Ocean waves approaching the shore can be described as sinusoidal periodic waves with amplitude 4.4 m and wavelength 3.8 m.Two swimmers are in the ocean, with swimmer C 2.4 m further offshore than swimmer D. If at some instant in time swimmer C is at the crest of the wave, how far below swimmer C will swimmer D be at the same time?Give your answer in metres.
Question
Ocean waves approaching the shore can be described as sinusoidal periodic waves with amplitude 4.4 m and wavelength 3.8 m.Two swimmers are in the ocean, with swimmer C 2.4 m further offshore than swimmer D. If at some instant in time swimmer C is at the crest of the wave, how far below swimmer C will swimmer D be at the same time?Give your answer in metres.
Solution
To solve this problem, we need to understand the properties of a sinusoidal wave. The distance between two points in a wave is given by the wavelength. In this case, the wavelength is 3.8 m.
The amplitude of the wave, which is the maximum displacement from the equilibrium position (or the "height" of the wave), is 4.4 m. This means that the crest of the wave is 4.4 m above the equilibrium position, and the trough (the lowest point) is 4.4 m below the equilibrium position.
Swimmer C is at the crest of the wave, so he is 4.4 m above the equilibrium position. Swimmer D is 2.4 m further offshore.
Since the wavelength is 3.8 m, a distance of 2.4 m represents 2.4/3.8 = 0.63, or 63% of the wavelength.
In a sinusoidal wave, a displacement of 50% of the wavelength from the crest would put you at the equilibrium position (0 m displacement), and a displacement of 100% of the wavelength from the crest would put you at the next crest (4.4 m displacement).
Since 63% is more than 50% but less than 100%, Swimmer D is somewhere between the equilibrium position and the next crest.
To find the exact position, we can use the sine function, which describes the displacement in a sinusoidal wave. The sine of 0 degrees is 0 (corresponding to the equilibrium position), and the sine of 90 degrees is 1 (corresponding to the crest).
We need to find the angle that corresponds to a displacement of 63% of the wavelength. This angle is 0.63 * 360 degrees = 226.8 degrees.
The sine of 226.8 degrees is -0.809, which means that Swimmer D is -0.809 * 4.4 m = -3.56 m from the crest.
Since Swimmer C is at the crest, Swimmer D is 3.56 m below Swimmer C.
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