Question

A swimmer in the ocean observes one day that the ocean surface waves are periodic and resemble a sine wave. The swimmer estimates that the vertical distance between the crest and the trough of each wave is approximately 0.45 m, and the distance between each crest is approximately 1.8 m. The swimmer counts that 12 waves pass every two minutes. Determine the simple harmonic wave function that would describes these waves.

Answer 1

The wave function describing the ocean waves with a vertical distance (amplitude) of 0.45 meters between crest and trough, wavelength of 1.8 meters, and a frequency of 0.1 Hz is y(x, t) = 0.225 m sin(π/0.9 x - 0.2π t).

Step-by-step explanation:

The swimmer observed sinusoidal waves and estimated a vertical distance (amplitude) of 0.45 meters and a wavelength of 1.8 meters. When 12 waves pass by every two minutes, the frequency is 0.1 Hz. The amplitude, as half the crest-to-trough distance, is 0.225 meters. The wave function for these waves can be represented by y(x, t) = A sin(kx - ωt), where A is the amplitude, k is the wave number (2π divided by the wavelength), and ω is the angular frequency (2π times the frequency).

The step-by-step creation of the wave function is as follows:

Calculate the wave number (k): k = 2π / 1.8 m = π/0.9 m^-1.Calculate the angular frequency (ω): ω = 2π * 0.1 Hz = 0.2π s^-1.Write the wave function: y(x, t) = 0.225 m * sin(π/0.9 m^-1 * x - 0.2π s^-1 * t).

Thus, a swimmer would model these waves with the wave function y(x, t) = 0.225 m sin(π/0.9 x - 0.2π t).