Waves notes

Progressive Waves and Basic Wave Terms

A wave is a disturbance that travels through a medium or space and transfers energy. The particles of the medium only oscillate about their mean positions. They do not travel with the wave over a long distance.

In a progressive wave, the disturbance moves continuously from one place to another. The particles have the same frequency but generally different phases. A wave may be mechanical, such as sound, or electromagnetic, such as light.

A sinusoidal progressive wave travelling along the positive x-axis may be written as y = A sin(ωt − kx). The sign changes if the wave travels in the opposite direction.

• Amplitude A is the maximum displacement of a particle from its mean position.
• Wavelength λ is the distance between two consecutive particles in the same phase, such as two consecutive crests or two consecutive troughs.
• Time period T is the time required for one complete vibration.
• Frequency f is the number of complete vibrations per second. Its unit is hertz, Hz.
• The relation between frequency and time period is fT = 1, or f = 1/T.
• Angular frequency is ω = 2πf = 2π/T.
• The phase difference between two points separated by distance x is Δφ = 2πx/λ.
• The distance between two particles in the same phase is λ, or any integral multiple of λ.
• A wave is described as a disturbance because it carries energy and information without carrying matter as a whole.

Wave Speed and Motion Related to Waves

Wave speed is the distance travelled by a wave disturbance per unit time. During one time period, a wave travels one wavelength. Therefore, the basic wave equation is v = fλ = λ/T.

When a wave enters another medium, its speed and wavelength may change because the properties of the medium change. Its frequency remains fixed by the source. Thus, the frequency does not change when a wave passes from one medium to another.

For a point moving in a circle, the linear speed is related to angular speed by v = rω. This circular motion is useful for understanding simple harmonic motion.

• The product of frequency and wavelength is the wave speed: v = fλ.
• If frequency changes to 4f while the wave remains in the same medium, its speed remains v and its wavelength becomes λ/4.
• During exactly one period of a tuning fork, the emitted sound travels one wavelength in air.
• For a wheel of circumference C making f revolutions per second, the linear speed of a point on its rim is v = fC.
• For a point on a rotating wheel, v = rω.
• For a wheel of diameter 4 m, radius is 2 m. If the rim speed is 16 m s−1, its angular speed is 8 rad s−1.
• The angular speed of Earth in its daily rotation is 2π radians per 24 hours, which is π/12 radians per hour.
• Frequency is determined by the source, while wave speed is mainly determined by the medium.

Speed of Sound and Related Effects

Sound is a longitudinal mechanical wave. It requires a material medium and cannot travel through vacuum. In air, sound travels through successive compressions and rarefactions.

The speed of sound depends on the properties of the medium and its temperature. It is generally greater in solids than in liquids and greater in liquids than in gases. In a gas, the speed increases as temperature increases.

The Doppler effect is the apparent change in frequency caused by relative motion between the source and the observer. It applies to sound and also to electromagnetic waves, including light.

• Sound cannot travel through vacuum because it is a mechanical wave.
• For a gas, the speed of sound is proportional to the square root of its absolute temperature, under suitable conditions.
• The speed of sound is not changed merely because the frequency of the source changes, provided the medium is unchanged.
• If a source and observer move towards each other, the observed frequency increases.
• If a source and observer move away from each other, the observed frequency decreases.
• A receding star shows a red shift because the detected light frequency is lower than the emitted frequency.
• For a slowly receding source of light, fobserved = femitted(c/(c + v)), where v is the recession speed.
• For an emitted frequency 4.57 × 10^14 Hz, recession speed 1.4 × 10^7 m s−1, and c = 3.0 × 10^8 m s−1, the detected frequency is approximately 4.36 × 10^14 Hz.
• An oscillating charge produces electromagnetic waves.

Superposition, Interference, Resonance, and Dispersion

The principle of superposition states that when two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements. After passing through one another, the waves continue independently.

Interference is the redistribution of intensity produced by the superposition of coherent waves. Constructive interference gives maximum intensity, while destructive interference gives minimum intensity.

Coherent sources have the same frequency and a constant phase difference. Two fine parallel slits illuminated by one source can act as coherent sources.

Dispersion or spreading of a light signal in an optical fibre can cause information received at the other end to be inaccurate. Different frequency components may travel with different speeds and arrive at different times.

