Unit 1: Superposition of Oscillations and Wave Motion

1. Linearity and the Superposition Principle

In classical mechanics, many physical systems exhibit linear behavior under small displacements. A physical system is said to be linear if its governing differential equation is linear. For a simple harmonic oscillator, the differential equation of motion is:

d²x/dt² + ω²x = 0

Since the variable x and its derivatives appear to the first power and are not multiplied together, this is a second-order linear homogeneous ordinary differential equation. Linearity leads directly to the Superposition Principle.

The Superposition Principle

The Superposition Principle states that when two or more physical disturbances (such as oscillations or waves) overlap in the same region of space, the resultant displacement at any point and any instant is equal to the algebraic (or vector) sum of the individual displacements caused by each oscillation independently.

Mathematically, if x1(t) and x2(t) are two independent solutions to a linear differential equation, then any linear combination of them is also a valid solution:

x(t) = c1 * x1(t) + c2 * x2(t)

Where c1 and c2 are arbitrary constants determined by initial conditions.

2. Superposition of Two Collinear Harmonic Oscillations

Collinear oscillations are those that take place along the exact same straight line (axis). We analyze two distinct cases based on their frequencies.

Case 1: Equal Frequencies

Consider two simple harmonic oscillations of the same frequency ω but different amplitudes and phases, acting along the x-axis:

x1 = A1 cos(ωt + φ1)

x2 = A2 cos(ωt + φ2)

According to the superposition principle, the resultant displacement x is:

x = x1 + x2 = A1 cos(ωt + φ1) + A2 cos(ωt + φ2)

Expanding the cosine terms using trigonometric identities:

x = A1 [cos(ωt)cos(φ1) - sin(ωt)sin(φ1)] + A2 [cos(ωt)cos(φ2) - sin(ωt)sin(φ2)]

x = cos(ωt) [A1 cos(φ1) + A2 cos(φ2)] - sin(ωt) [A1 sin(φ1) + A2 sin(φ2)]

To simplify, let us define two constants, R (resultant amplitude) and θ (resultant phase), such that:

R cos(θ) = A1 cos(φ1) + A2 cos(φ2) --- (Equation 1)

R sin(θ) = A1 sin(φ1) + A2 sin(φ2) --- (Equation 2)

Substituting these constants back into our displacement expression yields:

x = R cos(ωt)cos(θ) - R sin(ωt)sin(θ) = R cos(ωt + θ)

This shows that the resultant motion is also a simple harmonic motion along the same line, with the same frequency ω, but with a new amplitude R and initial phase θ.

Determining the Resultant Amplitude (R):

Squaring and adding Equation 1 and Equation 2:

R² (cos²(θ) + sin²(θ)) = [A1 cos(φ1) + A2 cos(φ2)]² + [A1 sin(φ1) + A2 sin(φ2)]²

R² = A1² + A2² + 2 A1 A2 [cos(φ1)cos(φ2) + sin(φ1)sin(φ2)]

R = sqrt(A1² + A2² + 2 A1 A2 cos(φ1 - φ2))

Determining the Resultant Phase (θ):

Dividing Equation 2 by Equation 1:

tan(θ) = [A1 sin(φ1) + A2 sin(φ2)] / [A1 cos(φ1) + A2 cos(φ2)]

Case 2: Different Frequencies (Beats)

When two collinear harmonic oscillations of slightly different frequencies superpose, they produce the phenomenon of beats. Beats are periodic variations in the intensity of sound or vibration heard at a distinct frequency.

Consider two oscillations with equal amplitudes A but slightly different angular frequencies ω1 and ω2:

x1 = A cos(ω1 t)

x2 = A cos(ω2 t)

Applying the superposition principle:

x = x1 + x2 = A [cos(ω1 t) + cos(ω2 t)]

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

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

Let us define the average angular frequency ω_avg = (ω1 + ω2) / 2 and the modulation frequency ω_mod = (ω1 - ω2) / 2. The expression simplifies to:

x = R(t) cos(ω_avg t)

Where the time-dependent amplitude R(t) is given by:

R(t) = 2 A cos((ω1 - ω2)t / 2)

Because ω1 is very close to ω2, the term cos(ω_avg t) oscillates rapidly, while the amplitude term R(t) varies slowly. This is an amplitude-modulated oscillation.

