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Chapter 16: Problem 48

You are traveling in a car toward a hill at a speed of \(40.0 \mathrm{mph}\). The car's horn emits sound waves of frequency \(250 . \mathrm{Hz},\) which move with a speed of \(340 . \mathrm{m} / \mathrm{s}\) a) Determine the frequency with which the waves strike the hill. b) What is the frequency of the reflected sound waves you hear? c) What is the beat frequency produced by the direct and the reflected sounds at your ears?

Short Answer

Expert verified
Answer: The beat frequency produced by the direct and reflected sounds at the ears is 1.93 Hz.

Step by step solution

01

Convert car speed to m/s

To convert the speed from mph to m/s, we'll use the conversion factor \(1\, \text {mph} = 0.447\, \text{m/s}\) Car speed = \(40.0\times 0.447\) = \(17.88\,\text{m/s}\) Now, we can proceed to solve for the frequencies and beat frequency.
02

Find the frequency with which the waves strike the hill

To find the frequency, we'll use the Doppler effect formula for approaching objects: \(f'=f\frac{v+v_0}{v}\) Where, \(f'\) is the apparent frequency, \(f\) is the emitted frequency, \(v\) is the speed of sound, \(v_0\) is the speed of the object (car). Plugging in the numbers: \(f' = 250\times \frac{340 + 17.88}{340}\) \(f'= 267.42\,\text{Hz}\) The frequency with which the waves strike the hill is \(267.42\,\text{Hz}\).
03

Find the frequency of the reflected sound waves you hear

Now, we will use the same Doppler effect formula but for receding objects. As the wave hits the hill and returns to the observer, the car is moving away from those waves. \(f''=f'\frac{v-v_0}{v}\) Where, \(f''\) is the frequency of the reflected waves. Plugging in the numbers: \(f'' = 267.42\times \frac{340 - 17.88}{340}\) \(f'' = 248.07\,\text{Hz}\) The frequency of the reflected sound waves you hear is \(248.07\,\text{Hz}\).
04

Calculate the beat frequency

The beat frequency is the absolute difference between the frequencies of the direct and reflected sounds: Beat frequency = \(|f''-f|\) Beat frequency = \(|248.07- 250|\) Beat frequency = \(1.93\, \mathrm{Hz}\) The beat frequency produced by the direct and reflected sounds at your ears is \(1.93\, \mathrm{Hz}\).

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Most popular questions from this chapter

Consider a sound wave (that is, a longitudinal displacement wave) in an elastic medium with Young's modulus \(Y\) (solid) or bulk modulus \(B\) (fluid) and unperturbed density \(\rho_{0}\). Suppose this wave is described by the wave function \(\delta x(x, t),\) where \(\delta x\) denotes the displacement of a point in the medium from its equilibrium position, \(x\) is position along the path of the wave at equilibrium, and \(t\) is time. The wave can also be regarded as a pressure wave, described by wave function \(\delta p(x, t),\) where \(\delta p\) denotes the change of pressure in the medium from its equilibrium value. a) Find the relationship between \(\delta p(x, t)\) and \(\delta x(x, t),\) in general. b) If the displacement wave is a pure sinusoidal function, with amplitude \(A\), wave number \(\kappa,\) and angular frequency \(\omega,\) given by \(\delta x(x, t)=A \cos (\kappa x-\omega t)\) what is the corresponding pressure wave function, \(\delta p(x, t) ?\) What is the amplitude of the pressure wave?

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