Two submarines A and B are traveling directly toward each other in still water at speeds vA = 8 m/s and vB = 10 m/s. In order to detect the presence of another submarine, submarine A emits sound waves of frequency ν0 = 1400 Hz. Find the frequency of the waves reflected from the submarine B perceived by a receiver located in submarine A.
Datos: speed of sound v = 1500 m/s.
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This problem is an application of the Doppler effect. In order to solve it we are going to split it in two parts. First, we will determine the frequency νB perceived by a receiver located in submarine B and then we will use this frequency as the initial frequency of the reflected waves that will reach submarine A.
Calculation of νB:
The general expression for Doppler shift is:
- |v|: is the speed of the waves with respect to the medium (water in this case).
- vm: is the speed of the medium with respect to Earth.
- vS: is the speed of the source emitting the waves.
- vR: is the speed of the observer (or listener).
- ν0: is the frequency of the waves emitted by the source.
The expression above has been derived assuming that the wavefronts, the source and the receiver are moving in the same direction. In order to apply the general expression to a particular problem, we must first determine what signs we need to use for the various speeds in the equation.
In this problem we are assuming still water, and thus vm = 0.
In the following figure are shown both submarines as well as the sound waves emitted by submarine A. In this first part of the problem submarine A plays the role of the source and submarine B is the receiver.
Since submarine B moves in the opposite direction to the wavefronts we have to change the sign of its speed vR in the expression for the frequency νB:
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In the second part of the problem submarine B is the source and it emits sound waves of frequency νB. Submarine A acts now as the receiver, as shown in the figure below.
Since submarine A is moving in the opposite direction to the wavefronts we have to change the sign of its velocity vR in the expression for the frequency νA perceived by a receiver located in it:
And substituting νB into the expression we get:
Last, substituting the numerical values into the above formula we get:
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