Problem 5 - University Physics with Modern Physics

Question

BIO UItrasound 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 \(\mathrm{kH} z\) 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 \(\mathrm{x}\) rays are utilized. For sharp imagery, the wave length 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

(a) 90.06 m, (b) 102066.67 Hz, (c) 1.372 cm, (d) 0.00440 m to 0.00879 m, (e) 6.2 MHz

Step 1: Understanding the Problem

Read carefully through the entire problem to identify the questions being asked and the given values for each part of the exercise. The key quantities to focus on are frequency, wavelength, and the speed of sound, as these will be necessary for calculations.

Step 2: Part (a) Whale Communication: Using the Speed and Frequency to find Wavelength

We use the equation for wave speed, \( v = f \lambda \), where \( v \) is the speed of sound in seawater (1531 \( \text{m/s}\)), \( f \) is the frequency (17 \( \text{Hz}\)), and \( \lambda \) is the wavelength. Rearrange the equation to solve for \( \lambda \):\[ \lambda = \frac{v}{f} = \frac{1531}{17} \approx 90.06 \text{ m} \].

Step 3: Part (b) Dolphin Clicks: Using Wavelength to find Frequency

Given that the wavelength \( \lambda = 1.5 \text{ cm} = 0.015 \text{ m} \), and the speed of sound in water \( v = 1531 \text{ m/s} \), use the wave speed equation, \( v = f \lambda \), to solve for the frequency \( f \): \[ f = \frac{v}{\lambda} = \frac{1531}{0.015} \approx 102066.67 \text{ Hz} \].

Step 4: Part (c) Dog Whistle: Finding Wavelength using Frequency

We use the wave speed equation, \( v = f \lambda \), where \( v = 343 \text{ m/s} \) (speed of sound in air), \( f = 25 \text{ kHz} = 25000 \text{ Hz} \). Solve for wavelength \( \lambda \):\[ \lambda = \frac{v}{f} = \frac{343}{25000} = 0.01372 \text{ m} = 1.372 \text{ cm} \].

Step 5: Part (d) Bats: Range of Wavelengths from Frequency Range

For frequencies \( f_1 = 39 \text{ kHz} \) and \( f_2 = 78 \text{ kHz} \), use the speed of sound in air, \( v = 343 \text{ m/s} \), to find the range of wavelengths. Calculate:For \( f_1 = 39 \text{ kHz} \):\[ \lambda_1 = \frac{343}{39000} \approx 0.00879 \text{ m} \].For \( f_2 = 78 \text{ kHz} \):\[ \lambda_2 = \frac{343}{78000} \approx 0.00440 \text{ m} \].Thus, the range is approximately 0.00440 m to 0.00879 m.

Step 6: Part (e) Sonograms: Frequency Required for Imaging a Tumor

To have a wavelength one-fourth the size of a 1.0 mm object, \( \lambda \approx \frac{1.0}{4} \text{ mm} = 0.25 \text{ mm} = 0.00025 \text{ m} \). Use \( v = 1550 \text{ m/s} \) for the speed of sound in tissue to solve for \( f \): \[ f = \frac{v}{\lambda} = \frac{1550}{0.00025} = 6200000 \text{ Hz} = 6.2 \text{ MHz} \].

Key Concepts

Frequency

Frequency in wave physics refers to how often the wave oscillates or completes a cycle in one second. It's measured in hertz (Hz). A higher frequency indicates more oscillations per second, while a lower frequency means fewer. This is crucial in understanding how different sounds or signals behave in various media.

To calculate frequency, you can use the formula:\[ f = \frac{v}{\lambda} \]where:

• \( f \) is the frequency.
• \( v \) is the speed of sound in the medium.
• \( \lambda \) is the wavelength.

For instance, if the wavelength in seawater is known, and the speed of sound in that medium is 1531 \(\text{m/s}\), you can easily find the frequency by rearranging the formula. Frequency is vital because different animals and devices utilize specific frequencies for communication and detection.

Wavelength

Wavelength is the distance between two consecutive points that are in phase on a wave. It is typically measured in meters (m) and is an essential aspect of wave physics and acoustics. Wavelength gives us insight into the "spread out" nature of waves and can vary greatly between different types of waves.

The relationship between wavelength, frequency, and the speed of sound is given by the formula \( v = f \lambda \). This relationship indicates that:

• For a given speed, a higher frequency results in a shorter wavelength.
• Conversely, a lower frequency corresponds to a longer wavelength.

For example, in the problem of whale communication, given a frequency of 17 Hz and a speed of sound of 1531 m/s in seawater, the wavelength can be found using \( \lambda = \frac{v}{f} \). Understanding wavelength helps in designing equipment in both human industries and understanding animal behaviors.

Speed of Sound

The speed of sound is how fast sound waves travel through a medium and is influenced by factors like temperature, medium type, and pressure. It is typically measured in meters per second (m/s). In air, the speed of sound is approximately 343 m/s, but it differs in other media, such as seawater or different tissues.

The formula to find the speed of sound is \( v = f \lambda \), which relates it to wavelength and frequency.

• In media like seawater, the speed of sound is often higher than in air due to its denser nature.
• Knowing the speed of sound is crucial for solving problems like those involving dolphin clicks or ultrasound used in medical imaging.

For specific applications, such as sonar in marine environments or ultrasound in medical fields, understanding variations in the speed of sound allows for precise calculations and solutions tailored to the specific medium and situation.