismic waves
medium to transfer
the energy travel (Earthquake waves)

2 (A) MECHANICAL

PROGRESSIVE WAVE

DEFINITION :
THE ONE IN WHICH THE WAVE PROFILE PROPAGATES
• The progressive waves have a definite speed called the
speed of propagation or wave speed. The direction of the
wave speed is always in the same direction of the wave
propagation

• There are two types of progressive wave,
a. Transverse progressive waves
b. Longitudinal progressive waves

PROGRESSIVE WAVE ≠ STANDING/STATIONARY WAVE

2 (A) MECHANICAL

2 (B) PROPERTIES OF WAVE
TRANSVERSE WAVES

DEFINITION :

A WAVE IN WHICH THE DIRECTION OF VIBRATIONS OF THE
PARTICLE IS PERPENDICULAR TO THE DIRECTION OF THE WAVE

PROPAGATION (WAVE SPEED) AS SHOWN IN FIGURE 2.1.

• Examples of the transverse waves are water waves, waves on a string (rope),
e.m.w. and etc…

• The transverse wave on the string can be shown in Figure 2.2.

2 (B) PROPERTIES OF

TRANSVERSE WAVES

direction of direction of the propagation
vibrations of wave

particle

Figure 2.1

Figure 2.2

2 (B) PROPERTIES OF

LONGITUDINAL WAVES

DEFINITION :
A WAVE IN WHICH THE DIRECTION OF VIBRATIONS OF THE PARTICLE IS PARALLEL TO

THE DIRECTION OF THE WAVE PROPAGATION (WAVE SPEED)
AS SHOWN IN FIGURE 2.3.

Examples of longitudinal waves are sound waves, waves in a spring, etc…

2 (B) PROPERTIES OF

LONGITUDINAL WAVES

particle direction of the propagation of wave
direction of vibrations
Figure 2.3

Figure 2.4

2 (B) PROPERTIES OF

2 (C) WAVE PARAMETERS

Distance between two
consecutive crests or

troughs

Highest point of the Motion of a wave
wave leaving its source

Maximum displacement of a Lowest point of the
wave in relation to its mean wave

position between two 2 (C) WAVES
consecutive crests or

troughs

SINUSOIDAL WAVES PARAMETER
• Figure 2.5 shows a periodic sinusoidal waveform.

Figure 2.5

2 (C) WAVES

WAVELENGTH, λ

• DEFINITION : DISTANCE BETWEEN TWO CONSECUTIVE PARTICLES (POINTS) WHICH
HAVE THE SAME PHASE IN A WAVE

• From the Figure 2.5,
• Particle B is in phase with particle C
• Particle P is in phase with particle Q

• Particle S and T are in phase
• The S.I. unit of wavelength is metre (m)

2 (C) WAVES

PERIOD, T

• DEFINITION : TIME TAKEN FOR A PARTICLE (POINT) IN THE WAVE TO
COMPLETE ONE CYCLE

• In this period, T the wave profile moves a distance of one wavelength, λ
• Thus

• Its unit is second (s)

FREQUENCY, f
• DEFINITION : THE NUMBER OF CYCLES (WAVELENGTH) PRODUCED IN ONE

SECOND
• Its unit is hertz (Hz) or s−1

2 (C) WAVES

AMPLITUDE, A • It moves a distance of λ in time T hence
• DEFINITION : THE MAXIMUM
• The S.I. unit: m s−1
DISPLACEMENT FROM THE • The value of wave speed is constant but the
EQUILIBRIUM POSITION TO THE
CREST OR TROUGH OF THE WAVE velocity of the particles vibration in wave is
MOTION. varies with time, t
• It is because the particles executes SHM where
WAVE SPEED, V the equation of velocity for the particle, vy is
• DEFINITION : THE DISTANCE

TRAVELLED BY A WAVE PROFILE PER
UNIT TIME
• Figure 2.6 shows a progressive wave
profile moving to the right

Figure 2.6

2 (C) WAVES

WAVE NUMBER, K
• DEFINITION :

• The S.I. unit of wave number is m−1

DISPLACEMENT, Y
• DEFINITION : THE DISTANCE

MOVED BY A PARTICLE FROM ITS
EQUILIBRIUM POSITION AT EVERY
POINT ALONG A WAVE

2 (C) WAVES

2 (D) EQUATION OF PROGRESSIVE WAVE

• Figure 2.7 shows a progressive wave profile moving to the right.

y (displacement)

x (distance from origin)
Figure 2.7

• From the Figure 2.7, consider x = 0 as a reference particle, hence the equation
of displacement for particle at x = 0 is given by

2 (D) EQUATION OF PROGRESSIVE WAVE

• Since the wave profile propagates to the right, thus the other particles will
vibrate

• For particle at point P,
• The vibration of particle is lag behind the vibration of particle at O by a
phase difference of φ radian.
• Thus the phase of particle at P is
• Therefore the equation of displacement for particle’s vibration at P is

2 (D) EQUATION OF PROGRESSIVE WAVE

• Figure 10.4 shows three particles in the wave profile
that propagates to the right

Figure 10.4

2 (D) EQUATION OF PROGRESSIVE WAVE

• From the Figure 10.10, when Δφ increases hence the distance
between two particle, x also increases. Thus

∝Phase difference distance from the origin
(Δφ )
(x)

• Then,

and

2 (D) EQUATION OF PROGRESSIVE WAVE

• Therefore the general equation of displacement for sinusoidal
progressive wave is given by

The wave propagates to the right :

The wave propagates to the left :

where
y (x,t) : displacement of the particle as a function of x and t
A : amplitude of the wave
ω : angular frequency
k : wave number
x : distance from the origin
t : time

