Problem 2 - University Physics with Modern Physics

Question

Provided the amplitude is sufficiently great, the human ear can respond to longitudinal waves over a range of frequencies from about 20.0 Hz to about 20.0 kHz. (a) If you were to mark the beginning of each complete wave pattern with a red dot for the long-wavelength sound and a blue dot for the short- wavelength sound, how far apart would the red dots be, and how far apart would the blue dots be? (b) In reality would adjacent dots in each set be far enough apart for you to easily measure their separation with a meter stick? (c) Suppose you repeated part (a) in water, where sound travels at 1480 m/s. How far apart would the dots be in each set? Could you readily measure their separation with a meter stick?

Step-by-Step Solution

Answer

In air, red dots are 17.15 m apart and blue dots are 0.01715 m apart. In water, red dots are 74.0 m apart, and blue dots are 0.074 m apart. Dots are too close for easy measurement with a meter stick for high frequencies.

Step 1: Understanding the Problem

We need to find the wavelength of sound waves at different frequencies, both at the extremes (20 Hz and 20 kHz), and determine if these wavelengths are easily measurable using a meter stick in air and water environments.

Step 1: Define the Speed of Sound in Air

In air, the speed of sound is approximately 343 m/s. We will use this speed to calculate the wavelengths of sound waves in air at given frequencies.

Step 2: Calculate Wavelength in Air for 20 Hz Frequency

The formula for wavelength \( \lambda \) is given by \( \lambda = \frac{v}{f} \), where \( v \) is the speed of sound and \( f \) is the frequency. For a frequency of 20 Hz:\[ \lambda_{20\,Hz} = \frac{343\,m/s}{20\,Hz} = 17.15\,m \]

Step 3: Calculate Wavelength in Air for 20 kHz Frequency

Apply the same formula to find the wavelength for a frequency of 20 kHz:\[ \lambda_{20\,kHz} = \frac{343\,m/s}{20000\,Hz} = 0.01715\,m \]

Step 4: Assess the Measureability in Air

Compare the wavelengths with a meter stick, which is 1 m long. The 17.15 m wavelength is much longer than the stick, easily measurable. The 0.01715 m wavelength is much shorter than the stick, making it difficult to measure accurately.

Step 5: Define the Speed of Sound in Water

In water, the speed of sound is approximately 1480 m/s. This value will be used to calculate the wavelengths for sound waves at the same frequencies as before, but now in water.

Step 6: Calculate Wavelength in Water for 20 Hz Frequency

Using the speed of sound in water:\[ \lambda_{20\,Hz} = \frac{1480\,m/s}{20\,Hz} = 74.0\,m \]

Step 7: Calculate Wavelength in Water for 20 kHz Frequency

Calculate the wavelength for a frequency of 20 kHz in water:\[ \lambda_{20\,kHz} = \frac{1480\,m/s}{20000\,Hz} = 0.074\,m \]

Step 8: Assess the Measureability in Water

In water, the 74.0 m wavelength is much longer than a meter stick. The 0.074 m wavelength is closer in size but still shorter than the meter stick, making it somewhat more measurable compared to in air but still challenging for precise measurement.

Key Concepts

Frequency

Frequency refers to how often the particles of the medium vibrate when a sound wave passes through it.
It is measured in hertz (Hz), where one hertz is equal to one cycle per second. When it comes to sound waves, frequency determines the pitch of the sound.
Sound waves with higher frequencies produce higher-pitched sounds. Conversely, lower frequencies produce lower-pitched sounds. Human beings can generally hear frequencies ranging from 20 Hz to 20,000 Hz (20 kHz).
High-frequency sound waves, like those at 20 kHz, have a very short wavelength and tend to be more difficult to measure with ordinary tools such as a meter stick. Lower-frequency sound waves, like 20 Hz, have longer wavelengths and are much more measurable.

Wavelength

Wavelength is the distance between two consecutive points in phase on a wave, such as crest to crest or trough to trough.
For sound waves, wavelength is essential for describing how sound travels through different environments.The wavelength inversely depends on frequency. This relationship is detailed in the equation:\[\lambda = \frac{v}{f}\]where \( \lambda \) is the wavelength, \( v \) is the speed of sound, and \( f \) is the frequency.
At lower frequencies like 20 Hz, wavelengths are longer, making them more discernible and easier to measure when they propagate through a medium, such as air or water.
At higher frequencies like 20 kHz, the wavelengths are much shorter, making them harder to detect with tools like meter sticks.

Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium. It varies based on the medium and its properties.
For instance, sound travels faster in water (1480 m/s) than in air (343 m/s). This happens because particles in water are more densely packed, allowing sound waves to transfer quickly from one particle to the next.
The speed of sound affects both frequency and wavelength since they are interconnected in the above equation. An increase in the speed of sound results in an increase in wavelength for a given frequency.
This is why sound waves of the same frequency are longer in water than in air, making measurement considerations differ based on the medium.

Longitudinal Waves

Longitudinal waves are types of waves where the particle movement is parallel to the direction of wave propagation.
Sound waves are a prime example of longitudinal waves. In longitudinal waves, particles of the medium compress and rarefy in the same direction the wave travels. This creates regions of high pressure (compressions) and low pressure (rarefactions) along the path of the wave.
These alternating regions allow sound waves to transmit energy across distances.
Understanding this movement is essential to comprehending how sound propagates through various media and how our ears perceive sound frequencies.

Medium of Propagation

The medium of propagation is crucial for the transmission of sound waves. It is the substance through which the sound waves travel.
Whether air, water, or solid, the properties of the medium highly influence the speed and characteristics of the sound waves. - **Air:** Sound moves at 343 m/s, allowing easier measurement for longer wavelengths but complicating it for shorter ones like 20 kHz. - **Water:** Sound travels at 1480 m/s. Here, even shorter wavelengths are slightly easier to measure compared to air. The density and elasticity of the medium are significant factors. In denser media like water, sound travels faster because particles are closer together, facilitating quicker energy transfer.
The choice of medium affects practical applications, such as determining whether precise measurement of wave properties can be conducted efficiently.