Beat Frequency:

The intensity is proportional to the square of the amplitude, R(t)². The amplitude R(t) reaches its maximum value of ±2A whenever:

cos((ω1 - ω2)t / 2) = ±1

This occurs when:

(ω1 - ω2)t / 2 = nπ => (2π f1 - 2π f2)t / 2 = nπ => t = n / (f1 - f2)

The time interval between two successive maxima (beat period T_beat) is:

T_beat = t_(n+1) - t_n = 1 / (f1 - f2)

The frequency of these periodic intensity variations is called the Beat Frequency (f_beat):

f_beat = 1 / T_beat = |f1 - f2|

Common Mistake: Students often write the beat frequency as (f1 - f2)/2 because of the factor of 2 in the cosine argument. Remember, because intensity depends on the square of the amplitude, both the positive and negative peaks of the envelope R(t) represent maximum intensity. Thus, the frequency of intensity peaks is exactly |f1 - f2|.

3. Superposition of N Collinear Harmonic Oscillations

When multiple co-directional waves overlap, we use systematic summation methods to evaluate the resultant motion. Two key scenarios are explored:

Case 1: Equal Phase Differences

Consider N collinear harmonic oscillations, each having the same amplitude 'a' and frequency 'ω', where each successive oscillation has a constant phase difference 'φ' relative to the preceding one:

x1 = a cos(ωt)

x2 = a cos(ωt + φ)

x3 = a cos(ωt + 2φ)

...

x_N = a cos(ωt + (N-1)φ)

The resultant displacement is the sum of these N terms:

x = a sum_[n=0]^[N-1] cos(ωt + nφ)

Using phasor addition or complex numbers (Euler's formula), the sum resolves to:

x = R cos(ωt + Θ)

Where the resultant amplitude R is:

R = a * [sin(Nφ/2) / sin(φ/2)]

And the resultant phase Θ is:

Θ = (N-1)φ/2

Physical Interpretation of Amplitude:

• Principal Maxima: When φ is very small or approaches 0 (or any multiple of 2π), the limit of sin(Nφ/2)/sin(φ/2) becomes N. Thus, R = N*a. The oscillations add constructively.
• Minima: The amplitude R becomes zero when sin(Nφ/2) = 0 but sin(φ/2) ≠ 0. This happens when Nφ/2 = mπ (where m is an integer not divisible by N).

Case 2: Equal Frequency Differences

Consider N collinear harmonic oscillations with equal amplitudes 'a', starting in phase, but having equal frequency increments 'Δω':

x1 = a cos(ωt)

x2 = a cos((ω + Δω)t)

x3 = a cos((ω + 2Δω)t)

...

x_N = a cos((ω + (N-1)Δω)t)

By mathematical analogy to the phase-difference derivation (replacing phase φ with Δω * t), the resultant displacement is:

x = R(t) cos(ω_avg t)

Where the time-dependent amplitude R(t) is:

R(t) = a * [sin(N Δω t / 2) / sin(Δω t / 2)]

And the average frequency is ω_avg = ω + (N-1)Δω/2. This describes a transient group of waves (wave packet), which is central to understanding wave pulse dispersion and group velocity in quantum mechanics and optics.

4. Superposition of Two Perpendicular Harmonic Oscillations

When a particle is simultaneously subjected to two simple harmonic motions acting at right angles (perpendicularly) to each other, its path of motion changes. We can analyze this analytically and graphically.

Analytical Method

Let the two mutually perpendicular SHMs be along the x and y axes:

x = A sin(ωt + δ) --- (Equation 1)

y = B sin(ωt) --- (Equation 2)

Here, A and B are the amplitudes, ω is the common angular frequency, and δ is the phase difference between the two motions. From Equation 2:

sin(ωt) = y/B

Using the identity cos²(ωt) + sin²(ωt) = 1:

cos(ωt) = sqrt(1 - y²/B²)

Expanding Equation 1:

x/A = sin(ωt)cos(δ) + cos(ωt)sin(δ)

Substitute the values of sin(ωt) and cos(ωt) into the expansion:

x/A = (y/B)cos(δ) + sqrt(1 - y²/B²)sin(δ)

x/A - (y/B)cos(δ) = sqrt(1 - y²/B²)sin(δ)

Squaring both sides to eliminate the square root:

[x/A - (y/B)cos(δ)]² = (1 - y²/B²)sin²(δ)

x²/A² + y²/B² cos²(δ) - 2(xy/AB)cos(δ) = sin²(δ) - y²/B² sin²(δ)

x²/A² + y²/B² (cos²(δ) + sin²(δ)) - 2(xy/AB)cos(δ) = sin²(δ)

x²/A² + y²/B² - 2(xy/AB)cos(δ) = sin²(δ)

This is the general equation of an oblique ellipse. The trajectory of the particle is constrained to this curve within a bounding box of dimensions 2A by 2B.

