This problem is killing me - literally

[KudosDude]

Duck season has arrived. A Harlequin duck (Histrionicus histrionicus) is bobbing vertically up and down due to a passing water wave. The duck bobs through a total vertical distance of 10.0 cm and completes 7.03 cycles during 10.4 s according to your Mickey Mouse watch. You also note that a particular crest of the wave travels the 8.99 m from the duck to your canoe during 5.05 s.

(a) What is the frequency of the water wave?
Hz
(b) What is the speed of the wave?
m/s
(c) What is the wavelength of the waves?
m
(d) Write a mathematical expression for the wavefunction (x, t) assuming it is a sinusoidal wave with the duck located at the origin for x and at a peak of the wave when t = 0 s.
(x, t) = cos( x - t)


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Please explain as well, don't just calculate. Thanks.

 

[mikelepore]

You are given so many cycles in so many seconds. What's that number in units of cycles per second?

You are given so many meters in so many seconds. What's that number in units of meters per second?

What vocabulary words are the names of the above variables?

What formula relates them to the wavelength?

 

[Moetasim]

I can understand your frustration with this problem. Let me break it down for you and provide a response to each question.

(a) The frequency of a wave is the number of complete cycles it completes in one second. In this case, the duck completes 7.03 cycles in 10.4 seconds. So, the frequency of the water wave is calculated as 7.03 cycles / 10.4 seconds = 0.676 Hz.

(b) The speed of a wave is the distance it travels in a given time. In this case, the crest of the wave travels 8.99 meters in 5.05 seconds. So, the speed of the wave is calculated as 8.99 meters / 5.05 seconds = 1.78 m/s.

(c) The wavelength of a wave is the distance between two consecutive peaks or troughs. In this case, we can use the formula wavelength = speed / frequency. So, the wavelength of the waves is calculated as 1.78 m/s / 0.676 Hz = 2.63 m.

(d) The wavefunction is a mathematical representation of the wave. In this case, assuming the wave is a sinusoidal wave, we can write the wavefunction as (x, t) = A * cos(kx - ωt), where A is the amplitude, k is the wave number (2π/λ), and ω is the angular frequency (2πf). We know the frequency (f = 0.676 Hz) and the wavelength (λ = 2.63 m) from our previous calculations. So, the wavefunction can be written as (x, t) = A * cos(2π/2.63 m * x - 2π * 0.676 Hz * t).

I hope this helps to clarify the problem for you. Remember, as a scientist, it is important to not just calculate but also understand the concepts behind the calculations. Keep exploring and learning!