Frequency difference to find a 20m whale

[bananabandana]

Homework Statement

Please see attachment for diagram. The two boats are coherent sources of sound waves (phase difference $\phi$) - i.e it's a double slit problem.

Prove the formula given for $y_{max}$.
Suppose the whale is 20m long. How large should the sonar frequency $f$ be so that the whale can always be detected? Assume $B=2.18\times 10^{9} Pa$ and $\rho =1.05 \times 10^{3} \ kg$

Homework Equations

$$ y_{max} = \frac{d \lambda (n+\frac{\phi}{2\pi})}{a} $$
Where $y_{max}$ is the $y$ value at which the two sources constructively interfere for a given depth, $d$. The depth of the whale (the tube shaped thing) is $d=350m$.

The phase velocity of the sonar waves, $v_{p}$ is given by:
$$ v_{p}=\sqrt{\frac{B}{\rho}} $$

The Attempt at a Solution

Is it sensible just to substitute $y_{max}=20$ and just do the algebra? ( I have already done the proof for $y_{max}$.)This would seem the obvious thing to do from the diagram ( and since the question is only a couple of marks). But I'm just wondering if this would really work. It says earlier in the question that the whale is moving horizontally - so I'm just thinking there might be a more complete way to approach the problem, but I don't know how to do it...

 

[tech99]

Homework Statement

Please see attachment for diagram. The two boats are coherent sources of sound waves (phase difference $\phi$) - i.e it's a double slit problem.
View attachment 81564
Prove the formula given for $y_{max}$.
Suppose the whale is 20m long. How large should the sonar frequency $f$ be so that the whale can always be detected? Assume $B=2.18\times 10^{9} Pa$ and $\rho =1.05 \times 10^{3} \ kg$

Homework Equations

$$ y_{max} = \frac{d \lambda (n+\frac{\phi}{2\pi})}{a} $$
Where $y_{max}$ is the $y$ value at which the two sources constructively interfere for a given depth, $d$. The depth of the whale (the tube shaped thing) is $d=350m$.

The phase velocity of the sonar waves, $v_{p}$ is given by:
$$ v_{p}=\sqrt{\frac{B}{\rho}} $$

The Attempt at a Solution

Is it sensible just to substitute $y_{max}=20$ and just do the algebra? ( I have already done the proof for $y_{max}$.)This would seem the obvious thing to do from the diagram ( and since the question is only a couple of marks). But I'm just wondering if this would really work. It says earlier in the question that the whale is moving horizontally - so I'm just thinking there might be a more complete way to approach the problem, but I don't know how to do it...

To locate the whale reliably, I suppose it must occupy one maximum of the radiation pattern. If it occupies more, its direction will be uncertain. If less, the return signal will be smaller than necessary. Notice that if the whale lies symmetrically across a pattern null, that will still be a null. The "head" will reflect one lobe and the "tail" the adjacent one, and adjacent lobes are in antiphase.