Mathematical derivation of wave Equation

As it is well known that if a medium is disturbed from its equilibrium position it may form a wave that may travel from one point to another. In geophysics the term wave forms the basis in many techniques such as seismic refraction and reflection, Electromagnetic techniques such as GPR. In Earthquake seismology seismic waves are responsible for transferring energy from one point to another. Primary when describing a wave there are basic physical properties used such as wave speed, Amplitude, Period just to mention a few. However, when we want to describe in more detail the position and motion of individual particles at particular time during wave propagation, we have to consider the term Wave function. 

You see! Today through this post I want you to know the basic mathematics description  behind  a wave. Let's go.

I will be more specific in describing this by using the transverse wave (string wave), since it is easier to carry it out as a simple experiment.

Basic periodic wave Terminologies and equation 

Amplitude (A): Is the maximum displacement of medium particles from their rest (equilibrium) mean position

Wave length: Is a distance between two successive crests or trough. It is denoted by the symbol Lambda (λ). Its SI unit is metre (m)

Frequency (f): Is the number of oscillations (cycles) completed per unit time. It represented by f and It's SI unit is Hertz (Hz)

f = No. Cycles/time 

f = 1/T

f = 2π/T

Also f = w/2π where W - Angular Velocity.

Periodic time (T): Is the time taken by a wave to complete one cycle.

T = 1/f

wave velocity (V): Is the rate of displacement moved by a wave.

V = X (y)/T where X - distance along X -  direction, y - distance along y - direction

Since displacement can be regarded as wavelength (λ)

V =  λ/T

But f = I/T 

Then V = λf......................(1)

This is the basic Periodic wave Equation.

WAve Number (k) : Is the reciprocal of wavelength ( λ) of a wave.

K = 1/λ, if we consider I cycle = 2π

Then the equation above became

K = 2π/λ

Then λ = 1/k and put in eqn 1 above

V = f/k

Also since w = 2πf, and f = V/λ

Then w = 2πV/λ

Then w = kV..........................(2)

It is another form of periodic wave Equation

Wave function : Is the function that describes the position (X) of any particle in medium at a given time ( t ).

Such as y = y ( X, t). The right part of this expression  is a Wave function.

I assume you have a basic idea on partial derivation, since our function has two variables (t and X), in order to see the effect of one variable we should differentiate while keeping constant (fixed) another variable, this is how partial derivative works.

Since wave on a string obey harmonic function with sinusoidal shape , their displacement can be given as

X = A coswt 

Then wave function ,  y (X, t) = A cos (wt + Q) 

Where Q - phase angle

At X= 0, y ( X = 0 , t) = A cos wt

While w = 2πf

y ( X = 0, t) = A cos 2πft........................(3)

We have to consider two cases such as below

CASE 1: For sinusoidal wave moving in + X - direction (right)

Displacement at time t, is in opposite as the motion of point X = 0

Then t = X/V will decrease  in eqn 3

y ( X, t) = A cos [2πf (t - X/V)]

Then y (X , t) = A cos 2πf (X/V - t).......................(4)

The equation above is sinusoidal wave moving in + X - direction

Other forms of equation 4

Since T = 1/f, and λ = V/f

Then, y ( X ,t) = A cos 2π ( X/λ - 1/T)..................(5)

Also from eqn 2, w = kV

y ( X, t) =A cos [kX - wt]....................(6)

CASE 2: When sinusoidal wave is Moving in - X - direction 

Here the displacement at time t, is the same as the motion of point X = 0.

Then , t = t + X/V

Then , y (X, t) = A cos 2πf (X/V + t)..........................(7)

It is a sinusoidal wave Moving in - X , direction.

Other forms will be the same as those in + X direction, only sign conversion is essential to distinguish them.

y ( X ,t) = A cos 2π ( X/λ + 1/T).............................(8)

y ( X, t) = A cos 2π (kX + wt)...........................(9)

Then general equation for sinusoidal wave moving in either ( ± ) x direction

y ( X, t) = A cos 2π ( kX ± wt)............................(10)

Since kX -  wt = constant (Phase, In radian)

Take dkX/dt - dwt/dt = d (constant)/dt

KdX/dt - w = 0

dX/dt = w/k

dX/dt = Phase Velocity.

Phase Velocity = w/k.......................(11)

Particles Velocity and Acceleration

Consider the equation 6, then find first Partial derivative to obtain the Transverse Velocity

y ( X, t) = A cos 2π ( kX - wt)

Vy ( X, t) =  ∂y ( x, t)/ ∂t = wA sin (kX - wt).............(12)

Take the second Partial derivative of eqn 6 to obtain the acceleration.

ay ( X, t) =  ∂2y (X, t)/ ∂t2  = - w2 Acos (kX - wt) 

Then ay (X, t) = - w2 y ( X, t).......................(13)

From equation 6, we can obtain the second Partial derivative (w.r.t X) as the curvature of the wave train.

 ∂y ( X, t)/ ∂X = Slope of the curve

 ∂2y(X,t)/∂X2 = - k2Acos (kX - wt)=- k2y (X,t).............(14)

You see! Equations 13 and 14, seem to be alike, divide these two equations such as equation 13/14.

 ( ∂2y (X, t)/∂t2)/ ( ∂2 y (X, t)/ ∂X2 ) = w2 / k2 = V2

Then, The equation 15  below is the Linear wave equation

 ∂2 y (X, t)/ ∂X2 = (1/V2) ( ∂2y (X, t)/ ∂t2).............(15)

All in all we have to be specific that the Linear wave equation above is a general expression used to describe some waves such as sound waves, string waves that propagate in non - dispersive Media. Also it can be used to derive Parameters related to Electromagnetic waves.

Thanks for your attention!

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