Propagation of Sound Waves

A. Production:

Sound is produced by mechanical vibrations of objects (tuning fork prongs, stretched strings, vocal cords). It travels as a disturbance carrying energy without the bodily transport of the medium particles.

B. The Bell Jar Experiment (Material Medium Requirement):

1. An electric bell is suspended inside an airtight glass bell jar connected to a rotary vacuum pump.
2. Initially, with air inside the jar, the hammer striking the gong produces a loud, clear sound.
3. When the vacuum pump is turned on and air is gradually evacuated, the sound becomes fainter and fainter, until it dies away into complete silence, even though the hammer is still seen striking the gong vigorously.
4. When air is readmitted, the sound instantly returns.
5. Conclusion: Sound requires a material medium for its propagation and cannot travel through a vacuum.

A. Longitudinal Waves:

A wave in which the individual particles of the medium vibrate back and forth parallel to the direction of wave propagation.

• Compression ($C$): Region where medium particles are crowded together $\implies$ High density and High pressure.
• Rarefaction ($R$): Region where medium particles are spread apart $\implies$ Low density and Low pressure.

B. Derivation of Wave Equation $V = f\lambda$:

• Wavelength ($\lambda$): The distance traveled by the wave during the time period ($T$) of one complete vibration: $\text{Distance} = \lambda$.
• Time taken $= T$.
• By definition: $\text{Wave Velocity } (V) = \frac{\text{Distance}}{\text{Time}} = \frac{\lambda}{T}$.
• Since frequency $f = \frac{1}{T}$:

A. Medium Elasticity & Density ($V = \sqrt{E / \rho}$):

Although solids have higher density, their modulus of elasticity ($E$) is enormously higher than liquids and gases. Speed in steel $\approx 5100\text{ m/s}$; in water $\approx 1500\text{ m/s}$; in air ($0^\circ\text{C}$) $\approx 332\text{ m/s}$; in air ($20^\circ\text{C}$) $\approx 343\text{ m/s}$.

B. Factors Affecting Speed of Sound in Air:

• Temperature: $V \propto \sqrt{T(\text{K})}$. In air, speed increases by approximately $\mathbf{0.61\text{ m/s}}$ for every $1^\circ\text{C}$ rise in temperature: $V_t = V_0 + 0.61t$.
• Humidity: Humid air containing water vapor ($ ext{H}_2 ext{O}$, molar mass $18$) is less dense than dry air ($ ext{N}_2 + ext{O}_2$, molar mass $29$). Since $V \propto \frac{1}{\sqrt{\rho}}$, sound travels faster in humid/moist air than in dry air.
• Pressure: Speed of sound in a gas is completely independent of atmospheric pressure (because when pressure changes, density changes proportionally at constant temperature, keeping $\frac{P}{\rho}$ constant).

Problem 1: A source produces $50$ crests and $50$ troughs in $0.5\text{ seconds}$. If the distance between a crest and its consecutive trough is $10\text{ cm}$, find: (i) Frequency, (ii) Wavelength, (iii) Wave velocity.

Solution:

1. Frequency $f$ is number of complete waves per second:

2. The distance between a crest and consecutive trough is half a wavelength ($\frac{\lambda}{2}$):

3. Wave velocity:

Problem 2: A SONAR signal sent from a research ship to the seabed returns in $1.6\text{ seconds}$. If the speed of sound in seawater is $1500\text{ m/s}$, calculate the depth of the sea.

Solution:

The sound travels to the ocean bed and reflects back, covering a total distance of $2d$ in time $t = 1.6\text{ s}$: