Wave

Wave is the disturbance( means a change in pressure, density or displacement of the particles of the medium about their equilibrium position) produced in the particles of the medium when an energy transfers through the medium.

Transverse wave
If the particle of the medium vibrate along the direction perpendicular to the direction of wave motionm then this tyoe if wave is called a transverse wave.
Longitudinal wave.
If the particle of the medium vibrate along the direction parallel to the direction of wave motion then this type of wave is called longitudinal wave.

Progressive wave
A wave in which the crest and trough( in transvers) or compression and refraction( in lonbgitudinal) travel is called progressive. They vary continiously.

Equation of Progressive wave
Let us consider a progressive wave originating from O travels along the positive x-axis. Let is have frequency f, wavelength λ amplitude a angular velocity V
Since the motion in the medium is simple harmonic motion, the displacement of the particle be given as,
y = a sinwt-----------1
Let us consider a point p at a distance x, from origin O where the wave lags. Let wave lags at P(i.e phase difference) by φ. Then the displacement equation will be
y = asin(wt- φ)-------------2
We know that at distance λ phase difference will be 2 so that the phase difference at distance x will be
φ = 2 x/ λ
equation 2 becomes
y = asin(wt-2 x/ λ)---------------------3
Again,
y = asin(wt-kx)---------------------4
Where k = 2 / λ = wave number
Again using w = 2 / λ we get
y = asin(2 / λ(vt-x))-------------5
These equation 3 4 and 5 are equation of progressive wave.

Particle velocity
The displacement of particle per unit time is called particle velocity we have the displacement of particle when a wave is travelling alon x-axis as
y = asin(wt-kx)---------------------1
Diffrentiating the equation wrt time we get
dy/dx = acos(wt-kx)w
dy/dt = awcos(wt-kx)------------2
Now
Differentiating equation wrt x we get
dy/dx = -akcos(wt-k)-----------3
we have w = 2(pi)f = 2(pi)V/ λ = kV
Equation 2 becomes
dy/dt = akvcos(wt-kx)-------------4
Now from equation 3 and 4 becomes
dy/dt = -v^2dy/dx-------------5
partical velocity = -(wave velocity) * slope of displacement curve at that point