Comprehensive Guide to Wave Motion and Acoustics

6. Wave Motion

Wave motion is a disturbance that propagates through a medium, transferring energy without the net transfer of matter. Waves are ubiquitous in nature, from the ripples on a pond to the propagation of light and sound. Understanding wave motion is crucial for many fields of science and engineering.

6.1 Progressive Waves

Progressive waves, also known as traveling waves, are waves that transmit energy from one point to another in the direction of wave propagation. In these waves, the disturbance moves through the medium. Examples include water waves and sound waves in air.

• Transverse Waves: The particles of the medium oscillate perpendicular to the direction of wave propagation. Examples: light waves, waves on a stretched string.
• Longitudinal Waves: The particles of the medium oscillate parallel to the direction of wave propagation. Examples: sound waves, P-waves in earthquakes.

6.2 Mathematical Description of a Wave

A simple harmonic progressive wave can be mathematically represented by a sinusoidal function. For a wave traveling in the positive x-direction, its displacement y at position x and time t can be given by:

y(x, t) = A sin(kx - ωt + φ)

• A: Amplitude (maximum displacement from equilibrium)
• k: Angular wave number (k = 2π/λ, where λ is wavelength)
• ω: Angular frequency (ω = 2πf, where f is frequency)
• φ: Phase constant
• λ: Wavelength (distance between two consecutive crests or troughs)
• f: Frequency (number of oscillations per unit time)
• T: Period (time for one complete oscillation, T = 1/f)
• v: Wave speed (v = fλ = ω/k)

The wave equation, a second-order linear partial differential equation, describes the propagation of waves:

∂²y/∂x² = (1/v²) ∂²y/∂t²

6.3 Stationary Waves

Stationary waves, or standing waves, are formed when two progressive waves of the same amplitude and frequency traveling in opposite directions superpose. In a stationary wave, the energy does not propagate; it remains localized. Key features include:

• Nodes: Points where the amplitude is always zero.
• Antinodes: Points where the amplitude is maximum.

The distance between two consecutive nodes or antinodes is λ/2. The distance between a node and an adjacent antinode is λ/4.

7. Mechanical Waves

Mechanical waves require a medium for their propagation. They transfer energy through the oscillation of particles in the medium. Examples include sound waves, water waves, and seismic waves.

7.1 Speed of Wave Motion; Velocity of Sound in Solid and Liquid

The speed of a mechanical wave depends on the properties of the medium. Generally, the stiffer the medium and the less dense it is, the faster the wave travels.

• In Solids: The velocity of a longitudinal wave (e.g., sound) in a solid rod is given by:

v = √(Y/ρ)

• where Y is Young's Modulus and ρ is the density of the solid.
• In Liquids: The velocity of a longitudinal wave in a liquid is given by:

v = √(B/ρ)

• where B is the Bulk Modulus and ρ is the density of the liquid.

7.2 Velocity of Sound in Gas

Newton first proposed that sound travels isothermally through a gas. According to his formula:

v = √(P/ρ)

• where P is the pressure and ρ is the density of the gas.

7.3 Laplace’s Correction

Experiments showed that Newton's formula underestimated the speed of sound in air. Laplace corrected this by proposing that the propagation of sound in a gas is an adiabatic process (no heat exchange with surroundings). The corrected formula is:

v = √(γP/ρ)

• where γ (gamma) is the adiabatic index or ratio of specific heats (C_p/C_v) for the gas. For air, γ ≈ 1.4. This correction significantly improved the theoretical value to match experimental results.

7.4 Effect of Temperature, Pressure, Humidity on Velocity of Sound

• Temperature: The velocity of sound in a gas is directly proportional to the square root of its absolute temperature (T).

v ∝ √T

• This means as temperature increases, the speed of sound increases. For every 1°C rise in temperature, the speed of sound in air increases by approximately 0.61 m/s.
• Pressure: At constant temperature, the velocity of sound in a gas is independent of pressure. This is because, while pressure (P) changes, the density (ρ) also changes proportionally, keeping the ratio P/ρ (and thus γP/ρ) constant.
• Humidity: The presence of water vapor (humidity) in air decreases the density of the air (since water vapor's molecular weight is less than that of dry air components like N2 and O2). A decrease in density, according to Laplace's formula, leads to an increase in the speed of sound. Thus, sound travels faster in humid air than in dry air.

8. Waves in Pipes and Strings

The formation of stationary waves in confined spaces like pipes and on stretched strings leads to musical sounds and forms the basis of many musical instruments.

8.1 Stationary Waves in Closed and Open Pipes

Organ pipes are cylindrical tubes that produce sound through the vibration of air columns, forming stationary waves.

• Closed Organ Pipe (one end closed, one end open):

A node forms at the closed end and an antinode at the open end.

Fundamental frequency (1st harmonic): L = λ/4 ⇒ f₁ = v/(4L)

Overtones are odd multiples of the fundamental frequency: 3rd harmonic (f₃ = 3f₁), 5th harmonic (f₅ = 5f₁), etc.

• Open Organ Pipe (both ends open):

Antinodes form at both open ends, with a node in the middle (for the fundamental).

Fundamental frequency (1st harmonic): L = λ/2 ⇒ f₁ = v/(2L)

Overtones are all integer multiples of the fundamental frequency: 2nd harmonic (f₂ = 2f₁), 3rd harmonic (f₃ = 3f₁), etc.

8.2 Harmonics and Overtones in Closed and Open Organ Pipes

Harmonics are frequencies that are integer multiples of the fundamental frequency. Overtones are frequencies higher than the fundamental. The 1st overtone is the next higher frequency after the fundamental, the 2nd overtone is the next, and so on.

