Unit 1: Oscillations and Wave Motion: An Exam-Ready Resource

Oscillation

Superposition of Two Collinear Harmonic Oscillations

The superposition principle is fundamental when dealing with multiple oscillations. It states that when two or more waves or oscillations of the same type overlap in a medium, the resultant displacement at any point and at any instant is the vector sum of the displacements due to each individual wave or oscillation.

Linearity and Superposition Principle: For linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually.

Oscillations having Equal Frequencies

When two collinear harmonic oscillations of the same frequency act simultaneously, their superposition results in a new simple harmonic oscillation of the same frequency. The amplitude and phase of the resultant oscillation depend on the amplitudes and phase difference of the original oscillations.

Consider two oscillations given by:

x₁(t) = A₁ sin(ωt + φ₁)

x₂(t) = A₂ sin(ωt + φ₂)

The resultant oscillation is x(t) = x₁(t) + x₂(t) = A sin(ωt + φ), where:

A = √[A₁² + A₂² + 2A₁A₂ cos(φ₂ - φ₁)]

tan φ = (A₁ sin φ₁ + A₂ sin φ₂) / (A₁ cos φ₁ + A₂ cos φ₂)

Important Observations:

• If φ₂ - φ₁ = 0 (in phase), A = A₁ + A₂ (maximum amplitude).
• If φ₂ - φ₁ = π (out of phase), A = |A₁ - A₂| (minimum amplitude).

Oscillations having Different Frequencies (Beats)

When two harmonic oscillations of slightly different frequencies (ω₁ and ω₂) and comparable amplitudes are superposed, the resultant amplitude varies periodically with time. This periodic variation in amplitude leads to a phenomenon called beats.

Consider two oscillations:

x₁(t) = A cos(ω₁t)

x₂(t) = A cos(ω₂t)

Using the superposition principle, x(t) = x₁(t) + x₂(t) = A [cos(ω₁t) + cos(ω₂t)]

Using the trigonometric identity cos C + cos D = 2 cos((C+D)/2) cos((C-D)/2):

x(t) = [2A cos((ω₁ - ω₂)/2 ⋅ t)] cos((ω₁ + ω₂)/2 ⋅ t)

The term in the square bracket represents the time-varying amplitude. The frequency of the amplitude variation is (ω₁ - ω₂)/2. A beat occurs when the amplitude reaches a maximum. This happens twice per cycle of the amplitude variation.

Beat Frequency (f_beat): The number of beats heard per second is the absolute difference between the frequencies of the two individual oscillations.

f_beat = |f₁ - f₂|

Real-world Application: Beats are used to tune musical instruments. When two instruments play the same note, and beats are heard, it means their frequencies are slightly different. By adjusting one instrument until the beats disappear, they can be perfectly in tune.

Superposition of Two Perpendicular Harmonic Oscillations

When two simple harmonic oscillations act perpendicular to each other, the resultant motion is generally complex and traces specific patterns known as Lissajous figures. This phenomenon can be analyzed using both graphical and analytical methods.

Graphical and Analytical Methods

Analytical Method:

Consider two perpendicular SHMs:

x(t) = A_x sin(ω_xt + φ_x)

y(t) = A_y sin(ω_yt + φ_y)

To find the resultant path, we eliminate 't' from these equations. The complexity of the resulting equation depends on the frequencies (ω_x, ω_y) and phase difference (φ_y - φ_x).

Graphical Method:

This method involves plotting the position (x, y) of the particle at various instances in time. By taking points from the individual x-t and y-t graphs and combining them, the path traced by the particle can be visualized.

Lissajous Figures with Equal and Unequal Frequency and their Uses

Lissajous Figures: These are the patterns generated when a point undergoes two independent harmonic motions perpendicular to each other.

Equal Frequencies (ω_x = ω_y = ω):

The shape of the Lissajous figure depends crucially on the phase difference (Δφ = φ_y - φ_x) between the two oscillations.

• Δφ = 0 (or 2π): The resultant path is a straight line passing through the origin with a positive slope (y = (A_y/A_x)x).
• Δφ = π (or 3π): The resultant path is a straight line passing through the origin with a negative slope (y = -(A_y/A_x)x).
• Δφ = π/2 (or 3π/2): If A_x = A_y, the path is a circle. If A_x ≠ A_y, the path is an ellipse.
• Other Phase Differences: Generally, the path is an ellipse.

Unequal Frequencies (ω_x ≠ ω_y):

When the frequencies are unequal, the Lissajous figures become more complex. The patterns are stable and closed only if the ratio of the frequencies (ω_y/ω_x) is a rational number (e.g., 1:2, 2:3). If the ratio is irrational, the pattern continuously changes and never repeats exactly.

The complexity of the figure increases with higher integer ratios. For example, a 1:2 ratio might result in a figure-eight pattern.

Uses of Lissajous Figures:

• Frequency Comparison: By observing the stable pattern, the ratio of unknown frequency to a known reference frequency can be determined. This is a common application in oscilloscopes.
• Phase Difference Measurement: For equal frequencies, the shape of the figure directly indicates the phase difference between the two signals.
• Signal Generator Testing: Used to check the accuracy and stability of signal generators.
• Artistic Displays: Sometimes used in abstract art and visual effects due to their intricate and beautiful patterns.

