Calculate the wavelength of extreme sounds (20 Hz and 20,000 Hz) in air, where the speed of sound is 343 meters per second.
Understand the Problem
The question is asking us to calculate the wavelength of sounds at the extremes of the human ear's frequency range (20 Hz and 20,000 Hz) while considering the speed of sound in air, which is given as 343 meters per second. The wavelength can be calculated using the formula: wavelength = speed / frequency.
Answer
The wavelength for 20 Hz is $17.15$ meters, and for 20,000 Hz, it is $0.01715$ meters.
Answer for screen readers
The wavelength of sounds for 20 Hz is approximately $17.15$ meters, and for 20,000 Hz, it is approximately $0.01715$ meters.
Steps to Solve
- Identify the wavelength formula
We are using the formula to calculate wavelength given by
$$ \text{wavelength} = \frac{\text{speed}}{\text{frequency}} $$
Here, "speed" is the speed of sound in air (343 m/s), and "frequency" will vary depending on the given frequencies (20 Hz and 20,000 Hz).
- Calculate the wavelength for 20 Hz
Now, we will substitute the speed of sound and the frequency of 20 Hz into the wavelength formula.
$$ \text{wavelength}_{20} = \frac{343 \text{ m/s}}{20 \text{ Hz}} $$
- Simplify the calculation
Calculating the above expression gives us:
$$ \text{wavelength}_{20} = 17.15 \text{ meters} $$
- Calculate the wavelength for 20,000 Hz
Next, we will substitute the speed of sound and the frequency of 20,000 Hz into the wavelength formula.
$$ \text{wavelength}_{20000} = \frac{343 \text{ m/s}}{20000 \text{ Hz}} $$
- Simplify the calculation
Calculating this expression we get:
$$ \text{wavelength}_{20000} = 0.01715 \text{ meters} $$
The wavelength of sounds for 20 Hz is approximately $17.15$ meters, and for 20,000 Hz, it is approximately $0.01715$ meters.
More Information
The human ear can detect sounds in a very wide range of frequencies, from very low (20 Hz) to very high (20,000 Hz). This means that the wavelengths of these sounds also vary significantly. At 20 Hz, the wavelength is long, allowing these low-frequency sounds to travel further. In contrast, at 20,000 Hz, the wavelength is very short, meaning these high-frequency sounds are more localized.
Tips
- Confusing frequency with wavelength: It's common to confuse the two concepts. Remember that frequency is how often a sound wave cycles per second, while wavelength is the distance between successive peaks of the wave.
- Incorrectly applying the formula: Always check that you're using the right formula and substituting the right values for speed and frequency.
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