Wave Equation

What is Harmonic Wave Equation?

Propagation of a disturbance ( a wave ) which travel through a medium due to vibrations or oscillations of medium particles about their mean position is called a wave motion. The waves, which are periodic in nature are called Harmonic Waves. Characteristics of wave motion can be expressed in the form of wave equation.

An equation representing the characteristics of a wave motion is called a Harmonic Wave Equation.

A general form of Harmonic Wave Equation can be written as –

y = A \sin ( \omega t - k x )

A wave is called a harmonic wave when the medium particles vibrate simple harmonically about their mean position. Equation which represents the characteristics of wave is called a wave equation.

Distance-Displacement Equation

Distance-displacement wave equation for a wave is derived from Distance-displacement curve. Consider about displacement-distance curve of a wave motion as shown in figure.

Suppose a plane progressive harmonic wave starts from the origin O and travels along the positive direction of X axis with wave velocity ( v ) .

The displacement of the particle at origin O i.e. at distance ( x = 0 ) at any instant ( t )  is given by –

y ( 0, t ) = A \sin \omega t

Now consider about a particle at point P at a distance ( x ) from O . The wave disturbance will reach at point P in time \left ( \frac {x}{v} \right )  seconds. This means that, the particle at point P will start vibrating after \left ( \frac {x}{v} \right )  seconds of the time at which particle at O had started vibrating.

Therefore, displacement of particle at point P at an instant ( t ) is equal to the displacement of the particle at O at time \left ( t - \frac {x}{v} \right )  seconds.

Displacement of particle at point P at any time ( t ) will be –

y ( x, t ) = A \sin \omega \left ( t - \frac { x }{ v } \right )

= A \sin \left ( \omega t - \frac { \omega }{ v } x \right ) ……….. (1)

This is the most common form of a wave equation called distance-displacement wave equation.

Distance-Wave number Equation

From distance-displacement wave equation, we will get –

y ( x, t ) = A \sin \left ( \omega t - \frac { \omega }{ v } x \right )

But, \quad \left ( \frac { \omega }{ v } \right ) = \left ( \frac { 2 \pi \nu }{ v } \right )

= \left ( \frac { 2 \pi }{ \lambda } \right ) = k ( A constant)

Quantity ( k ) is called angular wave number or propagation constant. Therefore, propagation constant of a wave is given by –

k = \left ( \frac { 2 \pi }{ \lambda } \right ) = \left ( \frac { \omega }{ v } \right )

Hence, wave equation is represented as –

y ( x, t ) = A \sin \left ( \omega t - k x \right ) ……… (2)

This is another form of a wave equation called distance-wave number wave equation.

Distance-Wavelength Equation

For a harmonic wave, we know that –

\omega = \left ( \frac { 2 \pi }{ T } \right )

Hence, rewriting wave equation (2), we will get –

y ( x, t ) = A \sin \left [ \left ( \frac { 2 \pi }{ T } \right ) t - \left ( \frac { 2 \pi }{ \lambda } \right ) x \right ]

= A \sin 2 \pi \left [ \left ( \frac { t }{ T } \right ) - \left ( \frac { x }{ \lambda } \right ) \right ]

= A \sin \left ( \frac { 2 \pi }{ T } \right ) \left [ t - \left ( \frac { x }{ \lambda } \right ) T \right ] ………. (3)

Distance-Velocity Equation

Velocity of wave motion is given by –

v = \left ( \frac { \lambda }{ T } \right )

Rewriting equation (3), we will get –

y ( x, t ) = A \sin \left ( \frac { 2 \pi }{ T } \right ) \left [ t - \left ( \frac { x }{ v } \right ) \right ]

= A \sin \left ( \frac { 2 \pi }{ T } \right ) \left ( \frac {1}{v} \right ) \left ( v t - x \right )

= A \sin \left ( \frac { 2 \pi }{ \lambda } \right ) \left ( v t - x \right ) ……… (4)

Equations (1), (2), (3) and (4) are different forms of wave equation which are used depending upon the given conditions.

