# 17.8 Shock waves  (Page 4/8)

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A bullet is fired and moves at a speed of 1342 mph. Assume the speed of sound is $v=340.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\text{.}$ What is the angle of the shock wave produced?

A speaker is placed at the opening of a long horizontal tube. The speaker oscillates at a frequency of f , creating a sound wave that moves down the tube. The wave moves through the tube at a speed of $v=340.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\text{.}$ The sound wave is modeled with the wave function $s\left(x,t\right)={s}_{\text{max}}\text{cos}\left(kx-\omega t+\varphi \right)$ . At time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ , an air molecule at $x=2.3\phantom{\rule{0.2em}{0ex}}\text{m}$ is at the maximum displacement of 6.34 nm. At the same time, another molecule at $x=2.7\phantom{\rule{0.2em}{0ex}}\text{m}$ has a displacement of 2.30 nm. What is the wave function of the sound wave, that is, find the wave number, angular frequency, and the initial phase shift?

$\begin{array}{}\\ \\ {s}_{1}=6.34\phantom{\rule{0.2em}{0ex}}\text{nm}\hfill \\ {s}_{2}=2.30\phantom{\rule{0.2em}{0ex}}\text{nm}\hfill \\ k{x}_{1}+\varphi =0\phantom{\rule{0.2em}{0ex}}\text{rad}\hfill \\ k{x}_{2}+\varphi =1.20\phantom{\rule{0.2em}{0ex}}\text{rad}\hfill \\ k\left({x}_{2}-{x}_{1}\right)=1.20\phantom{\rule{0.2em}{0ex}}\text{rad}\hfill \\ k=3.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}\hfill \\ \omega =1019.62\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}\hfill \\ {s}_{1}={s}_{\text{max}}\text{cos}\left(k{x}_{1}-\varphi \right)\hfill \\ \varphi =5.66\phantom{\rule{0.2em}{0ex}}\text{rad}\hfill \\ s\left(x,t\right)=6.30\phantom{\rule{0.2em}{0ex}}\text{nm}\text{cos}\left(3.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-1019.62\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t+5.66\right)\hfill \end{array}$

An airplane moves at Mach 1.2 and produces a shock wave. (a) What is the speed of the plane in meters per second? (b) What is the angle that the shock wave moves?

A 0.80-m-long tube is opened at both ends. The air temperature is $26\text{°}\text{C}\text{.}$ The air in the tube is oscillated using a speaker attached to a signal generator. What are the wavelengths and frequencies of first two modes of sound waves that resonate in the tube?

${v}_{\text{s}}=346.40\phantom{\rule{0.2em}{0ex}}\text{m/s}$ ;
$\begin{array}{}\\ \\ {\lambda }_{n}=\frac{2}{n}L\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{f}_{n}=\frac{{v}_{\text{s}}}{{\lambda }_{n}}\hfill \\ {\lambda }_{1}=1.60\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{f}_{1}=216.50\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \\ {\lambda }_{2}=0.80\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{f}_{1}=433.00\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \end{array}$

A tube filled with water has a valve at the bottom to allow the water to flow out of the tube. As the water is emptied from the tube, the length L of the air column changes. A 1024-Hz tuning fork is placed at the opening of the tube. Water is removed from the tube until the $n=5$ mode of a sound wave resonates. What is the length of the air column if the temperature of the air in the room is $18\text{°}\text{C?}$

Consider the following figure. The length of the string between the string vibrator and the pulley is $L=1.00\phantom{\rule{0.2em}{0ex}}\text{m}\text{.}$ The linear density of the string is $\mu =0.006\phantom{\rule{0.2em}{0ex}}\text{kg/m}\text{.}$ The string vibrator can oscillate at any frequency. The hanging mass is 2.00 kg. (a)What are the wavelength and frequency of $n=6$ mode? (b) The string oscillates the air around the string. What is the wavelength of the sound if the speed of the sound is ${v}_{s}=343.00\phantom{\rule{0.2em}{0ex}}\text{m/s}$ ?

a. $\begin{array}{}\\ \\ {\lambda }_{6}=0.40\phantom{\rule{0.2em}{0ex}}\text{m}\hfill \\ \\ \\ v=57.15\frac{\text{m}}{\text{s}}\hfill \\ {f}_{6}=142.89\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \end{array}$ ; b. ${\lambda }_{\text{s}}=2.40\phantom{\rule{0.2em}{0ex}}\text{m}$

Early Doppler shift experiments were conducted using a band playing music on a train. A trumpet player on a moving railroad flatcar plays a 320-Hz note. The sound waves heard by a stationary observer on a train platform hears a frequency of 350 Hz. What is the flatcar’s speed in mph? The temperature of the air is ${T}_{\text{C}}=22\text{°}\text{C}$ .

Two cars move toward one another, both sounding their horns $\left({f}_{s}=800\phantom{\rule{0.2em}{0ex}}\text{Hz}\right)$ . Car A is moving at 65 mph and Car B is at 75 mph. What is the beat frequency heard by each driver? The air temperature is ${T}_{C}=22.00\text{°}\text{C}$ .

$\begin{array}{}\\ \\ v=344.08\frac{\text{m}}{\text{s}}\hfill \\ {v}_{A}=29.05\frac{\text{m}}{\text{s}},\phantom{\rule{0.5em}{0ex}}{v}_{B}=33.52\phantom{\rule{0.2em}{0ex}}\text{m/s}\hfill \\ {f}_{A}=961.18\phantom{\rule{0.2em}{0ex}}\text{Hz,}\phantom{\rule{0.5em}{0ex}}\hfill \\ {f}_{B}=958.89\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \\ {f}_{A,\text{beat}}=161.18\phantom{\rule{0.2em}{0ex}}\text{Hz,}\phantom{\rule{0.5em}{0ex}}{f}_{B,\text{beat}}=158.89\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \end{array}$

Student A runs after Student B. Student A carries a tuning fork ringing at 1024 Hz, and student B carries a tuning fork ringing at 1000 Hz. Student A is running at a speed of ${v}_{A}=5.00\phantom{\rule{0.2em}{0ex}}\text{m/s}$ and Student B is running at ${v}_{B}=6.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\text{.}$ What is the beat frequency heard by each student? The speed of sound is $v=343.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\text{.}$

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