Sound Wave flash cards

1. 1.What is sound as a physical wave?

   Sound is a mechanical, longitudinal wave consisting of compressions and rarefactions that propagate through a material medium (solid, liquid, or gas). It requires a medium and cannot travel through vacuum.

   Hint: Mechanical + needs a medium.

2. 2.Why are sound waves in a gas classified as longitudinal?

   Because the particle oscillations of the medium are parallel to the direction of wave propagation. The gas cannot sustain shear, so only longitudinal (compressional) disturbances propagate.

   Hint: Particle motion vs propagation direction.

3. 3.What are compressions and rarefactions?

   A compression is a region of higher-than-normal pressure/density; a rarefaction is a region of lower-than-normal pressure/density. In a travelling wave they move outward at the speed of sound.

   Hint: High-pressure vs low-pressure zones.

4. 4.Give the audible frequency range for humans and the terms outside it.

   Audible range is about $20\ \text{Hz}$ to $20{,}000\ \text{Hz}$. Below $20\ \text{Hz}$ is infrasonic; above $20\ \text{kHz}$ is ultrasonic.

   Hint: 20 Hz to 20 kHz.

5. 5.Write the displacement wave equation for a plane sound wave travelling in $+x$.

   $s(x,t)=s_0\sin(\omega t-kx)$, where $s$ is the longitudinal displacement of a particle, $s_0$ the displacement amplitude, $\omega=2\pi f$, and $k=2\pi/\lambda$.

   Hint: Displacement s of a medium particle.

6. 6.State the one-dimensional linear wave equation obeyed by sound.

   $\dfrac{\partial^2 s}{\partial t^2}=v^2\dfrac{\partial^2 s}{\partial x^2}$, where $v$ is the speed of sound. Any function $s=f(x\pm vt)$ satisfies it.

   Hint: Second derivatives in t and x.

7. 7.How are angular frequency $\omega$, wave number $k$, and speed $v$ related?

   $v=\dfrac{\omega}{k}=f\lambda$. This holds for any non-dispersive wave, including sound in air.

   Hint: v = ω/k.

8. 8.What is the general expression for the speed of a longitudinal wave in a medium?

   $v=\sqrt{\dfrac{E}{\rho}}$, where $E$ is the relevant elastic modulus and $\rho$ the density. For fluids $E$ is the bulk modulus $B$; for solid rods it is Young's modulus $Y$.

   Hint: √(elasticity/inertia).

9. 9.Give the speed of sound in a fluid in terms of bulk modulus.

   $v=\sqrt{\dfrac{B}{\rho}}$, where $B$ is the bulk modulus and $\rho$ the density of the fluid.

   Hint: Bulk modulus over density.

10. 10.State Newton's formula for the speed of sound in a gas.

    Newton assumed sound propagation is isothermal, so $B=P$ (pressure). Then $v=\sqrt{\dfrac{P}{\rho}}$. For air at STP this gives about $280\ \text{m/s}$.

    Hint: Isothermal → B = P.

11. 11.Why did Newton's formula fail experimentally?

    It predicted $\approx 280\ \text{m/s}$ whereas the measured value is $\approx 332\ \text{m/s}$ — an error of about $16\%$. The isothermal assumption was wrong.

    Hint: 280 predicted vs 332 measured.

12. 12.What is Laplace's correction and what did he assume?

    Laplace argued the compressions/rarefactions are so rapid that the process is adiabatic, not isothermal. Then $B=\gamma P$, giving $v=\sqrt{\dfrac{\gamma P}{\rho}}$, where $\gamma=C_P/C_V$.

    Hint: Adiabatic → B = γP.

13. 13.Write the corrected (Laplace) speed of sound in a gas and evaluate for air.

    $v=\sqrt{\dfrac{\gamma P}{\rho}}$. For air $\gamma\approx1.4$, so $v\approx\sqrt{1.4}\times280\approx331\ \text{m/s}$, matching experiment.

    Hint: Multiply Newton's value by √γ.

14. 14.Express the speed of sound in an ideal gas in terms of temperature.

