Problem 37 - College Physics

Question

Ultrasound and infrasound. (a) Whale communication. Blue whales apparently communicate with each other using sound of frequency 17 \(\mathrm{Hz}\) , which can be heard nearly 1000 \(\mathrm{km}\) away in the ocean. What is the wavelength of such a sound in seawater, where the speed of sound is 1531 \(\mathrm{m} / \mathrm{s} ?\) (b) Dolphin clicks. One type of sound that dolphins emit is a sharp click of wavelength 1.5 \(\mathrm{cm}\) in the ocean. What is the frequency of such clicks? (c) Dog whistles. One brand of dog whistles claims a frequency of 25 \(\mathrm{kHz}\) for its product. What is the wavelength of this sound? (d) Bats. While bats emit a wide variety of sounds, one type emits pulses of sound having a frequency between 39 \(\mathrm{kHz}\) and 78 \(\mathrm{kHz}\) . What is the range of wavelengths of this sound? (e) Sonograms. Ultrasound is used to view the interior of the body, much as x rays are utilized. For sharp imagery, the wavelength of the sound should be around one-fourth (or less) the size of the objects to be viewed. Approximately what frequency of sound is needed to produce a clear image of a tumor that is 1.0 \(\mathrm{mm}\) across if the speed of sound in the tissue is 1550 \(\mathrm{m} / \mathrm{s} ?\)

Step-by-Step Solution

Answer

Wavelengths: (a) 90 m, (b) 102 kHz, (c) 1.372 cm, (d) 4.4-8.8 mm, (e) 6.2 MHz.

Step 1: Calculating Wavelength of Whale Communication

To find the wavelength \( \lambda \) of the sound used by whales, use the formula \( \lambda = \frac{v}{f} \), where \( v = 1531 \, \mathrm{m/s} \) is the speed of sound in seawater and \( f = 17 \, \mathrm{Hz} \) is the frequency. So, \( \lambda = \frac{1531}{17} = 90.06 \, \mathrm{m} \). The wavelength is approximately 90 meters.

Step 2: Calculating Frequency of Dolphin Clicks

Given the wavelength \( \lambda = 1.5 \, \mathrm{cm} = 0.015 \, \mathrm{m} \) in seawater with \( v = 1531 \, \mathrm{m/s} \), use the formula \( f = \frac{v}{\lambda} \). Thus, \( f = \frac{1531}{0.015} = 102066.67 \, \mathrm{Hz} \). The frequency is approximately 102 kHz.

Step 3: Calculating Wavelength of Dog Whistle

With frequency \( f = 25 \, \mathrm{kHz} = 25000 \, \mathrm{Hz} \) and assuming the speed of sound in air \( v = 343 \, \mathrm{m/s} \), use \( \lambda = \frac{v}{f} \). This gives \( \lambda = \frac{343}{25000} = 0.01372 \, \mathrm{m} \) or 1.372 cm.

Step 4: Calculating Range of Wavelengths for Bat Pulses

For frequencies between 39 kHz and 78 kHz, use the speed of sound in air, \( v = 343 \, \mathrm{m/s} \). The range of wavelengths is given by \( \lambda_{min} = \frac{343}{78000} = 0.004397 \, \mathrm{m} \) and \( \lambda_{max} = \frac{343}{39000} = 0.00879 \, \mathrm{m} \). Hence, the range is approximately 4.4 mm to 8.8 mm.

Step 5: Calculating Ultrasound Frequency for Sharp Imagery

For a tumor of 1.0 mm, we need ultrasound wavelengths of approximately \( \frac{1}{4} \times 1.0 = 0.25 \, \mathrm{mm} = 0.00025 \, \mathrm{m} \). The speed of sound in tissue is \( 1550 \, \mathrm{m/s} \), hence, \( f = \frac{v}{\lambda} = \frac{1550}{0.00025} = 6200000 \, \mathrm{Hz} = 6.2 \, \mathrm{MHz} \).

Key Concepts

Wave Frequency

Frequency is a key characteristic of a wave, describing how often the wave oscillates in a given period of time. It's often measured in hertz (Hz), which signifies cycles per second. For example, a wave frequency of 17 Hz means the wave completes 17 cycles every second. In physics, the formula to find frequency is:\[ f = \frac{v}{\lambda} \]where:

• \( f \) is the frequency of the wave.
• \( v \) is the speed of the wave through a medium, like air or water.
• \( \lambda \) is the wavelength, which we'll explore further below.

By rearranging this formula, we can find the frequency if we know the velocity and wavelength of the wave. This information is essential in various real-world applications, from understanding animal communication to developing ultrasound technology.

Wave Wavelength

Wavelength refers to the distance between two successive peaks (or troughs) of a wave. It's a key concept for understanding waves, appearing in phenomena from light to sound waves. We're usually dealing with meters (m) for wavelength, but it can be as small as nanometers.
Different wavelengths mean different sounds or colors, which is why they play a major role in communication and imaging technologies.
The basic formula for wavelength: \[ \lambda = \frac{v}{f} \]involves:

• \( \lambda \) being the wavelength.
• \( v \) as the velocity at which the wave travels through its medium.
• \( f \) as the frequency.

By understanding this relationship, you can calculate a characteristic that directly affects the behavior of a wave. This is especially important in applications like marine biology and sonar, where sonar waves must penetrate long distances in water.

Ultrasound Imaging

Ultrasound imaging utilizes sound waves with frequencies higher than the upper audible limit of human hearing. Used widely in the medical field, this technique helps view internal structures of the body in a safe, non-invasive way.
The key principle is simple:

• Ultrasound devices emit high-frequency sound waves.
• These waves travel through the body until they hit a boundary between tissues.
• The waves then reflect back, creating echoes collected by sensors.

By using wavelengths smaller than the objects being imaged, the detail of images improves. To produce sharp results, the emitted wavelength should be roughly a quarter of the size of the targeted structures. For example, imaging a 1.0 mm tumor requires sound waves of very short wavelengths like 0.25 mm, meaning extremely high frequencies.

Dolphin Communication

Dolphins are known for their advanced methods of communication. They use a variety of sound waves underwater, which can travel long distances. Among their vocal repertoire is a high-frequency click.
These clicks serve purposes like navigation, hunting, and social interaction. The speed of sound in water, combined with the frequency and wavelength of these clicks, allows dolphins to effectively "see" with sound, a process known as echolocation.
Here's how they use it:

• Emit high-frequency sound waves.
• Sound waves bounce off objects and return as echoes.
• Time taken for echoes to return tells the dolphin about size and distance of objects.

Understanding dolphins' wave interactions highlights how animals adapt to their environments and shows the practical applications of waves in biology.

Physics Problems

Physics problems involving wave properties can seem complex, but they often come down to applying key formulas for frequency and wavelength. By clearly understanding these concepts, solving real-world problems becomes manageable!A typical physics problem might ask you to calculate one property of a wave (wavelength or frequency) based on the others (speed and the remaining property).
For instance, the relationship \( v = f \lambda \)allows you to solve for any missing component:

• Given speed and frequency, find wavelength.
• Given speed and wavelength, find frequency.

With practice, these solutions become more intuitive, helping you uncover the dynamics of waves in numerous scenarios. Applying this knowledge can range from decoding animal communication to developing high-tech medical devices.