Unit 2: Wave Mechanics and Vibrations

1. Velocity of Transverse Vibrations of Stretched Strings

Conceptual Explanation

When a stretched string is plucked or disturbed, a transverse wave travels along its length. The speed at which this wave propagates depends directly on two mechanical properties of the string system: the tension pulling the string taut, and the mass distribution along the string. Tension acts as the restoring force trying to bring the string back to equilibrium, while the mass provides the inertia that resists acceleration.

Mathematical Derivation

Consider a small segment of a stretched string of length dl, having a linear mass density μ (mass per unit length). Let the tension in the string be T. Suppose the segment is curved into an arc of a circle of radius R during vibration, subtending a small angle 2θ at the center.

The tension forces T act tangentially at both ends of the segment. Resolving these forces:

• The horizontal components T * cos(θ) act in opposite directions and cancel each other out.
• The vertical components T * sin(θ) act inward toward the center, providing the necessary restoring force.

The total downward restoring force F is:

F = 2 * T * sin(θ)

Since θ is extremely small, we use the approximation sin(θ) ≈ θ. Therefore:

F ≈ 2 * T * θ

From the geometry of the circular arc, the angle is related to the arc length by 2θ = dl / R, which means θ = dl / (2 * R). Substituting this back into the force equation:

F = 2 * T * (dl / (2 * R)) = T * dl / R

The mass of this small segment of string is dm = μ * dl. The centripetal force required to keep this segment moving at speed v along its curved path is:

F_c = (dm * v²) / R = (μ * dl * v²) / R

At equilibrium, the restoring force matches the required centripetal force:

T * dl / R = (μ * dl * v²) / R

Simplifying the equation by canceling dl and R from both sides:

T = μ * v²

v = √(T / μ)

Practical Example

Consider a guitar string. To raise the pitch (frequency) of the string, a guitarist tightens the tuning peg. This increases the tension T, which in turn increases the wave velocity v. Because the fundamental wavelength remains fixed by the length of the string, the frequency must increase (since f = v / λ).

Common Mistakes

Using Total Mass: Students often mistakenly use the total mass of the string M in place of the linear mass density μ. Always divide the total mass by the total length of the string to find μ = M / L before calculating the velocity.

2. Velocity of Longitudinal Waves in a Fluid in a Pipe

Conceptual Explanation

Longitudinal waves, such as sound waves, propagate through a fluid (liquid or gas) inside a pipe via alternating compressions and rarefactions. When the fluid is compressed, its local pressure increases, acting as a restoring force that drives the propagation. The speed of these waves is governed by the volume elasticity (Bulk Modulus) of the fluid and its inertial resistance (density).

Formula

The velocity of a longitudinal wave in a fluid confined within a pipe is given by:

v = √(B / ρ)

Where:

• B is the Bulk Modulus of the fluid (measured in Pascals, Pa).
• ρ is the equilibrium density of the fluid (measured in kg/m³).

Important Observations

3. Newton’s Formula for Velocity of Sound and Laplace’s Correction

Newton's Formula

Sir Isaac Newton assumed that when sound waves travel through a gas, the temperature of the medium remains constant. Therefore, he treated the propagation of sound as an isothermal process.

For an isothermal process of an ideal gas, Boyle's Law states:

P * V = Constant

Differentiating both sides with respect to volume V:

P * dV + V * dP = 0

P * dV = -V * dP

P = -dP / (dV / V)

The term on the right side is the definition of the Isothermal Bulk Modulus (B_iso). Hence:

B_iso = P

Substituting this value into the general velocity formula v = √(B / ρ):

v = √(P / ρ)

Calculation for Air at STP

At Standard Temperature and Pressure (STP):

• Atmospheric pressure (P) = 1.013 * 10⁵ N/m²
• Density of air (ρ) = 1.293 kg/m³

Substituting these values into Newton's formula:

v = √(1.013 * 10⁵ / 1.293) ≈ 280 m/s

The Problem: The experimental value of the velocity of sound in air at STP is approximately 332 m/s. Newton's calculated value was about 16% lower than the actual value, representing a significant discrepancy.

