Harmonic Wave

Equation of a plane progressive harmonic wave

The continuous transfer of disturbance in the same medium along the same direction is called progressive wave motion. All the particles of the medium vibrate with the same amplitude and frequency. However the particles lag behind by certain phase difference than their preceeding particles. No particle is permanently at rest but vibrate with different phase. Consider a wave is travelling along the x-axis. Let \(a\) be the amplitude of vibration and \(y\) be the displacement of particles after time \(t\). Then, the equaation of motion of a particle at \(O\) is \(y=a \sin \omega t\) where \(\omega=2 \pi f \) is the angular velocity.Let, \(P\) be a particle at a distance \(x\) from the origin and the wave be travelling with velocity \(v\). then the particle \(P\) will start vibrating \(x/v\) seconds after the particle at \(O\) starts vibrating. then, if \(\phi\) be the phase difference between the two particles, the equation of motion of particle \(P\) is given by \(y=a\sin (2 \pi ft-\phi)\). We know, for path difference of \(\lambda\), phase difference is \(2\pi\). So, for path difference of \(x\), phase difference is \(2\pi /\lambda\), where, x is the distance between the particles at \(O\) and \(P\).\begin{align*} \therefore \phi=\frac{2\pi x}{\lambda} \end{align*}\begin{align*} and, y=a \sin(2\pi ft-\frac{2\pi x}{\lambda}) \end{align*}\begin{align*} or, y=a \sin 2 \pi (ft-x/\lambda) \end{align*}\begin{align*} or, y=a\sin2\pi(vt/\lambda-x/\lambda) \end{align*}\begin{align*} \therefore y=asin2\pi/\lambda(vt-x)-----1 \end{align*} This is the general equation of plane progressive wave of amplitude \(a\) propagating with velocity \(v\) along positive direction. For wave travelling in negative x- direction, the equation of wave is \(y=a\sin2\pi/\lambda(vt+x)\).

Particle velocity and wave velocity

Differentiating the equation of plane progressive wave motion with respect to t, we get \(particle \space velocity, u=\frac{\mathrm{dy} }{\mathrm{d} t}=\frac{2\pi v}{\lambda} a cos \frac{2\pi}{\lambda}(vt-x)---2\). If we differentiate the equation of pane progressive wave motion with respect to x, we get the slope of the displacement curve, also called strain or compression as \(\frac{\mathrm{dy} }{\mathrm{d} x}=-\frac{2\pi}{\lambda} a cos \frac{2\pi}{\lambda}(vt-x)---3\). From the above two equations, we obtain\begin{align*} u=\frac{\mathrm{dy} }{\mathrm{d} t}=-v\frac{\mathrm{dy} }{\mathrm{d} x}---4 \end{align*} Again, differentiating equation 2 with respect to t, we get, \begin{align*} \frac{\mathrm{d^2y} }{\mathrm{d} t^2}=-(\frac{2 \pi v}{\lambda})^2a \sin \frac{2\pi}{\lambda}(vt-x)---5\end{align*} Also, differentiating equation 3 with respect to x, we get,\begin{align*} \frac{\mathrm{d^2y} }{\mathrm{d} t^2}=-(\frac{2 \pi }{\lambda})^2a \sin \frac{2\pi}{\lambda}(vt-x)---6\end{align*} Comparing equations 5 and 6 we get,\begin{align*} \frac{\mathrm{d^2y} }{\mathrm{d} t^2}=v^2\frac{\mathrm{d^2y} }{\mathrm{d} x^2}---7\end{align*} Equation 7 is the differential equation of one dimensional progressive wave. The particle velocity \(u\) is maximum at mean position and minimum at extreme position however the wave velocity \(v\) is constant in a given medium.

References

Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.

Feynman, Richard P. The Feynman Lectures on Physics Volume 1. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.

Mathur, D S. Mechanics. New Delhi: S. Chand & Company Pvt. Ltd., 2015.

Young, Hugh D, Roger A Freedman and A Lewis Ford. University Physics. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.