Chapter 20: Problem 70

Question:         Two submarines are on ahead on collision course during maneuvers in the North Atlantic. The first sub is moving at $v_1 = 20.2\ {\rm km/h}$ and the second at $v_2 = 94.6\ {\rm km/h}$. The first sub sends out a sonar signal (sound wave in water) at $\nu_1 = 1030\ {\rm Hz}$. Sonar waves travel at $c=5470\ {\rm km/h}$.

1.

The second sub picks up the signal. What frequency $\nu_2$ does the second sonar detector hear?

2.

The first sub picks up the reflected signal. What frequency $\tilde\nu_1$ does the first sonar detector hear?

See fig. 27 (chap 20). The ocean is calm, assume no currents.

Solution:

Both the observer and source are moving, so

$$\nu_2 = \nu_1 \left( {c + v_2 \over c - v_1} \right)$$

$$= 1030 \ {5470 + 94.6 \over 5470 - 20.2} \ =\ 1051.7 \ {\rm Hz} \ \ \ .$$ (14)

The wave that is reflected back to the first sub has this frequency, and so we have

$$\tilde\nu_1 = \nu_2 \left( {c + v_1 \over c - v_2}\right)$$

$$= 1051.7\left( { 5470 + 20.2\over 5470 - 94.6}\right)$$

$$= 1074.16\ {\rm Hz} \ \ \ .$$ (15)