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 0.25 m, and the distance between each crest is approximately 1.1 m. The swimmer counts that 13 waves pass every two minutes. Determine the simple harmonic wave function that would describes these waves.

Answer 1

The simple harmonic wave function is
`y(x,t) = 0.125 sin (5.712 \ m^(-1) \ x - 0.6808 \ s^(-1)\ t )`

Step-by-step explanation:

Generally a sine wave is mathematically represented as

`y(x,t) =  A sin (k x - w t )`

Here A is the amplitude which is mathematically represented as

`A =  (Z)/(2)`

substituting 0.25 m for Z we have that

`A =  (0.25)/(2)`

`A = 0.125`

k is the wave number which is mathematically represented as

`k =  (2 \pi)/(\lambda)`

substituting 1.1 m for (wavelength ) we have

`k =  (2* 3,142)/(1.1)`

=>
`k =  5.712`

w is the angular frequency which is mathematically represented as

`w =  (2 \pi)/(T)`

Here T is the period which is mathematically represented as

`T =  \frac {t}{n}`

substituting 13 wave pass for n and
`t = 2 \ minutes =  120 \  s` for t

`T =  \frac {120}{13}`

`T =  9.230`

So

`w =  (2 * 3.142 )/( 9.230)`

`w =  0.6808 \  s^(-1)`

So

`y(x,t) = 0.125 sin (5.712 \ m^(-1) \ x - 0.6808 \ s^(-1)\ t )`