A water wave has the equation y=7.0sin[0.65(x-35.45t)], where x and y are measured in feet and t is measured in seconds. In miles per hour, what is the phase velocity (speed) of the wave?

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to remember the equation of the phase of a wave, such that:

`y(x,t) = A*sin(2pi(x/lambda - t/T))`

The problem provides the following equation `y=7.0sin[0.65(x-35.45t)]` , hence, you may identify the following factors, such that:

`{(A = 0.7),(2pi*t/T = 0.65*35.45 t),(0.65x = (2pix)/lambda):}`

Reducing duplicate factors in each of the equations above, yields

`{(A = 0.7),(T = (2pi)/(0.65*35.45)),(lambda = (2pi)/0.65):}`

`{(A = 0.7),(T = (2pi)/(0.65*35.45)),(lambda = (2pi)/0.65):}`

`{(A = 0.7),(T = 0.272),(lambda = 9.66):}`

You need to use the equation that relates the wavelength `lambda` , the velocity `v` and the period `T` , such that:

`v = lambda/T => v = 9.66/0.272 => v = 35.514 m/s`

You need to convert the units of measure in miles/hour, such that:

`v = 35.514*3600*10^(-3) (Km)/h => v = 127.8504 (Km)/h`

You know that `1Km ~~ 0.62137` miles, hence, you need to multiply the result by 0.62137, such that:

v = 127.8504* 0.62137(miles)/h => v = 79.442 miles/h

Hence, evaluating the phase velocity of the wave, under the given conditions, yields `v = 79.442` miles/h.