__What is Harmonic Wave Equation?__

Propagation of a disturbance ( a wave ) which travel through a medium due to vibrations or *oscillation*s 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 –**

- 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)
- 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 –

- 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
- 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 –

- Coefficients ( a ) and ( b ) are non zero constants.
- 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.