Q: A piston compresses water in a rigid cylinder, increasing the pressure by $$\Delta P = 50$$ MPa and decreasing the volume by $$2.25\%. $$What is the bulk modulus $$B of $$water?

$$B = \frac{\Delta P}{\Delta V / V} = \frac{50 \times 10^6$}{0.0225} \approx 2.22 \times 10^9$ Pa

Q: Given air at $$T = 300 K$$, $$\gamma = 1.4$$, $$k_B = 1.381 \times 10$$^{-23}$$ J/K, and molecular mass m = 4.81$$ \times $$10^{-26} kg$$, what is the speed of sound in air?

$$v = \sqrt{\frac{1.4 \times 1.381 \times 10$$^{-23}$$ \times 300}{4.81 \times 10$$^{-26}$$}} \approx 347 m/s$$

Q: If the density of air at 300 K is $$1.177 kg/m$^3$$ and the atmospheric pressure is $$1.013 \times 10^5$ Pa, what is the speed of sound in air using v$$ = \sqrt{\frac{\gamma P}{\rho}}$$ with $$\gamma = 1.4$$?

$$v = \sqrt{\frac{1.4 \times 1.013 \times 10^5$}{1.177}} \approx 347 m/s$$

Question 1

In the harmonic oscillator chain shown, what is the equation of motion for the nth mass, given that each mass $$m is $$connected by springs of stiffness $$K$$ and rest length $$a$$?

Accepted Answer

$$m \frac{d^2$ y_n}{dt^2$} = K(y_{n+1} - 2y_n + y_{n-1})$$

Question 2

Using dimensional analysis, what is the correct expression for the speed of a wave in a harmonic oscillator chain with spring constant $$K$$, mass $$m$$, and spacing $$a$$?

Accepted Answer

$$v = a \sqrt{\frac{K}{m}}$$

Question 3

Which of the following best describes the polynomial approximation of a function $$f(x)$$ near $$x_0 up to $$the second order?

Accepted Answer

$$f(x) \approx f(x_0$) + f'(x_0$)(x-x_0$) + \frac{1}{2}f''(x_0$)(x-x_0$)^2$

Question 4

In the continuous limit of the harmonic oscillator chain, what is the resulting wave equation for the displacement $$y(x,t)$$?

Accepted Answer

$$\frac{\$$partial^2$$ y}{\partial t^2$} = v^2$ \frac{\$$partial^2$$ y}{\partial x^2$}$$

Question 5

A slinky of mass $$m$$, equilibrium length $$L_0$$, spring constant $$K$$, and $$N$$ windings is stretched to length $$L. $$What is the speed of a longitudinal wave along the slinky?

Accepted Answer

$$v = \sqrt{\frac{K L^2$}{m}}$$

Question 6

If the speed of sound in a liquid is given by $$v = \sqrt{\frac{B}{\rho}}$$, what does $$B$$ represent?

Accepted Answer

The bulk modulus of the liquid

Question 7

A piston compresses water in a rigid cylinder, increasing the pressure by $$\Delta P = 50$$ MPa and decreasing the volume by $$2.25\%. $$What is the bulk modulus $$B of $$water?

Accepted Answer

$$B = \frac{\Delta P}{\Delta V / V} = \frac{50 	imes 10^6$}{0.0225} \approx 2.22 	imes 10^9$ Pa

Question 8

For a gas, the speed of sound is $$v = \sqrt{\frac{\gamma k_B T}{m}}. $$What does $$\gamma$$ represent?

Accepted Answer

The adiabatic index (heat capacity ratio)

Question 9

Which of the following statements is true about the speed of sound in air as temperature increases?

Accepted Answer

The speed of sound increases with temperature because $$v \propto \sqrt{T}$$

Question 10

Given air at $$T = 300 K$$, $$\gamma = 1.4$$, $$k_B = 1.381 	imes 10$$^{-23}$$ J/K, and molecular mass m = 4.81$$ 	imes $$10^{-26} kg$$, what is the speed of sound in air?

Accepted Answer

$$v = \sqrt{\frac{1.4 	imes 1.381 	imes 10$$^{-23}$$ 	imes 300}{4.81 	imes 10$$^{-26}$$}} \approx 347 m/s$$

Question 11

Why is the bulk modulus for sound waves in a gas given by $$B = \gamma P$$ rather than $$B = P$$?

Accepted Answer

Because sound waves are adiabatic processes, not isothermal

Question 12

If the density of air at 300 K is $$1.177 kg/m$^3$$ and the atmospheric pressure is $$1.013 	imes 10^5$ Pa, what is the speed of sound in air using v$$ = \sqrt{\frac{\gamma P}{ho}}$$ with $$\gamma = 1.4$$?

Accepted Answer

$$v = \sqrt{\frac{1.4 	imes 1.013 	imes 10^5$}{1.177}} \approx 347 m/s$$

Question 13

A sound wave travels through a liquid with bulk modulus $$B = 2.2 	imes 10^9$ Pa and density $$ho = 1000$$ kg/m^3$. What is the speed of sound in this liquid?

Accepted Answer

$$v = \sqrt{\frac{2.2 	imes 10^9$}{1000}} \approx 1483 m/s$$

Question 14

Which of the following best explains why sound travels faster in solids than in gases?

Accepted Answer

Solids have a much higher bulk modulus compared to their density

Question 15

If a longitudinal pulse travels the length $$L of a $$slinky at speed $$v$$, what is the time $$t$$ required for the pulse to travel from one end to the other?

Accepted Answer

$$t = \frac{L}{v}$$

Question 16

Which physical property is primarily responsible for the difference in the speed of sound between air and water?

Accepted Answer

Bulk modulus

Speed of Sound Waves: Harmonic Oscillator Chains and Sound in Materials

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