1© Manhattan Press (H.K.) Ltd. 8.4 Progressive wave equation.

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1© Manhattan Press (H.K.) Ltd. 8.4 Progressive wave equation

2 © Manhattan Press (H.K.) Ltd. Classification of waves 8.4 Progressive wave equation (SB p. 12) For a progressive wave, energy is transferred from the source and propagated outwards. The transverse waves and longitudinal waves are classified as progressive (or travelling) waves.

3 © Manhattan Press (H.K.) Ltd. Progressive wave equation 8.4 Progressive wave equation (SB p. 13) y = a sin  t This is equivalent to a SHM exhibiting along the y-axis. Go to More to Know 5 More to Know 5

4 © Manhattan Press (H.K.) Ltd. 8.4 Progressive wave equation (SB p. 13) The displacement y of particle P which is at a distance x from O can be written as: Progressive wave equation y = a sin(  t   ) y = a sin(  t  )

5 © Manhattan Press (H.K.) Ltd. 8.4 Progressive wave equation (SB p. 13) If a progressive wave moves from right to left, the oscillation of particle P leads that of particle O. Progressive wave equation y = a sin(  t + ) Go to Example 5 Example 5

6 © Manhattan Press (H.K.) Ltd. End

7 © Manhattan Press (H.K.) Ltd. Tsunami Tsunami (or giant tidal waves) has extremely long wavelength and can travel at hundreds of kilometres per hour. Because these giant waves are produced by earthquakes and volcanic eruptions under the ocean, they move deep water and thus transfer enormous amount of energy. Return to Text 8.4 Progressive wave equation (SB p. 12)

8 © Manhattan Press (H.K.) Ltd. Q: Q: A progressive wave is represented by the equation y = 0.1 sin(200πt  ) where x and y are in metres, t in seconds. (a) Find (i) the frequency (f ), (ii) the wavelength (λ), and (iii) the speed (v) of the wave. 8.4 Progressive wave equation (SB p. 14)

9 © Manhattan Press (H.K.) Ltd. Q: Q: (b) What is the phase difference between a point 0.25 m from O and another point 1.00 m from O? (c) Write down the wave equation for a progressive wave having twice the amplitude, twice the frequency and moving in the opposite direction in the same medium. Solution 8.4 Progressive wave equation (SB p. 14)

10 © Manhattan Press (H.K.) Ltd. Solution: Compare the equation y = 0.1 sin (200  t  ) with the wave equation y = a sin (  t  ). (a) (i)  = 200  f = = = 100 Hz (ii) =  = = 1.7 m 8.4 Progressive wave equation (SB p. 14)

11 © Manhattan Press (H.K.) Ltd. Solution (cont’d): (iii) v = f = 100  1.7 = 170 m s  1 (b) Distance between the two points = (1.0  0.25) m = 0.75 m One full wavelength corresponds to a phase difference of 2  radians.  A distance of 0.75 m corresponds to a phase difference = = = 2.77 rad 8.4 Progressive wave equation (SB p. 14)

12 © Manhattan Press (H.K.) Ltd. Solution (cont’d): (c) Amplitude of first wave = 0.1 m  Amplitude of second wave = 0.2 m Angular frequency,  = 2  f = 400  Wavelength, = = = = = Therefore, the required wave equation is: y = a sin(  t + ) = 0.2 sin(400  t + ) Return to Text 8.4 Progressive wave equation (SB p. 15)