Question

A bat is flitting about in a cave, navigating very effectively by the use of ultrasonic bleeps (short emissions of highfrequency sound lasting a millisecond or less and repeated several times a second). Assume that the sound emission frequency of the bat is \(39.2 \mathrm{kHz}\). During one fast swoop directly toward a flat wall surface, the bat is moving at \(8.58 \mathrm{~m} / \mathrm{s}\). Calculate the frequency of the sound the bat hears reflected off the wall.

Short answer

The frequency of the sound wave reflected by the wall that the bat hears while moving towards it at a speed of \(8.58 \mathrm{m/s}\) is approximately \(40.3 \mathrm{kHz}\).

Step-by-step solution

Identifying known values

The initial frequency \(f_0\) emitted by the bat is given as \(39.2 \mathrm{kHz}\) and the speed of the bat \(v_s\) is \(8.58 \mathrm{m/s}\). We also know that the speed of sound in air \(v_w\) is approximately \(343 \mathrm{m/s}\). We're solving for the frequency of the reflected sound \(f\).

Determine the direction of motion

The bat is moving towards the wall, therefore we will be using the formula for when the source is moving towards the observer. The formula in this case is \(f = f_0 \cdot \frac{{v_w + v_s}}{{v_w}}\).

Perform the calculation

Substitute the known values into the formula: \(f = 39.2 \cdot \frac{{343 + 8.58}}{{343}}\). The result should be the frequency of the sound wave reflected off the wall.

Key concepts

Understanding Sound Wave Frequency

Sound waves propagate through the air by creating vibrations in the surrounding medium. The frequency of a sound wave, measured in Hertz (Hz), refers to the number of these vibrations or oscillations that occur per second. It's an essential characteristic of sound that determines pitch. Higher frequencies produce higher pitches, which are perceived as more 'squeaky' to the human ear, while lower frequencies correspond to deeper, bass-like pitches. The concept of sound wave frequency can be illustrated by the image of ripples spreading out when a single stone is thrown into still water; the faster you throw stones (the higher the frequency), the more ripples (or waves) you create per second.

In the textbook exercise scenario, the bat emits a sound at a frequency of 39.2 kilohertz (kHz), indicating a very high number of oscillations every second. Humans can typically hear sounds ranging from 20 Hz to 20 kHz, so the bat's ultrasound is beyond human hearing. This high-frequency sound is ideal for echolocation because it provides the bat with detailed information about its environment.

The Role of Ultrasonic Bleeps in Navigation

Ultrasonic bleeps are sound waves that have frequencies beyond the upper limit of human hearing. Bats and other animals use these ultrasonic sounds for echolocation, navigating their environment by emitting these high-frequency sounds and listening to the echoes that return from objects around them.

The term 'ultrasonic' is derived from 'ultra', meaning beyond, and 'sonic', relating to sound. Since high-frequency sounds have shorter wavelengths, they can detect smaller objects and give precise information about the bat's surroundings—essential for an animal flying at high speeds and often in complete darkness.

In our textbook example, the short emissions of ultrasonic bleeps by the bat provide a continuous stream of acoustic information. As these bleeps reflect off surfaces and return to the bat's ears, they carry data on the location, size, texture, and even the movement of objects, aiding in the bat's flawless navigation through the complex environment of a cave.

Speed of Sound and its Relevance

The speed of sound is the rate at which sound waves propagate through a medium. For most educational and practical purposes, we consider the speed of sound in the air at sea level and at a typical room temperature to be around 343 meters per second (m/s). This speed can change based on factors like temperature, humidity, and the medium through which the sound is traveling; sound travels faster in liquids and even faster in solids compared to air.

Knowing the speed of sound is vital when calculating the Doppler effect, which describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. In the exercise, as the bat speeds towards the wall, the speed of sound assists us in determining how the frequency of the reflected ultrasonic bleeps would change. As the bat moves through the air, the sound waves it emits compress slightly in front of it, increasing the frequency of the waves that hit the wall and then bounce back to the bat. This phenomenon helps the bat gauge its distance from the wall with stunning accuracy and adjust its flight path accordingly.