TO BE NOTED –

1. If initial phase of particle at O is ( \phi _ 0 ) and the wave is moving in positive x direction, then general wave equation will be y ( x, t ) = A \sin ( \omega t - k x + \phi _ 0 ) ……… (6)
2. If initial phase of the particle at O is ( \phi _ 0 ) and the wave is moving in negative x direction, then general wave equation will be y ( x, t ) = A \sin ( \omega t + k x + \phi _ 0 ) ……… (7)

General Wave Equation

In wave motion, displacement ( y ) of particle can be defined in terms of position of particle ( x ) and time ( t ) .

Hence, displacement ( y )  is a function of ( x ) and ( t ) .

y = f ( x, t )

Expression [ y = f ( x, t ) ] will represent an wave equation, only if following two conditions are satisfied –

1. If \left [ \frac { \delta ^ 2 y }{ \delta t ^ 2 } \right ] = K \left [ \frac { \delta ^ 2 y }{ \delta x ^ 2 } \right ] \quad Where \quad K \ne 0
2. If displacement ( y ) is defined for all possible values of ( x ) and ( t ) .

EXAMPLE –

Clear understanding of the above conditions will be met by following example. Consider that a wave is propagating in positive ( x )  direction and the medium particles are oscillating with simple harmonic motion.

Then general equation of this wave can be written as –

y = A \sin ( \omega t - k x ) …….. (i)

By partial differentiation of equation (i) with respect to ( t ) , we will get –

\left ( \frac { \delta y }{ \delta t } \right ) = \omega A \cos ( \omega t - k x )

Differentiating once again, we will get –

\left ( \frac { \delta ^ 2 y }{ \delta t ^ 2 } \right ) = - \omega ^ 2 A \sin ( \omega t - k x ) …….. (ii)

By partial differentiation of equation (i) with respect to ( x ) , we will get –

\left ( \frac { \delta y }{ \delta x } \right ) = - k A \cos ( \omega t - k x )

Differentiating once again, we will get –

\left ( \frac { \delta ^ 2 y }{ \delta x ^ 2 } \right ) = - k ^ 2 A \sin ( \omega t - k x ) …….. (iii)

Dividing equation (ii) by equation (iii), we will get –

\left [ \frac { \delta ^ 2 y }{ \delta t ^ 2 } \right ] / \left [ \frac { \delta ^ 2 y }{ \delta x ^ 2 } \right ] = \left ( \frac { \omega ^ 2 }{ k ^ 2 } \right ) = K ^ 2

Where, \left ( \frac { \omega }{ k } \right ) = K (A constant)

Therefore, \quad \left [ \frac { \delta ^ 2 y }{ \delta t ^ 2 } \right ] = K \left [ \frac { \delta ^ 2 y }{ \delta x ^ 2 } \right ]

It is satisfying the given condition of wave equation.

Hence, the equation (i) is representing a wave equation in which the medium particles are executing simple harmonic motion.

Wave Velocity

General form of equation for harmonic wave is –

y = A \sin ( \omega t - k x )

For this type of wave equation –

K^2 = \left ( \frac { \omega ^ 2 }{ k ^ 2 } \right ) = \left ( \frac { \omega }{ k } \right ) ^ 2

= \left ( \frac { 2 \pi \nu }{ k } \right ) ^ 2

=  \left ( \nu \frac { 2 \pi }{ k } \right ) ^ 2 = \left ( \nu \lambda \right ) ^ 2 = v ^ 2

Therefore, \quad v^2 = \left ( \frac { \omega ^ 2 }{ k ^ 2 } \right )

Or, \quad v = \left ( \frac { \omega }{ k } \right )

Therefore, wave velocity \quad v = \left [ \frac {\text {Coefficient of t}}{\text {Coefficient of x}} \right ]

This relation can be remembered as a thumb rule for solutions of numerical problems related to wave motion.

Generic form of Wave Equation

General form of equation for harmonic wave is –

y = A \sin ( \omega t - k x )

This equation can be represented in a more generic form as –

y = f ( a x + b t )

In this equation, following two conditions must be satisfied –

1. Coefficients ( a ) and ( b ) are non zero constants.
2. Displacement ( y ) must be defined for every values of ( x ) and ( t ) .

Therefore, wave velocity will be –

v = \left ( \frac {b}{a} \right ) = Ratio of coefficient of time ( t ) and coefficient of displacement ( x ) This type of equation is more easier to remember.