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Chapter 12: Problem 39

A source of sound waves of frequency \(1.0 \mathrm{kHz}\) is stationary. An observer is traveling at 0.50 times the speed of sound. (a) What is the observed frequency if the observer moves toward the source? (b) Repeat if the observer moves away from the source instead.

Short Answer

Expert verified
Answer: The observed frequencies are 1.5 kHz when the observer moves towards the source, and 0.5 kHz when the observer moves away from the source.

Step by step solution

01

Identify the Doppler Effect Formula

The Doppler effect formula for sound waves is given by: $$f_{obs} = f_{src} \frac{v \pm v_{obs}}{v \mp v_{src}},$$ where \(f_{obs}\) is the observed frequency, \(f_{src}\) is the source frequency, \(v\) is the speed of sound, \(v_{obs}\) is the observer's speed, and \(v_{src}\) is the source's speed. In our case, the source is stationary, so \(v_{src} = 0\). The observer's speed is 0.50 times the speed of sound, which can be represented as \(v_{obs} = 0.50v\). #Step 2: Calculate Observed Frequency when the Observer moves toward the source#
02

Calculate observed frequency when the observer moves toward the source

When the observer moves toward the source, we use the formula with the plus sign for the numerator and the minus sign for the denominator. Plugging in the given values, we get: $$f_{obs1} = (1.0 \mathrm{kHz}) \frac{v + 0.50v}{v - 0} = (1.0 \mathrm{kHz}) \frac{1.50v}{v}$$ We can simplify the expression: $$f_{obs1} = 1.5 \times (1.0 \mathrm{kHz}) = 1.5 \mathrm{kHz}$$ So, the observed frequency when the observer moves toward the source is 1.5 kHz. #Step 3: Calculate Observed Frequency when the Observer moves away from the source#
03

Calculate observed frequency when the observer moves away from the source

When the observer moves away from the source, we use the formula with the minus sign for the numerator and the plus sign for the denominator. Plugging in the given values, we get: $$f_{obs2} = (1.0 \mathrm{kHz}) \frac{v - 0.50v}{v + 0} = (1 \mathrm{kHz}) \frac{0.50v}{v}$$ We can simplify the expression: $$f_{obs2} = 0.5 \times (1.0 \mathrm{kHz}) = 0.5 \mathrm{kHz}$$ So, the observed frequency when the observer moves away from the source is 0.5 kHz. To summarize, the observed frequencies for the given situation are: (a) 1.5 kHz when the observer moves towards the source, and (b) 0.5 kHz when the observer moves away from the source.

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Most popular questions from this chapter

(a) What is the pressure amplitude of a sound wave with an intensity level of \(120.0 \mathrm{dB}\) in air? (b) What force does this exert on an eardrum of area \(0.550 \times 10^{-4} \mathrm{m}^{2} ?\)
A periodic wave is composed of the superposition of three sine waves whose frequencies are \(36,60,\) and 84 Hz. The speed of the wave is 180 m/s. What is the wavelength of the wave? [Hint: The \(36 \mathrm{Hz}\) is not necessarily the fundamental frequency.]
At what frequency \(f\) does a sound wave in air have a wavelength of $15 \mathrm{cm},$ about half the diameter of the human head? Some methods of localization work well only for frequencies below \(f\), while others work well only above \(f\). (See Conceptual Questions 4 and 5 .)
A cello string has a fundamental frequency of \(65.40 \mathrm{Hz}\) What beat frequency is heard when this cello string is bowed at the same time as a violin string with frequency of \(196.0 \mathrm{Hz} ?\) [Hint: The beats occur between the third harmonic of the cello string and the fundamental of the violin. \(]\)
Two tuning forks, \(A\) and \(B\), excite the next-to-lowest resonant frequency in two air columns of the same length, but A's column is closed at one end and B's column is open at both ends. What is the ratio of A's frequency to B's frequency?
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