A water wave traveling in a straight line on a lake is described by the equation

Question

A water wave traveling in a straight line on a lake is described by the equation y(x, t) = (3.75cm)cos(.450cm^-1 x + 5.40s^-1 t) where y is the displacement perpendicular to the undisturbed surface of the lake.

a)How much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor?

b)What horizontal distance does the wave crest travel in that time?

c)What is the wave number?

d)What isthe number of waves per second that pass the fisherman?

e)How fast does a wave crest travel past the fisherman?

f)What is the maximum speed of his cork floater as the wave causes it to bob up and down?

Explanation / Answer

From the wave equation,

Amplitude(A) = 3.75 cm

Angular frequency() = 5.40 rad/s

Wavenumber(k) = 0.45 cm-1

(a) time taken for one complete wave pattern to go past a fisherman in a boat at anchor = 1/frequency

t = 2/ = 2/5.40 = 1.164 seconds

(b) Horizontal distance covered by wave in that time = Wavelength

= 2/wavenumber = 2/0.45 = 13.963 cm

(c) Wavenumber = k = 0.45 cm-1

(d) No of waves passing per second = Frequency of wave

f = /2 = 5.4/2 = 0.859 Hz = 0.859 vibrations per second

(e) Wave crest speed = Wavelegth/Time period

= Wavelength x frequency

= 13.963cm x 0.859Hz

= 11.994 cm/s

12 cm/s

(f) Max speed of cork floater = Max magnitude of y(x,t)/t i.e. particle velocity

Vparticle|max = Amplitude x Angular frequency

= A = 3.75cm x 5.4 rad/s

= 20.25 cm/s

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