Longitudinal Wave and Progressive Wave
Longitudinal Wave
If the vibrating particles of the medium oscillate to the same direction of propagation of the wave is called a longitudinal wave. Example: sound wave travelling in air. A longitudinal wave can travel in solid, liquid and gas.
Propagation of Longitudinal Wave
To understand the propagation of longitudinal waves in a medium considers nine particles named 1, 2, 3, 4, 5, 6, 7, 8, 9 of the medium lying at equal distances at their mean positions. The wave travels from left to right and the particles vibrate about their mean positions. After T/8 seconds, the particle 1 goes to the right and completes 1/8^{th} of its vibration. The disturbance reaches to the particle 2. After T/4 seconds, the particle 1 has reached its extreme right position and completes 1/4^{th} of its vibration. The disturbance reaches to the particle 3. The process continues. The waves reach to particle 9. Here 1 and 9 are again in the same phase. Here particles 1, 5 and 9 are at their mean positions. The particles 1 and 3 are close to the particle 2. This is the position of condensation. Similarly, particles 9 and 8 are close to the particle 7. This is also the position of condensation or compression. On the other hand, particles 4 and 6 are far away from the particle 5. This is the position of rarefaction. Hence in a longitudinal wave motion, condensations (compressions) and rarefactions are alternately formed.
Properties of Longitudinal Waves
 The particles of the medium vibrate simple harmonically along the direction of propagation of the wave.
 All particles have same amplitude, frequency and period.
 There is a gradual change in phase between the successive particles.
 The velocity of each particle is maximum at its mean position and zero at extreme points.
 When the particles move in the same direction as the propagation of the wave, it is in the region of compression but when they move in the opposite direction to the direction of propagation of the wave, it is in the region of rarefaction.
 When the particle is at the mean position, it is a region of maximum compressions or rarefaction.
 All particles vibrating in phase will be at a distance equal to nλ, where n = 1, 2, 3, 4, 5, etc. It means minimum distance between two vibrating particles in phase is equal to one wavelength.
 Due to repeated periodic motion of the particles, compressions and rarefactions are produced alternately.
Comparison between Transverse and Longitudinal Waves..
S.N.  Transverse Wave  Longitudinal Wave 
1.  Particles of the medium vibrate perpendicular to the direction of wave propagation.  Particles of the medium vibrate in the same direction to the wave propagation. 
2.  Alternate crest and trough.  Alternate compression and rarefaction. 
3.  It can travel in solid and surface of a liquid medium.  It can travel through solid, liquid and gas medium. 
4.  It can travel in a vacuum.  It cannot travel through a vacuum. 
5.  There are pressure and density variation. Examples: waves in the surface of liquid, wave in solid.  There pressure and density become maximum at compression and minimum at rarefaction. Examples: sound wave travelling in air. 
Progressive Wave
If the wave travels from one region to another region is called a progressive wave. Transverse wave and longitudinal wave are both progressive wave.
Consider a progressive wave (transverse wave ) is travelling on a medium. Ï´ be the point of wave starting. At any time ‘t’, the displacement of the wave at point ‘O’ is given by equation,
$$ y = a\sin \omega t \dots (i) $$
\begin{align*} \text {Where, y} & = \text {displacement of wave} \\ a &= \text {amplitude of wave} \\ \omega &= \text {angular velocity of wave} \\ t &= \text {time taken by wave} \\ \end{align*}
The disturbance travels later at point P than point O. So, particles at point P vibrates simple harmonically after certain time. Let particle ‘P’ is at distance x from the point o. the distance travelled by the wave in one complete oscillate is equal to λ. i.e λ = wavelength of the wave
\begin{align*} \text {For displacement,} \lambda \: \text {Phase angle} = 2\pi \\ \text {1 Phase angle} = \frac {2\pi}{\lambda } \\ \text {x phase angle} = \frac {2\pi }{\lambda } . x \\ \end{align*}
\begin{align*}\text {when the wave reaches at point P then the equation of wave is} \\ y = a\sin (\omega t  \phi ) \\ \phi = \text {phase angle} = \frac {2\pi }{\lambda } . x \\ \therefore y &= a \sin (\omega t  \frac {2\pi }{\lambda } . x ) \dots (ii) \\ \text {or,} \: y &= a\sin (\omega t – k.x) \dots (iii) \\ \end{align*}
\begin{align*} \text {Where,} \: \frac {2\pi }{\lambda } \\ \text {is the propagation constant, it is also called wave number.} \\ \text {since,} \: \omega &= \frac {2\pi }{T} \\ \text {then,} \: y = a\sin \left (\frac {2\pi }{T} . t  \frac {2\pi }{\lambda } . x\right ) \\ y &= a\sin 2\pi(\frac tT  \frac {x}{\lambda }) \dots (iv) \\ \text {Since,} \: \omega = 2\pi f = 2\pi \frac {v}{\lambda } \: \text {Then, from equation} \: (ii), \text {we can write,} \\ y &= a\sin\left (2\pi .\frac {v}{\lambda } . t  \frac {2\pi }{\lambda } \right ). x\\ y &= a \sin \frac {2\pi }{\lambda } (vt – x) \dots (v) \\ \end{align*}
\begin{align*} \text {Equations} \: (ii) , (iii) , (iv) \text {and} \: (v) \\ \text {are the general equation for progressive wave.} \\ \text {If the progressive wave travels from right to left direction.}\\ \text {Then,} \\ y &= a \sin \frac {2\pi }{\lambda } (vt + x) \dots (vi) \\ \end{align*}
Differential Equation of Wave Motion
\begin{align*} \text {The equation of wave is} \\ y &= a\sin \frac {2\pi }{\lambda } (vt – x) \dots (i) \\ \text {Differentiating equation} \: (i) \: \text {with respect to t, we get} \\ \frac {dy}{dt} &= \frac {2\pi v}{\lambda } a \cos \frac {2\pi }{\lambda }(vt – x) \\ \text {and again differentiating,} \\ \frac {d^2y}{dt^2} &= \frac {4\pi ^2 v^2}{\lambda ^2} a \sin \frac {2\pi }{\lambda } (vt – x) \dots (ii) \\ \text {When the equation} \: (i) \: \text {is differentiated with respect to x, we get} \\ \frac {dy}{dx} &= \frac {2\pi }{\lambda } a \cos \frac {2\pi }{\lambda }(vt – x) \\ \frac {d^2y}{dt^2} &= \frac {4\pi ^2 }{\lambda ^2} a \sin \frac {2\pi }{\lambda } (vt – x) \dots (iii) \\ \text {From equation} \: (ii) \: \text {and equation} \: (iii), \: \text {we have} \\ \frac {d^2y}{dt^2} = v^2 \frac {d^2y}{dx^2} \dots (iv) \\ \text {which is the differential wave equation.} \\ \end{align*}
Reference
Manu Kumar Khatry, Manoj Kumar Thapa, et al. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.
S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.
If the vibrating particles of the medium oscillate to the same direction of propagation of the wave is called a longitudinal wave.
If the wave travels from one region to another region is called a progressive wave.
Transverse wave and longitudinal wave are both progressive wave.

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Maniraj thapaderivation of progressive wave 
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