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# Important Formulae: Wave Motion Notes | EduRev

## JEE : Important Formulae: Wave Motion Notes | EduRev

``` Page 1

Important Formulae
1.  In any type of wave, oscillations of a physical quantity y are produced at one place and these
oscillations (along with energy and momentum) are transferred to other places also.
2.  Classification of Waves
A wave may be classified in following three ways
First A transverse wave is one in which oscillations of y are perpendicular to wave velocity.
Electromagnetic waves are transverse in nature. A longitudinal wave is one in which oscillations
of y are parallel to wave velocity. Sound waves are longitudinal in nature.
Second Mechanical waves require medium for their propagation. Sound waves are mechanical in
nature. Non-mechanical waves do not require medium for their propagation. Electromagnetic
waves are non-mechanical in nature.
Third Transverse string wave is one dimensional. Transverse waves on the surface of water are
two dimensional. Sound wave due to a point source is three dimensional.
3.  Wave Equation
In any wave equation value of y is a function of position and time. In case of one dimensional
wave position can be represented by one co-ordinate (say x) only. Hence,
y = f(x, t) Only those functions of x and t represent a wave equation which satisfy following
condition.

22
22
yy
(cons tan t)
xt
??
?
??

Here constant =
2
1
v

where v is the wave speed. All functions of x and t of type,
y = f(ax ? bt)
satisfy above mentioned condition of wave equation, provided value of y should be finite for any
value of t. If y(x, t) function is of this type, then following two conclusions can be drawn.
(i) Wave speed
coefficient of t b
v
coefficient of x a
??
(ii) Wave travels along positive x-direction. If ax and bt have opposite signs and it travels along
negative x-direction if they have same signs.
4.  Plane Progressive Harmonic Wave
If oscillations of y are simple harmonic in nature then wave is called plane progressive harmonic
wave. General equation of this wave is,
y = A sin ( ?t ? kx ? ?)
or y = A cos ( ?t ? kx ? ?)
In these equations,
(i) A is amplitude of oscillation,
(ii) ?

is angular frequency,
Page 2

Important Formulae
1.  In any type of wave, oscillations of a physical quantity y are produced at one place and these
oscillations (along with energy and momentum) are transferred to other places also.
2.  Classification of Waves
A wave may be classified in following three ways
First A transverse wave is one in which oscillations of y are perpendicular to wave velocity.
Electromagnetic waves are transverse in nature. A longitudinal wave is one in which oscillations
of y are parallel to wave velocity. Sound waves are longitudinal in nature.
Second Mechanical waves require medium for their propagation. Sound waves are mechanical in
nature. Non-mechanical waves do not require medium for their propagation. Electromagnetic
waves are non-mechanical in nature.
Third Transverse string wave is one dimensional. Transverse waves on the surface of water are
two dimensional. Sound wave due to a point source is three dimensional.
3.  Wave Equation
In any wave equation value of y is a function of position and time. In case of one dimensional
wave position can be represented by one co-ordinate (say x) only. Hence,
y = f(x, t) Only those functions of x and t represent a wave equation which satisfy following
condition.

22
22
yy
(cons tan t)
xt
??
?
??

Here constant =
2
1
v

where v is the wave speed. All functions of x and t of type,
y = f(ax ? bt)
satisfy above mentioned condition of wave equation, provided value of y should be finite for any
value of t. If y(x, t) function is of this type, then following two conclusions can be drawn.
(i) Wave speed
coefficient of t b
v
coefficient of x a
??
(ii) Wave travels along positive x-direction. If ax and bt have opposite signs and it travels along
negative x-direction if they have same signs.
4.  Plane Progressive Harmonic Wave
If oscillations of y are simple harmonic in nature then wave is called plane progressive harmonic
wave. General equation of this wave is,
y = A sin ( ?t ? kx ? ?)
or y = A cos ( ?t ? kx ? ?)
In these equations,
(i) A is amplitude of oscillation,
(ii) ?

is angular frequency,

2
T , 2 f
?
? ? ? ?
?
and
1
f
T2
?
??
?

(iii) k is wave number,

2
k
?
?
?

( ? ? wavelength)
(iv) Wave speed vf
k
?
? ? ?
(v) ?

is initial phase angle and
(vi) ( ?t ? kx ? ?)

is phase angle at time fat coordinate x.
5.  Particle Speed (v
p
) and Wave Speed (v) in Case of Harmonic Wave
(i) y = f(x,t) where x and fare two variables. So,
p
y
v
t
?
?
?

(ii) In harmonic wave, particles are in SHM. Therefore, all equations of SHM can be applied for
particles also.
(iii) Relation between v
p
and v

p
y
v v.
t
?
??
?

6.  Phase Difference ( ? ?)
Case I ? ? = ?(t
1
~ t
2
)  or  ? ? =
2
T
?
?t
= phase difference of one particle at a time interval of ?t.
Case II ? ? = k(x
1
~ x
2
) =
2 ?
?
.?x
= phase difference at one time between two particles at a path difference of ?x.
7.  Energy Density (u), Power (P) and Intensity (I) in Harmonic Wave
(i) Energy density
22
1
uA
2
? ??

= energy of oscillation per unit volume
(ii) Power
22
1
P A Sv
2
? ??

= energy transferred per unit time.
(iii) Intensity
22
1
l A v
2
? ?? = energy transferred per unit time per unit area.