• Closed Pipe: Only odd harmonics (f₁, 3f₁, 5f₁, ...) are present. The first overtone is the 3rd harmonic.
• Open Pipe: All harmonics (f₁, 2f₁, 3f₁, ...) are present. The first overtone is the 2nd harmonic.

8.3 End Correction in Pipes

The antinode at the open end of an organ pipe is not exactly at the end but slightly outside the pipe due to the inertia of the air. This small distance is called the end correction (e). The effective length (L_eff) of the pipe is:

• For a closed pipe: L_eff = L + e
• For an open pipe: L_eff = L + 2e (one end correction for each open end)

Typically, e ≈ 0.6r, where r is the radius of the pipe.

8.4 Velocity of Transverse Waves along a Stretched String

The velocity (v) of a transverse wave on a stretched string is determined by the tension (T) in the string and its linear mass density (μ = mass per unit length):

v = √(T/μ)

8.5 Vibration of String and Overtones

When a stretched string fixed at both ends vibrates, it forms stationary waves. Nodes are at the fixed ends. The possible wavelengths are:

λ_n = 2L/n

where L is the length of the string and n = 1, 2, 3, ... is the harmonic number.

The corresponding frequencies are:

f_n = v/λ_n = (n/2L)√(T/μ)

• Fundamental frequency (n=1): f₁ = (1/2L)√(T/μ)
• 1st overtone (n=2, 2nd harmonic): f₂ = 2f₁
• 2nd overtone (n=3, 3rd harmonic): f₃ = 3f₁

8.6 Laws of Vibration of Fixed String

These laws describe how the fundamental frequency of a vibrating string depends on its physical properties:

1. Law of Length: For a given tension and linear density, the fundamental frequency is inversely proportional to the length of the string (f ∝ 1/L).
2. Law of Tension: For a given length and linear density, the fundamental frequency is directly proportional to the square root of the tension (f ∝ √T).
3. Law of Linear Density: For a given length and tension, the fundamental frequency is inversely proportional to the square root of the linear mass density (f ∝ 1/√μ).

9. Acoustic Phenomena

Acoustics is the branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound, and infrasound.

9.1 Sound Waves: Pressure Amplitude

Sound waves are longitudinal mechanical waves that propagate as compressions and rarefactions in a medium. These correspond to regions of higher and lower pressure than the ambient pressure. The maximum change in pressure from the equilibrium pressure is called the pressure amplitude (ΔP_max).

Pressure amplitude is related to displacement amplitude (s_max) by:

ΔP_max = Bks_max

• where B is the bulk modulus and k is the angular wave number.

9.2 Characteristics of Sound: Intensity; Loudness, Quality and Pitch

Sound has several subjective and objective characteristics:

• Intensity (Objective): The average rate at which energy is transported by the wave per unit area perpendicular to the direction of propagation. Measured in W/m².

I = (1/2) ρvω²s_max² = (ΔP_max)² / (2ρv)

• Loudness (Subjective): The physiological sensation of sound intensity. It depends on intensity and the sensitivity of the ear. Measured in decibels (dB).

L = 10 log₁₀(I/I₀)

• where I₀ is the threshold of hearing (10^-12 W/m²).
• Pitch (Subjective): The perception of the frequency of a sound. Higher frequency means higher pitch.
• Quality or Timbre (Subjective): The characteristic that distinguishes sounds of the same pitch and loudness produced by different sources. It depends on the presence and relative intensities of overtones (harmonics).

9.3 Doppler’s Effect

The Doppler effect is the apparent change in frequency of a wave (sound or light) due to the relative motion between the source and the observer. When the source and observer are moving towards each other, the perceived frequency increases (higher pitch). When they are moving away, the perceived frequency decreases (lower pitch).

f' = f [(v ± v_o) / (v ± v_s)]

• f': Apparent frequency
• f: Original frequency of the source
• v: Speed of sound in the medium
• v_o: Speed of the observer (plus if moving towards source, minus if moving away)
• v_s: Speed of the source (minus if moving towards observer, plus if moving away)

Applications include radar guns, medical ultrasound, and astronomical observations (redshift/blueshift).

10. Nature and Propagation of Light

Light is an electromagnetic wave, meaning it does not require a medium for its propagation. It exhibits both wave-like and particle-like properties (wave-particle duality).

10.1 Huygen’s Principle

Huygens' principle, proposed by Christiaan Huygens in 1678, is a method for analyzing wave propagation. It states that:

• Every point on a wavefront may be considered as a source of secondary spherical wavelets that spread out in the forward direction at the speed of light.
• The new wavefront at a later time is the envelope of all these secondary wavelets.

This principle can be used to explain reflection, refraction, diffraction, and interference of light.

10.2 Reflection and Refraction according to Wave Theory

Huygens' principle provides a classical explanation for the laws of reflection and refraction.

• Reflection: When a plane wavefront strikes a plane reflecting surface, each point on the wavefront acts as a source of secondary wavelets. By constructing the envelope of these wavelets, it can be shown that the angle of incidence equals the angle of reflection (Law of Reflection). The frequency, wavelength, and speed of light remain unchanged.
• Refraction: When a plane wavefront passes from one medium to another, the speed of light changes. Due to the change in speed, the wavelets in the second medium will travel different distances in the same amount of time. This bending of the wavefront results in refraction. Huygens' principle leads to Snell's Law:

n₁ sinθ₁ = n₂ sinθ₂

• where n₁ and n₂ are the refractive indices of the two media, and θ₁ and θ₂ are the angles of incidence and refraction, respectively. The frequency of light remains constant during refraction, but its wavelength and speed change.