Question

The \(15.8-\mathrm{kHz}\) whine of the turbines in the jet engines of an aircraft moving with speed \(193 \mathrm{~m} / \mathrm{s}\) is heard at what frequency by the pilot of a second craft trying to overtake the first at a speed of \(246 \mathrm{~m} / \mathrm{s}\) ?

Short answer

The frequency of the turbines as heard by the pilot in the second aircraft is approximately \(32.27 kHz\) or \(32270 Hz\).

Step-by-step solution

Identify given quantities

From the problem, we can identify the following values:- The frequency of the turbines in the aircraft, \(f_s = 15.8 ~kHz\) or \(15800 ~Hz\).- The speed of the aircraft producing the sound, \(v_s = 193 m/s\).- The speed of the aircraft overhearing the sound or the speed of the observer, \(v_o = 246 m/s\).- The speed of sound in air at sea level, \(v = 343 m/s\).

Apply the Doppler effect formula

Next we apply the Doppler effect formula for sound which is given by:\[ f_o = f_s \frac{v + v_o}{v - v_s} \]where - \(f_o\) is observed frequency- \(f_s\) is the source frequency - \(v\) is the speed of sound - \(v_o\) is the speed of the observer- \(v_s\) is the speed of the source.

Substitute the known values into the formula and solve

Inserting the known values from Step 1 into the formula, we get:\[ f_o = 15800 Hz \frac{343 m/s + 246 m/s}{343 m/s - 193 m/s} \]Solving the calculation on the right gives a value of approximately 32.27 kHz or \(32270 Hz\).

Key concepts

Sound Frequency

In the realm of acoustics, sound frequency is a fundamental concept that refers to the number of sound waves passing a point in one second, measured in Hertz (Hz). Frequency determines the pitch of a sound: higher frequencies produce higher pitches, while lower frequencies result in lower pitches. For example, a deep bass tone might have a frequency of around 30 Hz, whereas a high-pitched whistle might have a frequency of over 10,000 Hz.

Understanding sound frequency is crucial when discussing the Doppler effect, which occurs when there is a relative motion between a sound source and an observer. The Doppler effect causes the observed frequency to differ from the source frequency. In the context of a moving aircraft with turbine whines of 15.8 kHz, this frequency is what the source (the aircraft's turbines) emits. However, depending on the motion of the observer, this frequency can be perceived as higher or lower than 15.8 kHz due to the Doppler shift. If the observer is moving towards the source, the frequency seems higher, and if moving away, it seems lower.

Relative Motion in Physics

The concept of relative motion plays a pivotal role when analyzing situations involving the Doppler effect. Relative motion in physics refers to the calculation of the motion of an object with respect to another moving object. It is an important concept as it helps us understand how velocity is not absolute but is dependent on the observer’s frame of reference. For instance, if you're on a train passing another train moving in the same direction but at a slower speed, the other train may appear to be moving backwards relative to your position.

In our exercise focusing on the Doppler effect, the relative motion between the two aircraft is critical. The observer's aircraft is moving faster than the sound source's aircraft. The speeds of both the source (\(193 m/s\)) and the observer (\(246 m/s\)) must be considered relative to the medium through which the sound travels—which in this case, is the air. Relative speeds affect how the sound frequency emitted by the source is perceived by the observer.

Speed of Sound

Lastly, the speed of sound is a vital element in the study of sound waves and the Doppler effect. The speed of sound is the distance that sound waves travel through an elastic medium per unit of time. It's primarily dependent on the medium's properties, such as temperature, humidity, and pressure. In the air at sea level, the speed of sound is approximately 343 meters per second (m/s), often used as a standard value in calculations.

In our exercise, the speed of sound (\(v\)) establishes the baseline speed at which the turbine's sound waves travel to the pilot of the overtaking aircraft. When the source and observer are in motion, the Doppler effect formula incorporates the speed of sound to determine how those speeds affect the frequency perceived by the observer. The fact that both the source and observer are moving at speeds less than the speed of sound means we are dealing with subsonic conditions, allowing us to use the standard Doppler shift formula where the speed of sound remains constant for the given atmospheric conditions.