What is a progressive wave? Derive wave equation for a progressive wave.

Progressive Wave: A wave that travels from one region of medium to another region carrying energy in the form of crest and trough or compression and rarefaction is called the progressive wave. Both transverse and longitudinal waves are progressive waves.E.g., water waves, sound waves, lightwave, etc. are progressive waves. The motion of the progressive wave is given as below:

Equation of Progressive Wave: 

Let us consider a wave that is traveling from left to right as shown in the figure, the displacement of the vibrating particle in the medium is given by,

y = a sin ωt -------- (i)

Where a is amplitude, t is time, ω = 2πf is the angular frequency of the vibration and f is the frequency of vibration. If φ be the phase angle of the particle P at a distance x from O, then the displacement equation is given by 

y = a sin (ωt - φ) -------- (ii)

Since for a path difference, λ phase difference is \(\frac{2\mathrm\pi}\lambda x\)
i.e., \(\varphi=\frac{2\mathrm\pi}\lambda x\)
Now,

\(y=a\sin\left(\omega t-\frac{2\pi}\lambda x\right)\)

\(y=a\sin2\pi\left(\frac tT-\frac x\lambda\right)\)

\(y=a\sin\left(\frac{2\pi}Tt-\frac{2\pi}\lambda x\right)\)

\(y=a\sin\left(\omega t-kx\right)\) -------- (iii) [∵ \(k=\frac{2\pi}\lambda\) , a wave number or wave vector ]

If the wave is travelling from right to left, then the displacement of the particle is given by,

\(y=a\sin2\pi\left(\frac tT+\frac x\lambda\right)\) ---------- (iv)

These equations (iii) and (iv) are the plane progressive wave equations.