Engineers designing an underwater acoustic communication system must ensure that two…
MATHEMATICS · Multi-part · A Level · Mathematics · Quantities and units in mechanics · Quantities and units in mechanics
Problem
Engineers designing an underwater acoustic communication system must ensure that two different signaling tones deliver equal energy per second to a distant receiver located in stationary seawater. The tones travel along identical paths, and each is a sinusoidal pressure wave generated by a submerged transducer.
• Tone 1 has a frequency of 800Hz and a pressure amplitude A1.
• Tone 2 has a pressure amplitude three times larger, A2=3A1.
The design team is advised that, within the operating range of their equipment, the acoustic power transmitted by a tone is directly proportional to its frequency and is independent of its pressure amplitude.
(a) Derive an algebraic expression for the ratio of the transmitted powers P2/P1 in terms of the frequencies f1 and f2, based on the advisory relationship.
(b) Using your expression, determine the frequency f2 required for Tone 2 so that P2=P1. Provide your answer with appropriate units.
(c) Critically evaluate the advisory statement that power is independent of pressure amplitude by discussing one physical circumstance in which this statement might fail for underwater sound. Support your argument with qualitative reasoning.
(d) Design a brief experimental procedure (no more than three ordered steps) that the engineers could implement in the field to test whether acoustic power in seawater truly depends only on frequency for the amplitude range of their tones. Your procedure should specify the key measurements to be taken and how the resulting data would confirm or refute the advisory statement.