    Using $P=\rho RT/M$: $v=\sqrt{\dfrac{\gamma RT}{M}}$, where $T$ is absolute temperature and $M$ the molar mass. So $v\propto\sqrt{T}$.

    Hint: v = √(γRT/M).

15. 15.How does the speed of sound in a gas depend on pressure at constant temperature?

    It is independent of pressure. Since $P/\rho=RT/M$ is fixed at constant $T$, increasing $P$ increases $\rho$ proportionally and $v$ is unchanged.

    Hint: P and ρ change together.

16. 16.How does the speed of sound depend on temperature?

    $v\propto\sqrt{T}$ (absolute temperature). Near room temperature $v$ increases by about $0.61\ \text{m/s}$ per $^\circ\text{C}$ rise in air.

    Hint: √T; ≈0.61 m/s per °C.

17. 17.How does the speed of sound depend on the molar mass of a gas?

    $v\propto\dfrac{1}{\sqrt{M}}$ at fixed temperature. Lighter gases (e.g. hydrogen, helium) carry sound faster than heavier gases.

    Hint: Lighter gas → faster sound.

18. 18.How does humidity affect the speed of sound in air?

    Moist air is less dense than dry air (water vapour, $M=18$, replaces $N_2/O_2$), so $\rho$ decreases and $v$ increases with humidity.

    Hint: Water vapour lowers density.

19. 19.Compare typical speeds of sound in air, water, and steel.

    Roughly: air $\approx340\ \text{m/s}$, water $\approx1480\ \text{m/s}$, steel $\approx5000\ \text{m/s}$. Sound is fastest in solids and slowest in gases.

    Hint: Solids > liquids > gases.

20. 20.Write the pressure wave corresponding to $s=s_0\sin(\omega t-kx)$.

    The excess pressure is $p=p_0\cos(\omega t-kx)$ with $p_0=Bk\,s_0$. Pressure is $90^\circ$ out of phase with displacement.

    Hint: Take p = −B ∂s/∂x.

21. 21.State the phase relationship between the displacement wave and the pressure wave.

    They are $90^\circ$ ($\pi/2$) out of phase. Where displacement is maximum, excess pressure is zero, and where displacement is zero, pressure is maximum.

    Hint: Quarter-cycle apart.

22. 22.Relate pressure amplitude $p_0$ to displacement amplitude $s_0$.

    $p_0=Bk\,s_0=B\dfrac{\omega}{v}s_0=\rho v\omega\,s_0$ (using $B=\rho v^2$). Pressure amplitude grows with frequency for fixed $s_0$.

    Hint: p₀ = ρvωs₀.

23. 23.Define the intensity of a sound wave.

    Intensity is the average power transmitted per unit area normal to the propagation direction: $I=\dfrac{P_{avg}}{A}$, measured in $\text{W/m}^2$.

    Hint: Power per unit area.

24. 24.Give the intensity in terms of pressure amplitude and displacement amplitude.

    $I=\dfrac{p_0^2}{2\rho v}=\tfrac{1}{2}\rho v\,\omega^2 s_0^2$. Intensity is proportional to the square of the amplitude.

    Hint: I ∝ p₀² ∝ s₀².

25. 25.How does the intensity from a point source vary with distance?

    For a point source radiating uniformly, $I\propto\dfrac{1}{r^2}$ (inverse-square law), so amplitude $\propto1/r$. Energy spreads over a sphere of area $4\pi r^2$.

    Hint: Spread over 4πr².

26. 26.What is the standard reference intensity for sound level, and why?

    $I_0=10^{-12}\ \text{W/m}^2$, the approximate threshold of human hearing at $1\ \text{kHz}$. All decibel levels are referenced to it.

    Hint: Threshold of hearing.

27. 27.Define the sound intensity level in decibels.

    $\beta=10\log_{10}\!\left(\dfrac{I}{I_0}\right)\ \text{dB}$, where $I_0=10^{-12}\ \text{W/m}^2$. It is a logarithmic measure of loudness.

    Hint: β = 10 log(I/I₀).

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