A water wave traveling in a straight line on a lake is described by the equation y(x , t) = (2.75 cm) cos(0.410 rad/cm x + 6.20 rad/s 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, and what horizontal distance does the wave crest travel in that time? (b) What are the wave number and the number of waves per second that pass the fisherman? (c) How fast does a wave crest travel past the fisherman, and what is the maximum speed of his cork floater as the wave causes it to bob up and down?

Solution 10E Step 1 of 7: For the given water wave traveling in a straight line on a lake is described by the equation y(x , t) = (2.75 cm) cos(0.410 rad/cm x + 6.20 rad/s t) From the above equation, Amplitude, A= 2.75 cm 1 Wave number, k=0.410 cm 1 Angular frequency, =6.2 s Step 2 of 7: (a) How much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor, and what horizontal distance does the wave crest travel in that time To calculate the time taken for one complete wave pattern to go past a fisherman in a boat at anchor Using T= f where f is frequency Using f= T= 2 2 Substituting =6.2 s1 T= 2× 6.2 s T=1.012 sec Therefore, time taken to reach the fisherman is 1.012 s. Step 3 of 7: To calculate the horizontal distance the wave crest travel in that time Using = 2 k 1 2 Substituting k=0.410 cm = 0.410 cm = 15.31 cm Therefore, the wave crest travels 15.31 cm in that time. Step 4 of 7: (b) What are the wave number and the number of waves per second that pass the fisherman 1 Wave number, k=0.410 cm To calculate the number of waves per second that pass the fisherman, Using f= 2 1 Substituting =6.2 s 1 f= 2×3.14 f= 0.987 Hz Therefore, the number of waves that passes fishermen in one second is 0.987 Hz.