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

© Manhattan Press (H.K.) Ltd. 1

8.4 8.4

Progressive wave Progressive wave equationequation


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 The transverse waves and longitudinal waves are classified as progressive (or waves are classified as progressive (or travelling) waves.travelling) waves.


Progressive wave equation


y = a sint

This is equivalent to a SHM This is equivalent to a SHM exhibiting along the exhibiting along the yy-axis.-axis.

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More to Know 5More to Know 5



The displacement The displacement yy of particle of particle PP which is which is at a distance at a distance xx from from OO can be written as: can be written as:


y = a sin(t )

y = a sin(t )



If a progressive wave moves from right If a progressive wave moves from right to left, the oscillation of particle to left, the oscillation of particle PP leads leads that of particle that of particle OO..


y = a sin(t + )Go to

Example 5Example 5


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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.

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Q: Q: A progressive wave is represented by the

equationy = 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.

1720 x



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



Solution:Solution:

Compare the equation y = 0.1 sin (200t ) with the wave

equation y = a sin (t ).

(a) (i) = 200

f =

=

= 100 Hz

(ii) =

= = 1.7 m

1720 x

x2

2

2200

x2

101717

20 x



Solution (cont’d):Solution (cont’d):

(iii) v = f

= 100 1.7

= 170 m s1

(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

750 2 .

750 71

2 ..



Solution (cont’d):Solution (cont’d):

(c) Amplitude of first wave = 0.1 m

Amplitude of second wave = 0.2 m

Angular frequency, = 2f = 400

Wavelength, = = =

= =

Therefore, the required wave equation is:

y = a sin(t + )

= 0.2 sin(400t + )

fv

200170

2017

x2 2017

2 x17

40 x

x2

1740 x

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