Graphical Method

The graphical method constructs the resulting trajectory by plotting the position coordinates (x, y) at equal time increments. Two reference circles of radii A and B are drawn. The uniform circular motion is projected onto the horizontal and vertical axes respectively. By tracking intersections for corresponding phase angles, the exact path can be traced visually.

5. Lissajous Figures and Their Uses

The curves traced out by a particle subjected to two mutually perpendicular simple harmonic oscillations are called Lissajous Figures.

Lissajous Figures with Equal Frequencies (1:1 Ratio)

The shape of the curve depends entirely on the phase difference δ between the two perpendicular oscillations:

Lissajous Figures with Unequal Frequencies

When the frequencies are in simple ratios (e.g., 1:2, 1:3, 2:3), the figures become more complex, forming closed loops. For example, with a 1:2 frequency ratio, the equations are:

x = A sin(2ωt + δ)

y = B sin(ωt)

Depending on δ, this system yields a "figure-of-eight" shape, a parabola, or open loop variations.

Rule of Tangents: The frequency ratio of the two perpendicular oscillations can be determined directly from their Lissajous figure using the formula:

(Frequency of X-oscillation) / (Frequency of Y-oscillation) = (Number of intersections of the curve with a vertical tangent line) / (Number of intersections of the curve with a horizontal tangent line)

Uses of Lissajous Figures

• Frequency Measurement: By feeding an unknown frequency signal into one axis of an oscilloscope (CRO) and a known reference frequency into the other, the unknown frequency can be calculated using the tangent ratio.
• Phase Difference Determination: The phase shift between two electrical signals can be calculated by measuring the dimensions of the resulting ellipse on a screen (sin(δ) = y_intercept / y_max).
• Acoustic Testing: Used to calibrate audio equipment, tuning forks, and analyze the phase properties of stereo audio signals.

6. Wave Motion and Plane Progressive Waves

Wave motion is a mechanism of energy and momentum propagation through a medium (or vacuum) without any net transport of the matter itself. A Plane Progressive Wave (or travelling wave) is a wave that travels continuously in a given direction through a medium with constant amplitude.

Mathematical Representation of a Progressive Wave

If a simple harmonic wave propagates in the positive x-direction, the displacement 'y' of a particle at position 'x' at time 't' is given by:

y(x, t) = A sin(ωt - kx)

Where:

• A: Amplitude (maximum displacement of particles).
• ω: Angular frequency, equal to 2πf or 2π/T.
• k: Wave number (propagation constant), equal to 2π/λ.
• λ: Wavelength (distance between two consecutive points in the same phase).
• v: Wave velocity, equal to ω/k = f * λ.

If the wave propagates in the negative x-direction, the sign changes:

y(x, t) = A sin(ωt + kx)

The 1D Classical Wave Differential Equation

Let us take the partial derivatives of the wave equation y(x, t) = A sin(ωt - kx) with respect to time 't' and position 'x'.

First derivatives:

∂y/∂t = ω A cos(ωt - kx) (Particle velocity, v_p)

∂y/∂x = -k A cos(ωt - kx) (Slope or spatial gradient)

Second derivatives:

∂²y/∂t² = -ω² A sin(ωt - kx) = -ω² y

∂²y/∂x² = -k² A sin(ωt - kx) = -k² y

Dividing the second-order partial equations:

(∂²y/∂t²) / (∂²y/∂x²) = -ω² y / (-k² y) = ω²/k²

Since the wave velocity is v = ω/k, we get v² = ω²/k². Substituting this value:

∂²y/∂t² = v² (∂²y/∂x²)

Rearranging into the standard form of the One-Dimensional Differential Wave Equation:

∂²y/∂x² = (1/v²) (∂²y/∂t²)

Any physical function f(x - vt) or g(x + vt) satisfies this linear differential equation, proving they represent valid travelling waves.

7. Pressure of a Longitudinal Wave

In a longitudinal wave (like sound), particles of the medium oscillate back and forth parallel to the direction of wave propagation. This motion creates alternating regions of high pressure (compressions) and low pressure (rarefactions).

