# 17.8 Shock waves  (Page 3/8)

 Page 3 / 8

## Summary

• The Mach number is the velocity of a source divided by the speed of sound, $M=\frac{{v}_{\text{s}}}{v}.$
• When a sound source moves faster than the speed of sound, a shock wave is produced as the sound waves interfere.
• A sonic boom is the intense sound that occurs as the shock wave moves along the ground.
• The angle the shock wave produces can be found as $\text{sin}\phantom{\rule{0.2em}{0ex}}\theta =\frac{v}{{v}_{\text{s}}}=\frac{1}{M}.$
• A bow wake is produced when an object moves faster than the speed of a mechanical wave in the medium, such as a boat moving through the water.

## Key equations

 Pressure of a sound wave $\text{Δ}P=\text{Δ}{P}_{\text{max}}\text{sin}\left(kx\mp \omega t+\varphi \right)$ Displacement of the oscillating molecules of a sound wave $s\left(x,t\right)={s}_{\text{max}}\text{cos}\left(kx\mp \omega t+\varphi \right)$ Velocity of a wave $v=f\lambda$ Speed of sound in a fluid $v=\sqrt{\frac{\beta }{\rho }}$ Speed of sound in a solid $v=\sqrt{\frac{Y}{\rho }}$ Speed of sound in an ideal gas $v=\sqrt{\frac{\gamma RT}{M}}$ Speed of sound in air as a function of temperature $v=331\frac{\text{m}}{\text{s}}\sqrt{\frac{{T}_{\text{K}}}{273\phantom{\rule{0.2em}{0ex}}\text{K}}}=331\frac{\text{m}}{\text{s}}\sqrt{1+\frac{{T}_{\text{C}}}{273\text{°}\text{C}}}$ Decrease in intensity as a spherical wave expands ${I}_{2}={I}_{1}{\left(\frac{{r}_{1}}{{r}_{2}}\right)}^{2}$ Intensity averaged over a period $I=\frac{⟨P⟩}{A}$ Intensity of sound $I=\frac{{\left(\text{Δ}{p}_{\text{max}}\right)}^{2}}{2\rho v}$ Sound intensity level $\beta \left(dB\right)=10\phantom{\rule{0.2em}{0ex}}{\text{log}}_{10}\left(\frac{I}{{I}_{0}}\right)$ Resonant wavelengths of a tube closed at one end ${\lambda }_{n}=\frac{4}{n}L,\phantom{\rule{0.5em}{0ex}}n=1,3,5\text{,…}$ Resonant frequencies of a tube closed at one end ${f}_{n}=n\frac{v}{4L},\phantom{\rule{0.5em}{0ex}}n=1,3,5\text{,…}$ Resonant wavelengths of a tube open at both ends ${\lambda }_{n}=\frac{2}{n}L,\phantom{\rule{0.5em}{0ex}}n=1,2,3\text{,…}$ Resonant frequencies of a tube open at both ends ${f}_{n}=n\frac{v}{2L},\phantom{\rule{0.5em}{0ex}}n=1,2,3\text{,…}$ Beat frequency produced by two waves that differ in frequency ${f}_{\text{beat}}=|{f}_{2}-{f}_{1}|$ Observed frequency for a stationary observer and a moving source ${f}_{\text{o}}={f}_{\text{s}}\left(\frac{v}{v\mp {v}_{\text{s}}}\right)$ Observed frequency for a moving observer and a stationary source ${f}_{\text{o}}={f}_{\text{s}}\left(\frac{v±{v}_{\text{o}}}{v}\right)$ Doppler shift for the observed frequency ${f}_{\text{o}}={f}_{\text{s}}\left(\frac{v±{v}_{\text{o}}}{v\mp {v}_{\text{s}}}\right)$ Mach number $M=\frac{{v}_{s}}{v}$ Sine of angle formed by shock wave $\text{sin}\phantom{\rule{0.2em}{0ex}}\theta =\frac{v}{{v}_{s}}=\frac{1}{M}$

## Conceptual questions

What is the difference between a sonic boom and a shock wave?

Due to efficiency considerations related to its bow wake, the supersonic transport aircraft must maintain a cruising speed that is a constant ratio to the speed of sound (a constant Mach number). If the aircraft flies from warm air into colder air, should it increase or decrease its speed? Explain your answer.

The speed of sound decreases as the temperature decreases. The Mach number is equal to $M=\frac{{v}_{\text{s}}}{v},$ so the plane should slow down.

When you hear a sonic boom, you often cannot see the plane that made it. Why is that?

## Problems

An airplane is flying at Mach 1.50 at an altitude of 7500.00 meters, where the speed of sound is $v=343.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\text{.}$ How far away from a stationary observer will the plane be when the observer hears the sonic boom?

A jet flying at an altitude of 8.50 km has a speed of Mach 2.00, where the speed of sound is $v=340.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\text{.}$ How long after the jet is directly overhead, will a stationary observer hear a sonic boom?

$\begin{array}{}\\ \\ \theta =30.02\text{°}\hfill \\ {v}_{\text{s}}=680.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\hfill \\ \text{tan}\phantom{\rule{0.2em}{0ex}}\theta =\frac{y}{{v}_{\text{s}}t},\phantom{\rule{0.5em}{0ex}}t=21.65\phantom{\rule{0.2em}{0ex}}\text{s}\hfill \end{array}$

The shock wave off the front of a fighter jet has an angle of $\theta =70.00\text{°}$ . The jet is flying at 1200 km/h. What is the speed of sound?

A plane is flying at Mach 1.2, and an observer on the ground hears the sonic boom 15.00 seconds after the plane is directly overhead. What is the altitude of the plane? Assume the speed of sound is ${v}_{\text{w}}=343.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\text{.}$

$\begin{array}{}\\ \\ \text{sin}\phantom{\rule{0.2em}{0ex}}\theta =\frac{1}{M},\phantom{\rule{0.5em}{0ex}}\theta =56.47\text{°}\hfill \\ y=9.31\phantom{\rule{0.2em}{0ex}}\text{km}\hfill \end{array}$

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