• Constructive interference occurs when path difference is nλ and phase difference is 2nπ.
• Destructive interference occurs when path difference is (n + 1/2)λ and phase difference is (2n + 1)π.
• For the nth dark fringe from the centre, the phase difference is (2n + 1)π, or equivalently (n + 1/2) × 2π when n starts from zero. If the stated convention uses phase in half cycles, it is represented as (n + 1/2)π radians.
• Amplitude is independent of frequency, wavelength, and wave speed. The wave relation v = fλ connects speed, frequency, and wavelength, but amplitude is not fixed by these quantities.
• Resonance occurs when the frequency of an external periodic force becomes equal or very close to the natural frequency of an oscillator.
• At resonance, the amplitude of forced vibration becomes maximum, subject to damping.
• MRI works on the principle of resonance.

Stationary Waves

A stationary wave is formed by the superposition of two waves having the same frequency, wavelength, and amplitude, travelling in opposite directions. The pattern does not travel through the medium.

In a stationary wave, some particles remain permanently at rest. These positions are called nodes. Positions of maximum vibration are called antinodes. The separation between adjacent nodes or adjacent antinodes is λ/2.

The particles between a node and the next antinode vibrate with different amplitudes. All particles in one loop are in phase, while particles in neighbouring loops are in opposite phase.

• At a node, displacement is always zero and the particle of the medium has zero velocity.
• At an antinode, displacement and particle speed have their maximum possible values for the stationary wave.
• The distance between a node and the consecutive antinode is λ/4.
• The distance between two consecutive nodes is λ/2.
• The distance between two consecutive antinodes is λ/2.
• There is no net transfer of energy along an ideal stationary wave.
• A stationary wave has nodes and antinodes, whereas a progressive wave has a continuously travelling disturbance.
• The wave pattern of a stationary wave remains fixed, but particles between nodes continue to oscillate.

Organ Pipes and Sound in Air Columns

An organ pipe is an air column that produces sound by resonance. The natural frequencies depend on the length of the pipe and whether its ends are open or closed.

For an open pipe, both ends are displacement antinodes. The fundamental mode has an antinode at each end and a node at the centre. The effective wavelength of the fundamental is 2L, so f1 = v/2L.

For a pipe closed at one end, the closed end is a displacement node and the open end is a displacement antinode. The fundamental mode has length equal to one quarter of a wavelength, so f1 = v/4L.

• Fundamental frequency of an open pipe is fopen = v/2L.
• Fundamental frequency of a pipe closed at one end is fclosed = v/4L.
• For equal lengths, the ratio of fundamental frequencies of an open pipe to a one-end-closed pipe is 2:1.
• An open pipe supports all harmonics: f, 2f, 3f, and so on.
• A pipe closed at one end supports only odd harmonics: f, 3f, 5f, and so on.
• In an open pipe, the fundamental wavelength is 2L.
• In a closed pipe, the fundamental wavelength is 4L.
• End correction may slightly increase the effective length of a real pipe.

Simple Harmonic Motion and Oscillations

Simple harmonic motion is an oscillatory motion in which acceleration is directly proportional to displacement from the mean position and is directed towards the mean position. Its defining equation is a = −ω²x.

Examples include the projection of uniform circular motion on a diameter, small oscillations of a simple pendulum, and a mass attached to a spring. The motion is periodic, so it repeats after a fixed time period.

For a mass spring system, T = 2π√(m/k), where m is mass and k is the spring constant. For a simple pendulum undergoing small oscillations, T = 2π√(l/g).

• Displacement in SHM can be written as x = A sin(ωt + φ) or x = A cos(ωt + φ).
• Angular frequency and frequency are related by ω = 2πf.
• At the mean position, displacement is zero, speed is maximum, and acceleration is zero.
• At an extreme position, displacement and acceleration are maximum, while speed is zero.
• The total energy of an ideal harmonic oscillator remains constant.
• For a spring oscillator, total energy is E = 1/2 kA².
• Potential energy is maximum at the extreme positions, while kinetic energy is maximum at the mean position.
• In damped oscillation, both amplitude and total mechanical energy decrease with time.
• For a damped spring system, damping ratio is ζ = b/(2√mk). For m = 0.5 kg, k = 50 N m−1, and b = 2 kg s−1, ζ = 0.2.
• For a pendulum in a train accelerating horizontally, the effective gravitational acceleration is greater than g. Therefore, its time period decreases.
• The angular acceleration of a rotating body is tangential in direction, while centripetal acceleration is radial. Their angle is 90°.