Question

A series of parallel linear water wave fronts are traveling directly toward the shore at 15.5 cm/s on an otherwise placid lake. A long concrete barrier that runs parallel to the shore at a distance of 3.10 m away has a hole in it. You count the wave crests and observe that 75.0 of them pass by each minute, and you also observe that no waves reach the shore at ±62.3cm from the point directly opposite the hole, but waves do reach the shore everywhere within this distance.Part AHow wide is the hole in the barrier?Part BAt what other angles do you find no waves hitting the shore?Enter your answers numerically separated by commas.

Answer 1

Answer:

0.629 m

$23.22^{\circ}, 36.26^{\circ}, 52.05^{\circ}, 80.3^{\circ}$

Explanation:

For destructive interference we have the case

$sin\theta=\dfrac{m\lambda}{a}$

m = 1,2,3.....

Frequency is given by

$f=\dfrac{75}{60}=1.25\ Hz$

Wavelength

$\lambda=\dfrac{v}{f}\\\Rightarrow \lambda=\dfrac{15.5}{1.25}\\\Rightarrow \lambda=12.4\ cm$

The angle

$\theta=tan^{-1}\dfrac{0.623}{3.1}\\\Rightarrow \theta=11.36^{\circ}$

The width is

$a=\dfrac{\lambda}{sin\theta}\\\Rightarrow a=\dfrac{0.124}{sin11.36}\\\Rightarrow a=0.629\ m$

The width of the hole is 0.629 m

For destructive interference

$sin\theta_2=\dfrac{2\lambda}{a}\\\Rightarrow \theta_2=sin^{-1}\dfrac{2\times 0.124}{0.629}\\\Rightarrow \theta_2=23.22^{\circ}$

$\theta_3=sin^{-1}\dfrac{3\times 0.124}{0.629}\\\Rightarrow \theta_3=36.26^{\circ}$

$\theta_4=sin^{-1}\dfrac{4\times 0.124}{0.629}\\\Rightarrow \theta_4=52.05^{\circ}$

$\theta_5=sin^{-1}\dfrac{5\times 0.124}{0.629}\\\Rightarrow \theta_5=80.3^{\circ}$

The angles are $23.22^{\circ}, 36.26^{\circ}, 52.05^{\circ}, 80.3^{\circ}$