Sound Intensity and Standing Sound Waves: Study Notes

Sound Intensity and Standing Sound Waves

Energy Transfer in a Sound Wave

Sound waves transfer energy through the oscillatory motion of particles in a medium. Each small fluid element undergoes simple harmonic motion, described by the displacement function:

• Displacement function: $y(x, t) = A \sin(kx - \omega t)$

• Energy of a harmonic oscillator: $\Delta E = \frac{1}{2} \Delta m \omega^2 A^2$

• Mass of element: $\Delta m = \rho S \Delta x$

• Energy transfer rate (intensity): $I = \frac{\Delta E}{S \Delta t}$, where $\Delta t = \frac{\Delta x}{v}$

• Intensity in terms of amplitude: $I = \frac{1}{2} \omega^2 A^2 \rho v$

Example: The energy carried by a sound wave is proportional to the square of its amplitude and frequency.

Pressure Oscillations in a Sound Wave

Sound waves cause oscillations in pressure as well as displacement. The excess pressure in a small fluid element is derived from the change in volume due to the wave:

• Volume change: $\Delta V = S (y(x+a, t) - y(x, t))$

• For small $a$: $\Delta V = V \frac{\partial y}{\partial x}$

• Excess pressure: $\Delta P(x, t) = -B \frac{\Delta V}{V} = -B \frac{\partial y}{\partial x} = -BkA \cos(kx - \omega t)$

• Pressure amplitude: $\Delta P_{max} = BkA$

Example: The pressure oscillations are out of phase with displacement oscillations.

Intensity of a Sound Wave

The intensity of a sound wave can be expressed in terms of both displacement amplitude and pressure amplitude. The speed of sound relates these quantities:

• Intensity (displacement): $I = \frac{1}{2} \omega^2 A^2 \rho v$

• Pressure amplitude: $\Delta P_{max} = BkA$

• Speed of sound: $v = \frac{\omega}{k} = \sqrt{\frac{B}{\rho}}$

• Intensity (pressure): $I = \frac{v}{2B} \Delta P_{max}^2 = \frac{1}{2} \rho v \Delta P_{max}^2 = \frac{1}{2} \sqrt{\rho B} \Delta P_{max}^2$

• Intensity in decibels: $\beta = (10 \text{ dB}) \log_{10} \frac{I}{I_0}$, $I = I_0 10^{\beta/(10 \text{ dB})}$

• Human hearing range: 20 Hz to 20 kHz

• Threshold of hearing: $I_0 = 10^{-12} \text{ W/m}^2 = 0 \text{ dB}$

• Pain threshold: $1 \text{ W/m}^2 = 120 \text{ dB}$

Example: A sound wave with higher amplitude or frequency carries more energy and is perceived as louder.

Pressure and Displacement in a Sound Wave

Pressure and displacement oscillations in a sound wave are related but occur out of phase. Displacement maxima correspond to pressure minima and vice versa:

• Displacement: $y(x, t) = A \cos(kx - \omega t)$

• Excess pressure: $\Delta P(x, t) = -B \frac{\partial y}{\partial x} = \Delta P_{max} \sin(kx - \omega t)$

• Pressure amplitude: $\Delta P_{max} = ABk$

• Pressure maxima/minima: Zero displacement

• Displacement maxima/minima: Zero excess pressure

Boundary Conditions in Sound Waves

Standing waves form in tubes due to reflections at boundaries. The nature of the boundary (open or closed) determines the location of nodes and antinodes:

• Displacement nodes: Pressure antinodes

• Closed end: Displacement node, pressure antinode

• Open end: Displacement antinode, pressure node

Normal Modes in a Tube with Both Ends Open

In a tube open at both ends, standing waves have pressure nodes at the ends. The allowed wavelengths and frequencies are determined by the tube length:

• Excess pressure: $\Delta P(x, t) = \Delta P_{max} \sin(kx) \cos(\omega t)$

• Displacement: $y(x, t) = A \cos(kx) \cos(\omega t)$

• Pressure nodes: $x = 0, L$

• Allowed wavelengths: $L = n \frac{\lambda}{2}$, $\lambda = \frac{2L}{n}$, $n = 1, 2, 3, ...$

• Frequency in nth mode: $f_n = \frac{v}{\lambda_n} = n f_1$

• Fundamental frequency: $f_1 = \frac{v}{2L}$

Normal Modes in a Tube with One End Open, One End Closed

For a tube with one open and one closed end, the boundary conditions change, resulting in different allowed wavelengths and frequencies:

• Excess pressure: $\Delta P(x, t) = \Delta P_{max} \sin(kx) \cos(\omega t)$

• Displacement: $y(x, t) = A \cos(kx) \cos(\omega t)$

• Pressure node at $x = 0$: Displacement node at $x = L$

• Allowed wavelengths: $L = \left(n + \frac{1}{2}\right) \frac{\lambda}{2}$, $\lambda = \frac{4L}{2n+1}$, $n = 0, 1, 2, ...$

• Frequencies: $f_n = \frac{v}{\lambda_n}$ with odd $n = 1, 3, 5, ...$

• Fundamental frequency: $f_1 = \frac{v}{4L}$

Exercises

1. Exercise 1: What is the amplitude of pressure oscillations in a sound wave of intensity 50 dB under normal conditions?

2. Exercise 2: The intensity of sound is 20 dB at a 15 m distance from the source. How close should you move to the source for the sound level to be 60 dB?

3. Exercise 3: A standing wave in a pipe has a frequency of 440 Hz. The next higher harmonic has a frequency of 660 Hz. The speed of sound = 340 m/s. Determine the fundamental frequency and the length of the pipe (a) if its both ends are open; (b) if its one end is open and the other closed.

4. Exercise 4: At what frequency does a 3-meter long chimney moan?

Additional info: These exercises reinforce the concepts of sound intensity, decibel scale, and standing wave modes in tubes.