Derivation of the Pressure Wave Equation

Let the displacement of particles in a medium be along the x-direction, given by:

s(x, t) = s_0 sin(ωt - kx)

Consider a small volume element of the medium of cross-sectional area 'S' and thickness 'dx'. The initial volume is V = S * dx. Under the influence of the wave, the left boundary of this element shifts by s, and the right boundary shifts by s + ds. The change in volume (dV) of this element is:

dV = S * (∂s/∂x) dx

The volume strain is defined as the change in volume per unit initial volume:

Volume Strain = dV / V = [S * (∂s/∂x) dx] / [S * dx] = ∂s/∂x

From the definition of Bulk Modulus (B) of a medium:

B = - (Excess Pressure) / (Volume Strain) = - ΔP / (∂s/∂x)

Therefore, the excess pressure (or pressure deviation ΔP) is:

ΔP = - B * (∂s/∂x)

Differentiating s(x, t) with respect to x:

∂s/∂x = -k * s_0 cos(ωt - kx)

Substitute this back into the pressure equation:

ΔP = B * k * s_0 cos(ωt - kx)

Let P_0 represent the maximum pressure amplitude, where P_0 = B * k * s_0. Since the wave speed in a medium of density ρ is v = sqrt(B/ρ), we can write B = ρ * v². Thus:

P_0 = ρ * v² * k * s_0

The pressure wave equation is written as:

ΔP(x, t) = P_0 cos(ωt - kx) = P_0 sin(ωt - kx + π/2)

Crucial Physical Observation: The displacement wave s(x, t) is a sine function, whereas the pressure wave ΔP(x, t) is a cosine function. This means there is a phase difference of π/2 (90 degrees) between the displacement and pressure of a longitudinal wave. Compressions (maximum pressure) occur where particle displacement is zero, and maximum displacement occurs where pressure change is zero.

8. Energy and Intensity of a Progressive Wave

As a wave propagates, it carries energy through the medium. This energy is a combination of kinetic energy (due to particle velocities) and potential energy (due to the elastic deformation of the medium).

1. Kinetic Energy Density (K_d)

Consider a small element of volume dV = S * dx and mass dm = ρ * dV = ρ * S * dx. The kinetic energy (dK) of this element is:

dK = 1/2 * dm * (∂y/∂t)²

Using the wave equation y(x, t) = A sin(ωt - kx), the particle velocity is ∂y/∂t = ω A cos(ωt - kx). Substituting this:

dK = 1/2 (ρ * S * dx) * (ω² A² cos²(ωt - kx))

The kinetic energy per unit volume (kinetic energy density, K_d) is:

K_d = dK / (S * dx) = 1/2 * ρ * ω² * A² * cos²(ωt - kx)

2. Potential Energy Density (U_d)

The work done in deforming the medium results in potential energy. The potential energy density (U_d) of a volume element is:

U_d = 1/2 * ρ * v² * (∂y/∂x)²

Since ∂y/∂x = -k A cos(ωt - kx) and v = ω/k:

U_d = 1/2 * ρ * (ω/k)² * (-k A cos(ωt - kx))² = 1/2 * ρ * ω² * A² * cos²(ωt - kx)

For a progressive wave, the instantaneous kinetic energy density is equal to the instantaneous potential energy density.

3. Total Energy Density (E_d)

The total energy density is the sum of the kinetic and potential energy densities:

E_d = K_d + U_d = ρ * ω² * A² * cos²(ωt - kx)

The average total energy density (E) over a complete cycle is found by taking the time average of cos²(ωt - kx), which is 1/2:

E = 1/2 * ρ * ω² * A²

4. Wave Intensity (I)

Intensity (I) is defined as the amount of wave energy flowing per unit time across a unit area perpendicular to the direction of wave propagation.

I = (Total Energy) / (Area * Time) = (Energy Density * Volume) / (Area * Time)

Since Volume = Area * length = Area * (v * t), where v is wave speed:

I = (E * S * v * t) / (S * t) = E * v

Substitute the average energy density E into this relation:

I = 1/2 * ρ * v * ω² * A²

Substituting ω = 2πf:

I = 2 * π² * ρ * v * f² * A²

Key Observations for Exams:

• Intensity is directly proportional to the square of the Amplitude: I ∝ A²
• Intensity is directly proportional to the square of the Frequency: I ∝ f²
• Intensity is proportional to the density of the medium (ρ) and the wave velocity (v).