Waves and A Fisherman on a boat

[Punkyc7]

a water wave is traveling in a straight line on a lake is described by the equation
y(x,t)=.0375mcos(.0045 m^-1 (x) + 5.4 s^-1 (t))

How much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor, and what is the horizontal distance does the wave crest travel in that time.

My question is about the second part of the question.

for the first part I got 2Pi/5.4=t

so for the horizontal distance is it just .0045*((2Pi)/5.4)

 

[ideasrule]

No, because 0.0045 m^-1 isn't the speed of the wave (and it doesn't have the right units).

0.0045 m^-1 is k and wavelength is 2*pi/k. Is it intuitive that one wavelength passes the fisherman in the time it takes a complete cycle to go past?

 

[Punkyc7]

So was my time right?

so if lambda = 2*pi/k

would that mean the distance would be 1396.263?

 

[ideasrule]

Yes.

 

[rwisz]

m

I would like to clarify and provide some additional information about the equation given for the water wave. The equation provided, y(x,t) = 0.0375m cos(0.0045 m^-1 x + 5.4 s^-1 t), is the equation for a cosine wave that describes the displacement of the surface of the water at a certain point (x) and time (t). The amplitude of the wave is 0.0375m and the frequency is 5.4 s^-1. The term 0.0045 m^-1 represents the wave number, which is related to the wavelength of the wave.

Now, to answer the first part of the question, you have correctly calculated the time it takes for one complete wave pattern to pass a fisherman at anchor. This is represented by 2π/5.4 = t, which gives us a time of approximately 1.16 seconds.

For the second part of the question, we need to find the horizontal distance traveled by the wave crest in that time. To do this, we can use the wave speed equation, v = λf, where v is the wave speed, λ is the wavelength, and f is the frequency. We know the frequency is 5.4 s^-1, and we can calculate the wavelength using the wave number, λ = 2π/k, where k is the wave number. So, we have λ = 2π/0.0045 = 1394.4 m. Now, using the wave speed equation, we can calculate the wave speed, which is approximately 7500 m/s. Finally, we can calculate the horizontal distance traveled by the wave crest using the wave speed and time, d = vt = (7500 m/s)(1.16 s) = 8700 m.

Therefore, in summary, for a fisherman at anchor, it takes approximately 1.16 seconds for one complete wave pattern to pass, and the wave crest travels a horizontal distance of approximately 8700 meters in that time. It is important to note that this is an idealized scenario and does not take into account factors such as wind and other disturbances that may affect the movement of the wave.