A swimming duck paddles the water with its feet once every 1.6 s, producing surface waves with this period. The duck is moving at constant speed in a pond where the speed of surface waves is 0.32 m/s, and the crests of the waves ahead of the duck are spaced 0.12 m apart. (a) What is the duck’s speed? (b) How far apart are the crests behind the duck?

a) As duck is the source, therefore its frequency is:

$f_S=\frac{1}{T}=\frac{1}{1.6\mathrm{s}}=0625\mathrm{Hz}$

The distance between the wave's crests represents the wavelength of that wave, and because the source is moving, the waves in front of the source will have a shorter wavelength than the wavelength of the waves behind the source.

As a result, the wavelength in front of the moving duck is defined as;

${\lambda }_{\mathrm{in}\mathrm{front}}=\frac{v-v_S}{f_s}$

As, v=0.32m/s and source frequency are 0.625Hz,

So, the speed of a duck $v_s$is

$v_s=v-\left({\lambda }_{infront}\times f_s\right)$

Putting the values;

role="math" localid="1664338092359" $$\begin{gathered} {\mathrm{v}}_{\mathrm{s}}=0.32\mathrm{m}/\mathrm{s}-\left[\left(0.12\mathrm{m}\right)\times \left(0.625\mathrm{Hz}\right)\right] \\ {\mathrm{v}}_{\mathrm{s}}=0.25\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the speed of a duck is${\mathrm{v}}_{\mathrm{s}}=0.25\mathrm{m}/\mathrm{s}$

b) The wavelength of the waves behind the duck can be given as the distance between the crests behind the duck, so the equation is;

${\lambda }_{behind}=\frac{v+v_S}{f_s}$

Here $v_s$is taken positive because the source is moving away.

On putting the values,

$$\begin{gathered} {\mathrm{\lambda }}_{\mathrm{behind}}=\frac{\left(0.32\mathrm{m}/\mathrm{s}\right)+\left(0.25\mathrm{m}/\mathrm{s}\right)}{0.625\mathrm{Hz}} \\ {\mathrm{\lambda }}_{\mathrm{behind}}=0.91\mathrm{m} \end{gathered}$$