Wave Motion

Transverse Waves on a String

A transverse wave is a wave in which the oscillations of the medium particles are perpendicular to the direction of energy propagation of the wave.

On a string, when one end is plucked, the disturbance travels along the string. The string particles move up and down (perpendicular to the string's length), while the wave itself propagates along the length of the string.

The speed (v) of a transverse wave on a stretched string depends on the tension (T) in the string and its linear mass density (μ = mass per unit length).

v = √(T/μ)

Properties:

• Amplitude (A): Maximum displacement of a particle from its equilibrium position.
• Wavelength (λ): The spatial period of the wave; the distance over which the wave's shape repeats.
• Frequency (f): The number of complete oscillations per unit time, or the number of wave cycles passing a point per second.
• Period (T): The time taken for one complete oscillation or one complete wavelength to pass a point (T = 1/f).
• Wave Speed (v): The speed at which the wave disturbance propagates through the medium (v = fλ).

Travelling and Standing Waves on a String

Travelling Waves (Progressive Waves)

A travelling wave, also known as a progressive wave, is a wave that carries energy from one point to another without any net transport of matter. It appears to move through the medium.

The general equation for a sinusoidal travelling wave moving in the positive x-direction is:

y(x,t) = A sin(kx - ωt + φ)

• A is the amplitude.
• k is the angular wave number (k = 2π/λ).
• ω is the angular frequency (ω = 2πf).
• φ is the initial phase constant.

A wave moving in the negative x-direction is represented as y(x,t) = A sin(kx + ωt + φ).

Standing Waves (Stationary Waves)

A standing wave is formed when two identical travelling waves moving in opposite directions superpose. Unlike travelling waves, standing waves do not transfer energy. They appear to stand still, with fixed positions where the amplitude is always zero (nodes) and positions where the amplitude is maximum (antinodes).

Formation: Typically, standing waves are formed when a wave and its reflection interfere. On a string fixed at both ends, a wave travels to one end, reflects, and superposes with the incoming waves.

Characteristics:

• Nodes: Points on the string that remain stationary (zero displacement) at all times.
• Antinodes: Points on the string where the amplitude of oscillation is maximum.
• The distance between two consecutive nodes or two consecutive antinodes is λ/2.
• The distance between a node and an adjacent antinode is λ/4.

Normal Modes of a String

For a string fixed at both ends, only certain specific patterns of standing waves can be formed. These specific patterns, or modes of vibration, are called normal modes, and each corresponds to a specific resonant frequency.

Boundary Conditions: Since the string is fixed at both ends, these points must always be nodes (zero displacement). If the length of the string is L, then nodes must exist at x=0 and x=L.

For a standing wave, the distance between two consecutive nodes is λ/2. Thus, the length L must be an integer multiple of λ/2:

L = n (λ_n / 2)

where n = 1, 2, 3, ... is an integer representing the mode number.

From this, the wavelengths of the normal modes are:

λ_n = 2L / n

The frequency (f_n) of each normal mode can be found using the wave speed v = fλ:

f_n = v / λ_n = n (v / 2L)

Substituting v = √(T/μ):

f_n = (n / 2L) √(T/μ)

• Fundamental Frequency (First Harmonic, n=1): f₁ = (1 / 2L) √(T/μ). This is the lowest possible frequency, corresponding to the simplest standing wave pattern with one antinode in the middle.
• Harmonics/Overtones: The frequencies f_n are integer multiples of the fundamental frequency (f_n = n ⋅ f₁). They are often called harmonics. The (n-1)^th overtone corresponds to the n^th harmonic.

Example: For n=2, f₂ = 2f₁ (second harmonic, first overtone), with two antinodes and one node in the middle.

Group Velocity

In a dispersive medium, different frequency components of a wave travel at different phase velocities. When waves of slightly different frequencies superpose, they form wave packets or groups. The speed at which this overall shape of the wave packet, or the envelope of the wave, travels is called the group velocity.

Group Velocity (v_g): The velocity with which the envelope of a wave packet propagates through a medium. It represents the speed at which energy and information are transported by the wave.

For a wave packet formed by two waves with angular frequencies ω and ω+Δω, and wave numbers k and k+Δk:

v_g = dω / dk

Physical Significance: In many physical situations, it is the group velocity, not the phase velocity, that corresponds to the speed at which energy or information is transmitted.

Phase Velocity

The phase velocity (or phase speed) is the speed at which a specific phase of a wave (e.g., a crest, a trough, or a zero crossing) propagates through space.

Phase Velocity (v_p): The speed at which a point of constant phase on the wave propagates.

For a sinusoidal wave described by y(x,t) = A sin(kx - ωt + φ), the phase velocity is given by:

v_p = ω / k

where ω is the angular frequency and k is the angular wave number.

Relationship with wavelength and frequency: Since ω = 2πf and k = 2π/λ, the phase velocity can also be written as:

v_p = (2πf) / (2π/λ) = fλ

Comparison of Phase and Group Velocity: