Vibrations and Waves Legacy Problem #13 Guided Solution
Problem*
Humpback whales are known to produce a collection of elaborate and repeating sounds with frequencies ranging from 20 Hz to 10 kHz. The sound waves travel through water at speeds of approximately 1400 m/s. Determine the wavelengths of the waves at the lower and the upper end of this frequency range.
Audio Guided Solution
This problem pertains to the sounds created by humpback whales, which range in frequency from 20 Hz at the low end on up to 10 kHz at the high end. They travel through seawater at 1,400 m per second in order to determine the wavelengths of these waves. We need to use the wave equation V equals f times lambda, where V is the speed of 1,400 m per second, f is the given frequencies and must be in units of hertz, and wavelength will then come out in units of meters. So using 20 Hz as our low frequency, we can find the wavelength of these waves if we rearrange the equation as wavelength equals V divided by f. And then we divide 1,400 m per second by 20 Hz, and we get 70 m. And then on the high end, we have 10 kHz sounds. KHz is not the preferred unit for use in the wave equation, so we should change that to hertz, and that would be 10,000 Hz, one with four zeros after it. Now if we use the wave equation again and we arrange it to lambda, or wavelength, equal V divided by f, we can divide the 1,400 m per second by the 10,000 Hz, and we'll get 0.14 m.
Solution
20 Hz sounds have a wavelength of 70 m.
10 kHz sounds have a wavelength of 0.14 m.
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities and record in an organized manner, often times they can be recorded on the diagram itself. Equate given values to the symbols used to represent the corresponding quantity (e.g., \(\descriptive{v}{v,velocity} = 12.8 \unit{\meter\per\second}\), \(\descriptive{λ}{λ,wave length} = 4.52 \unit{m}\), \(\descriptive{f}{f,frequency} = \colorbox{gray}{Unknown}\)).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read About It!
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