Question

The speed of sound in dry air at \(20^{\circ} \mathrm{C}\) is \(343 \mathrm{~m} / \mathrm{s}\) and the lowest frequency sound wave that the human ear can detect is approximately \(20 \mathrm{~Hz}\) (a) What is the wavelength of such a sound wave? (b) What would be the frequency of electromagnetic radiation with the same wavelength? (c) What type of electromagnetic radiation would that correspond to? [Section 6.1]

Short answer

The wavelength of the sound wave with a frequency of 20 Hz is 17.15 meters. The frequency of the electromagnetic radiation with the same wavelength is approximately 1.75 x 10^7 Hz. This corresponds to radio waves in the electromagnetic spectrum.

Step-by-step solution

Finding the wavelength of the sound wave

Let's calculate the wavelength of the sound wave. We know the speed of sound in dry air at 20°C is 343 m/s and the frequency is 20 Hz. We can use the formula for the relationship between speed (v), frequency (f), and wavelength (λ): v = fλ Rearranging the formula to find the wavelength: λ = v / f Now, plug in the values: λ = 343 m/s ÷ 20 Hz λ = 17.15 m The wavelength of the sound wave is 17.15 meters.

Finding the frequency of the electromagnetic radiation

The speed of electromagnetic radiation, such as light, in a vacuum is approximately 3.00 x 10^8 m/s. We can use the same formula v = fλ to find the frequency (f) of the electromagnetic radiation with the same wavelength. Rearranging the formula to solve for frequency: f = v / λ Plug in the values: f = (3.00 x 10^8 m/s) ÷ 17.15 m f ≈ 1.75 x 10^7 Hz The frequency of the electromagnetic radiation with the same wavelength is approximately 1.75 x 10^7 Hz.

Identifying the type of electromagnetic radiation

To determine the type of electromagnetic radiation, we need to compare the wavelength 17.15 m with the electromagnetic spectrum ranges: - Radio waves: > 1 m - Microwaves: 1 m to 1 mm - Infrared: 1 mm to 700 nm - Visible light: 700 nm to 400 nm - Ultraviolet: 400 nm to 10 nm - X-rays: 10 nm to 0.001 nm - Gamma rays: < 0.001 nm The wavelength of 17.15 m falls within the radio wave range. Therefore, the electromagnetic radiation with the same wavelength is a radio wave.

Key concepts

Speed of Sound

The speed of sound is a fascinating topic that helps us understand how sound waves travel through the air. At 20°C, the speed of sound in dry air is approximately 343 meters per second (m/s). This speed can vary with changes in temperature, air pressure, and humidity, but 343 m/s is accepted as a standard under normal conditions.
To grasp this concept, think of sound as a series of vibrations moving through the air. It acts somewhat like ripples on a water surface. When a sound is made, it causes molecules in the air to vibrate and pass the energy onto neighboring molecules, forming a wave.

• Sound speed is influenced mainly by the medium it travels through. For instance, sound travels faster in water than in air.
• The formula to calculate sound speed is crucial: v = f\lambda where \(v\) is the speed of sound, \(f\) is the frequency, and \(\lambda\) is the wavelength.

Understanding how fast sound waves travel helps us dissect many phenomena, from thunder following lightning to the pitch variation in music.

Electromagnetic Spectrum

The electromagnetic spectrum is an all-encompassing term for the range of all types of electromagnetic radiation. This spectrum includes everything from radio waves, which have long wavelengths, to gamma rays, which have very short wavelengths.
Electromagnetic radiation travels at the speed of light, approximately 3.00 x 10^8 meters per second in a vacuum. The different types of radiation are categorized by their wavelength and frequency.

• Radio waves, microwaves, and infrared are longer wavelengths.
• Visible light is the tiny part of the spectrum that our eyes can detect.
• Ultraviolet, X-rays, and gamma rays have shorter wavelengths.

Each type serves various purposes, such as communication, medical imaging, and scientific research. Understanding this spectrum helps connect the dots between different technological applications and natural phenomena.

Wavelength of Sound

Wavelength is a crucial concept when studying sound. It represents the distance between consecutive points of a wave, like crest to crest. For sound waves, the wavelength helps determine the sound's pitch alongside its frequency.
In our example, to find the wavelength of a sound wave with a frequency of 20 Hz and the speed of sound as 343 m/s, we apply the formula: \(\lambda = \frac{v}{f}\) where \(\lambda\) is the wavelength. Substituting the known values: \(\lambda = \frac{343}{20}\), resulting in a wavelength of 17.15 meters.

• Lower frequency waves have longer wavelengths and tend to produce deeper, bass sounds.
• Higher frequencies have shorter wavelengths, resulting in higher-pitched sounds.

An understanding of wavelength allows us to appreciate how musical instruments function and how sound travels in various environments.

Frequency of Electromagnetic Radiation

Frequency in electromagnetic radiation refers to the number of wave cycles that pass a point in a given period, often a second. In our universe, all forms of electromagnetic radiation travel at the speed of light in a vacuum, 3.00 x 10^8 m/s. This speed remains constant while the frequency and wavelength vary inversely.
For electromagnetic radiation with the same wavelength as a 20 Hz sound wave (17.15 m), its frequency can be solved using the formula: \(f = \frac{v}{\lambda}\).
Plugging the values: \(f = \frac{3.00 \times 10^8}{17.15}\), giving approximately 1.75 x 10^7 Hz.

• Long wavelengths equate to lower frequencies, finding use in wireless communication, like radio broadcasts.
• Shorter wavelengths possess higher frequencies, invaluable in technologies such as X-ray imaging.

Comprehending frequency is pivotal in fields ranging from telecommunications to health to scientific research, each using different parts of the spectrum strategically.