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3a – Suppose one has a wave that travels at a speed of 1125 feet per second (about the speed of a sound wave). Suppose we can produce waves with the following different Wavelengths: i- Wavelength of 1 foot; ii- Wavelength of 10 feet, iii- Wavelength of 0.1 foot; iv- Wavelength of 4.25 feet. Calculate the Frequency for each of the Wavelengths given in i through iv. In solving this problem remember that the (Wavelength) X (Frequency) = (the Speed of the wave). The Frequency is given in waves per second. What does Wavelength and Frequency in part d have to do with playing the piano or other musical instruments? 3b – Suppose one has a wave that travels at a speed of 3 X 10^8 meters per second (the speed of light). Suppose we can produce waves with the following different Wavelengths: i- Wavelength of 10 centimeter (.1 meters) – microwave radiation ii- Wavelength of 10^(-5) meters – infrared radiation iii- Wavelength of 6 x 10^(-7) meters – green in the visible iv- Wavelength of 10^(-8) meters – ultraviolet radiation v- Wavelength of 10 ^(12) meters – gamma rays Calculate the Frequency for each of the Wavelengths given in i through v. In solving this problem remember that the (Wavelength) X (Frequency) = (the Speed of the wave). The Frequency is given in waves per second.
The Correct Answer and Explanation is:
To solve this problem, we can use the fundamental wave equation:Speed of the wave=Wavelength×Frequency\text{Speed of the wave} = \text{Wavelength} \times \text{Frequency}Speed of the wave=Wavelength×Frequency
Where:
- Speed of the wave is the speed at which the wave travels (in this case, the speed of light or the speed of sound),
- Wavelength is the distance between successive wave peaks,
- Frequency is how many waves pass a given point per second.
We can rearrange the equation to find Frequency:Frequency=Speed of the waveWavelength\text{Frequency} = \frac{\text{Speed of the wave}}{\text{Wavelength}}Frequency=WavelengthSpeed of the wave
3a. Speed of sound = 1125 feet per second
Given Wavelengths:
- i. Wavelength of 1 foot
- ii. Wavelength of 10 feet
- iii. Wavelength of 0.1 foot
- iv. Wavelength of 4.25 feet
Let’s calculate the frequencies for each wavelength.
- For Wavelength of 1 foot:
Frequency=11251=1125 waves per second\text{Frequency} = \frac{1125}{1} = 1125 \text{ waves per second}Frequency=11125=1125 waves per second
- For Wavelength of 10 feet:
Frequency=112510=112.5 waves per second\text{Frequency} = \frac{1125}{10} = 112.5 \text{ waves per second}Frequency=101125=112.5 waves per second
- For Wavelength of 0.1 foot:
Frequency=11250.1=11250 waves per second\text{Frequency} = \frac{1125}{0.1} = 11250 \text{ waves per second}Frequency=0.11125=11250 waves per second
- For Wavelength of 4.25 feet:
Frequency=11254.25=264.7 waves per second\text{Frequency} = \frac{1125}{4.25} = 264.7 \text{ waves per second}Frequency=4.251125=264.7 waves per second
Relevance to playing musical instruments: In musical instruments, the wavelength of the sound waves determines the pitch we hear. A shorter wavelength (higher frequency) results in a higher pitch (e.g., a high note on a piano). Conversely, a longer wavelength (lower frequency) results in a lower pitch (e.g., a low note). So, understanding the relationship between wavelength and frequency is essential for tuning and playing instruments.
3b. Speed of light = 3 x 10^8 meters per second
Given Wavelengths:
- i. Wavelength of 10 centimeters (0.1 meters) – microwave radiation
- ii. Wavelength of 10^(-5) meters – infrared radiation
- iii. Wavelength of 6 x 10^(-7) meters – green in the visible spectrum
- iv. Wavelength of 10^(-8) meters – ultraviolet radiation
- v. Wavelength of 10^12 meters – gamma rays
Let’s calculate the frequencies for each wavelength.
- For Wavelength of 0.1 meters (microwave radiation):
Frequency=3×1080.1=3×109 waves per second (3 GHz)\text{Frequency} = \frac{3 \times 10^8}{0.1} = 3 \times 10^9 \text{ waves per second (3 GHz)}Frequency=0.13×108=3×109 waves per second (3 GHz)
- For Wavelength of 10^(-5) meters (infrared radiation):
Frequency=3×10810−5=3×1013 waves per second\text{Frequency} = \frac{3 \times 10^8}{10^{-5}} = 3 \times 10^{13} \text{ waves per second}Frequency=10−53×108=3×1013 waves per second
- For Wavelength of 6 x 10^(-7) meters (green light in the visible spectrum):
Frequency=3×1086×10−7=5×1014 waves per second\text{Frequency} = \frac{3 \times 10^8}{6 \times 10^{-7}} = 5 \times 10^{14} \text{ waves per second}Frequency=6×10−73×108=5×1014 waves per second
- For Wavelength of 10^(-8) meters (ultraviolet radiation):
Frequency=3×10810−8=3×1016 waves per second\text{Frequency} = \frac{3 \times 10^8}{10^{-8}} = 3 \times 10^{16} \text{ waves per second}Frequency=10−83×108=3×1016 waves per second
- For Wavelength of 10^12 meters (gamma rays):
Frequency=3×1081012=3×10−4 waves per second\text{Frequency} = \frac{3 \times 10^8}{10^{12}} = 3 \times 10^{-4} \text{ waves per second}Frequency=10123×108=3×10−4 waves per second
Summary of Results:
- i. Microwave radiation: 3 x 10^9 Hz
- ii. Infrared radiation: 3 x 10^13 Hz
- iii. Green light: 5 x 10^14 Hz
- iv. Ultraviolet radiation: 3 x 10^16 Hz
- v. Gamma rays: 3 x 10^-4 Hz
These results show that waves with shorter wavelengths have higher frequencies. This principle also applies in the context of electromagnetic waves, where each region (radio waves, microwaves, infrared, visible, ultraviolet, and gamma rays) corresponds to specific wavelengths and frequencies.
