For the harmonic travelling wave y = 2 cos 2π (10t–0.0080x + 3.5) where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of,

(a) 4 m

(b) 0.5 m

(c)

(d) (at a given instant of time)

(e) What is the phase difference between the oscillation of a particle located at x = 100cm, at t = T s and t = 5 s?

Given:

The equation of the wave y = 2cos[2π(10t-0.0080x+3.5)]

(a) For the wave separated by a distance of 4 m the equation of vibration will be,

or

and the phase

Comparing this equation to the original equation

and the phase

the phase difference therefore

(b) For a separation of 0.5 m, proceeding as above,

and the phase

Comparing this equation to the original equation its phase

the phase difference therefore

(c) The wavelength of the equation can be found by comparing the equation to the general equation for the wave,

we have, k = 0.016π and ω =20π. We know that k is related to the wavelength as

where λ is the wavelength. Therefore for k = 0.016π we have

For a separation of or 0.625 m,

and the phase

Comparing this equation to the original equation its phase

the phase difference therefore

(d) For a separation of or 0.938 m, proceeding as above

and the phase

Comparing this equation to the original equation its phase

the phase difference therefore

(e) At x = 100 cm the equation of the wave becomes,

We know that ω is and time period T are related as

but ω = 20π,

At t = T= 0.1 s the equation of the wave is given by,

and the phase

At t = 5 s the equation of the wave is given by,

and the phase

the phase difference therefore