= energy transferred per unit time.
8.  Longitudinal Wave
(i) There are three equations associated with any longitudinal wave
y(x,t), ? ?

(x. t)and ? ? (x,t)
(ii) y represents displacement of medium particles from their mean position parallel to direction of
wave velocity.
Page 3

Important Formulae
1.  In any type of wave, oscillations of a physical quantity y are produced at one place and these
oscillations (along with energy and momentum) are transferred to other places also.
2.  Classification of Waves
A wave may be classified in following three ways
First A transverse wave is one in which oscillations of y are perpendicular to wave velocity.
Electromagnetic waves are transverse in nature. A longitudinal wave is one in which oscillations
of y are parallel to wave velocity. Sound waves are longitudinal in nature.
Second Mechanical waves require medium for their propagation. Sound waves are mechanical in
nature. Non-mechanical waves do not require medium for their propagation. Electromagnetic
waves are non-mechanical in nature.
Third Transverse string wave is one dimensional. Transverse waves on the surface of water are
two dimensional. Sound wave due to a point source is three dimensional.
3.  Wave Equation
In any wave equation value of y is a function of position and time. In case of one dimensional
wave position can be represented by one co-ordinate (say x) only. Hence,
y = f(x, t) Only those functions of x and t represent a wave equation which satisfy following
condition.

22
22
yy
(cons tan t)
xt
??
?
??

Here constant =
2
1
v

where v is the wave speed. All functions of x and t of type,
y = f(ax ? bt)
satisfy above mentioned condition of wave equation, provided value of y should be finite for any
value of t. If y(x, t) function is of this type, then following two conclusions can be drawn.
(i) Wave speed
coefficient of t b
v
coefficient of x a
??
(ii) Wave travels along positive x-direction. If ax and bt have opposite signs and it travels along
negative x-direction if they have same signs.
4.  Plane Progressive Harmonic Wave
If oscillations of y are simple harmonic in nature then wave is called plane progressive harmonic
wave. General equation of this wave is,
y = A sin ( ?t ? kx ? ?)
or y = A cos ( ?t ? kx ? ?)
In these equations,
(i) A is amplitude of oscillation,
(ii) ?

is angular frequency,

2
T , 2 f
?
? ? ? ?
?
and
1
f
T2
?
??
?

(iii) k is wave number,

2
k
?
?
?

( ? ? wavelength)
(iv) Wave speed vf
k
?
? ? ?
(v) ?

is initial phase angle and
(vi) ( ?t ? kx ? ?)

is phase angle at time fat coordinate x.
5.  Particle Speed (v
p
) and Wave Speed (v) in Case of Harmonic Wave
(i) y = f(x,t) where x and fare two variables. So,
p
y
v
t
?
?
?

(ii) In harmonic wave, particles are in SHM. Therefore, all equations of SHM can be applied for
particles also.
(iii) Relation between v
p
and v

p
y
v v.
t
?
??
?

6.  Phase Difference ( ? ?)
Case I ? ? = ?(t
1
~ t
2
)  or  ? ? =
2
T
?
?t
= phase difference of one particle at a time interval of ?t.
Case II ? ? = k(x
1
~ x
2
) =
2 ?
?
.?x
= phase difference at one time between two particles at a path difference of ?x.
7.  Energy Density (u), Power (P) and Intensity (I) in Harmonic Wave
(i) Energy density
22
1
uA
2
? ??

= energy of oscillation per unit volume
(ii) Power
22
1
P A Sv
2
? ??

= energy transferred per unit time.
(iii) Intensity
22
1
l A v
2
? ?? = energy transferred per unit time per unit area.
= energy transferred per unit time.
8.  Longitudinal Wave
(i) There are three equations associated with any longitudinal wave
y(x,t), ? ?

(x. t)and ? ? (x,t)
(ii) y represents displacement of medium particles from their mean position parallel to direction of
wave velocity.
(iii) From y (x, t) equation, we can make ? ?

(x, t) or ? ?

(x, t) equations by using the fundamental
relation between them,
? ? =
y
B.
x
?
?
and ? ? = -
y
.
x
?
?
?

(iv) ? ?
0
= BAk and ? ?
0
= ?Ak
(v) ? ? (x. t)and ? ? (x.t) are in same phase. But y (x.t) equation has a phase difference of
2
?
with rest two equations.
9.  Wave Speed
(i) Speed of transverse wave on a stretched wire
TT
v
S
??
??

(ii) Speed of longitudinal wave
E
v ?
?

(a) In solids, E = Y = Young's modulus of elasticity

Y
v ?
?

(b) In liquids, E = B = Bulk modulus of elasticity

B
v ?
?

(c) In gases, according to Newton,
E = B
T
= Isothermal bulk modulus of elasticity= ?

p
v ?
?

But results did not match with this formula. Laplace made correction in it. According to him, E =
B
S
- Adiabatic bulk modulus of elasticity = ? ?

p RT kT
v
Mm
? ? ?
? ? ?
?

10.  Effect of Temperature, Pressure and Relative Humidity in Speed of Sound in Air (or in a Gas)
(i) With temperature v ? T
(ii) With pressure Pressure has no effect on speed of sound as long as temperature remains
constant.
(iii) With relative humidity With increase in relative humidity in air, density decreases. Hence,
speed of sound increases.
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