2 (D) EQUATION OF PROGRESSIVE WAVE

• Some of the reference books, use other general equations
of displacement for sinusoidal progressive wave:

The wave propagates to the right :
The wave propagates to the left :

2 (D) EQUATION OF PROGRESSIVE WAVE

2 (F) PARTICAL VIBRATIONAL VELOCITY

Partical vibrational velocity, vy

The value of wave speed is constant but the velocity of the particles
vibration in wave is varies with time, t

It is because the particles executes SHM where the equation of velocity
for the particle, vy is

2 (D) WAVE PROPAGATION VELOCITY

Equation of a particle’s velocity in wave

By differentiating the displacement equation of the wave, thus

and

where

The velocity of the particle, vy varies with time but the wave velocity ,v is constant
thus

2 (H) WAVE PROPAGATION VELOCITY

WAVE SPEED, v
DEFINITION : THE DISTANCE TRAVELLED BY A WAVE PROFILE PER UNIT TIME.

Figure below shows a progressive wave profile moving to the right.

It moves a distance of λ in time T hence

2 (D) WAVE PROPAGATION VELOCITY

Displacement graphs of the wave

Displacement Displacement, y
graphs of the wave against
distance, x (y-x)
y∝x
y∝t

From both Displacement, y
together, the against time, t
wave speed
(y-t)
can be
determined

y-x graph

• The graph shows the displacement of all the particles in the
wave at any particular time, t

• For example, consider the equation of the wave is
• At time, t = 0 , thus

● Curve C → the displacement of all particles at t = 0 s
● Curve D → the displacement of all particles at t = t1 after the wave

propagates by distance x1

y-t graph

• The graph shows the displacement of any one particle in the
wave at any particular distance, x from the origin

• For example, consider the equation of the wave is

For the particle at x = 0, the equation of the particle is given
by
hence the displacement-time graph is

• For the particle at x = 0.25λ, the equation of the particle is
given by
and

hence the y-t graph is

EXAMPLE 10.2
A progressive wave is represented by the equation

where y and x are in centimeter and t in seconds.
a) Determine the angular frequency, the wavelength, the

period, the frequency and the wave speed
b) Sketch the displacement against distance graph for

progressive wave above in a range of 0 ≤ x ≤ λ at time, t = 0 s
c) repeat question (b) but for time, t = 0.5T
d) Sketch the displacement against time graph for the

particle at x = 0 in a range of 0 ≤ t ≤ T
e) Sketch the displacement against time graph for the

particle at x = 0.5λ in a range of 0 ≤ t ≤ T.

Solution : with
a. By comparing
and
thus
i.

ii.

iii. The period of the motion is

Solution:
a. iv. The frequency of the wave is given by

v. By applying the equation of wave speed thus

b. At time, t = 0 s, the equation of displacement as a function of
distance, x is given by

Solution:
b. Therefore the graph of displacement, y against distance, x in

the range of 0 ≤ x ≤ λ is

Solution:
c. First method :

At time, t = 0.5T and T = 2 s thus t = 1 s.

Therefore the equation of displacement as a function of
distance is given by

34

Solution:

c. Therefore the graph of displacement, y against distance,
x in the range of 0 ≤ x ≤ λ is

c. Second method :

By referring to the y-x graph for t = 0 s

In time, the distance travelled by the wave is

Hence move the y-axis to the left by amount , because

from the equation given the wave propagates to the right.

Solution :

c. Therefore the graph of displacement, y against distance, is

RULES
If the wave → to the left

→shift the y-axis to the right

If the wave → to the right

→shift the y-axis to the left

Solution:

d. The particle at distance, x = 0 , the equation of displacement as
a function of time, t is given by

Hence the displacement, y against time, t graph is

Solution:
e. First method :

The particle at distance, x = 0.5λ and λ = 2 cm thus
x = 1 cm. Therefore the equation of displacement as a function
of time, t is given by

OR

Solution:

e. Therefore the graph of displacement, y against time, t in
the range of 0 ≤ t ≤ T is

e. Second method :

By referring to the y-t graph for x = 0

Particle at,
The time taken by the wave to travel this distance is

Hence move the y-axis to the left by amount ,

because the wave propagates to the left.

Solution :

e. Therefore the graph of displacement, y against time, t is

RULES
If the wave → to the left

→shift the y-axis to the right

If the wave → to the right

→shift the y-axis to the left

EXAMPLE 10.3

Figure 10.5

Figure 10.5 shows a displacement, y against distance, x graph after
time, t for the progressive wave which propagates to the right with a

speed of 50 cm s−1.
a. Determine the wave number and frequency of the wave.
b. Write the expression of displacement as a function of x and t for

the wave above.

Solution:
a. From the graph,

By using the formula of wave speed, thus
b. The expression is given by

Where y and x in metres and t in seconds

EXERCISE 10.1:
1. A sinusoidal wave of frequency 500 Hz has a speed of 350 m

s-1. Determine
a) the distance between two particles on the wave that have phase
difference π/3 radians,
b) the phase difference between two displacements at a certain
point at times 1.00 ms apart.

ANS: 11.7 cm ; π radians
2. A wave travelling along a string is described by

where y in cm, x in m and t is in seconds. Determine
a) the amplitude, wavelength and frequency of the wave.
b) the velocity with which the wave moves along the string.
c) the displacement of a particle located at x = 22.5 cm and

t = 18.9 s.
ANS: 0.327 cm, 8.71 cm, 0.433 Hz; 0.0377 m s−1; −0.192 cm