Laplace's Correction

In 1816, the French physicist Pierre-Simon Laplace corrected Newton's assumption. Laplace noted that the compressions and rarefactions of a sound wave occur so rapidly that heat does not have sufficient time to flow out of the compressed regions (which warm up) or into the rarefied regions (which cool down). Furthermore, air is a very poor conductor of heat. Therefore, the process is not isothermal, but adiabatic.

For an adiabatic process, the governing equation is:

P * V^γ = Constant

Where γ (gamma) is the ratio of specific heats (C_p / C_v). Differentiating both sides:

V^γ * dP + γ * P * V^{(γ - 1)} * dV = 0

Divide the entire equation by V^{(γ - 1)}:

V * dP + γ * P * dV = 0

γ * P * dV = -V * dP

γ * P = -dP / (dV / V)

This expression defines the Adiabatic Bulk Modulus (B_ad):

B_ad = γ * P

Substituting this corrected modulus into the wave velocity formula yields the Newton-Laplace Formula:

v = √(γ * P / ρ)

Calculation with Laplace's Correction

For air (which is primarily composed of diatomic gases like Nitrogen and Oxygen), γ ≈ 1.4. Using the same values for pressure and density:

v = √(1.4 * 1.013 * 10⁵ / 1.293) ≈ 331.6 m/s

This calculated value matches the experimentally measured speed of sound of 332 m/s with remarkable accuracy.

4. Superposition of Two Harmonic Waves and Standing Waves

The Principle of Superposition

When two or more waves travel through the same medium simultaneously, the resultant displacement of any particle at a given instant is the algebraic sum of the individual displacements caused by each wave separately:

y = y_1 + y_2 + ... + y_n

Standing (Stationary) Waves

Standing waves are formed when two identical progressive harmonic waves of the same amplitude and frequency travel in opposite directions through a medium and superimpose upon each other. They are called "standing" because the wave pattern does not propagate through space; instead, the energy remains localized between nodes.

Analytical Treatment for a Fixed End

Consider a wave traveling to the right along a string fixed at x = 0. The incident wave is described by:

y_1 = A * sin(k * x - ω * t)

When this wave reflects from the fixed boundary, it undergoes a phase change of π (180 degrees) because the boundary must remain stationary. The reflected wave traveling to the left is represented by:

y_2 = -A * sin(k * x + ω * t)

Applying the principle of superposition, the resultant displacement is:

y = y_1 + y_2 = A * [sin(k * x - ω * t) - sin(k * x + ω * t)]

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

y = -2 * A * sin(k * x) * cos(ω * t)

This equation represents a standing wave. The term 2 * A * sin(k * x) is the amplitude of vibration, which varies as a function of position x.

Nodes (Points of Zero Amplitude)

Nodes occur where the amplitude is zero:

sin(k * x) = 0

k * x = n * π

Since k = 2 * π / λ:

(2 * π / λ) * x = n * π

x = n * λ / 2

For n = 0, 1, 2, 3.... Nodes are spaced at intervals of λ / 2.

Antinodes (Points of Maximum Amplitude)

Antinodes occur where the amplitude is at its maximum value:

|sin(k * x)| = 1

k * x = (2 * n + 1) * π / 2

(2 * π / λ) * x = (2 * n + 1) * π / 2

x = (2 * n + 1) * λ / 4

For n = 0, 1, 2, 3.... Antinodes are also spaced at intervals of λ / 2, and are positioned halfway between adjacent nodes.

Analytical Treatment for a Free End

When a wave reflects from a free boundary, it does not undergo any phase change. The incident and reflected waves are:

y_1 = A * sin(k * x - ω * t)

y_2 = A * sin(k * x + ω * t)

Superimposing these waves:

y = y_1 + y_2 = A * [sin(k * x - ω * t) + sin(k * x + ω * t)]

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

y = 2 * A * cos(k * x) * sin(ω * t)

Here, the position-dependent amplitude is 2 * A * cos(k * x).

Nodes (Free End)

cos(k * x) = 0

x = (2 * n + 1) * λ / 4

Antinodes (Free End)

|cos(k * x)| = 1

x = n * λ / 2

5. Phase and Group Velocities and Their Relations

Phase Velocity (v_p)

Phase velocity is the speed at which a single monochromatic wave wave-front of constant phase propagates through space.

v_p = ω / k

Where ω is the angular frequency (2 * π * f) and k is the wave number (2 * π / λ).

Group Velocity (v_g)

In real physical systems, waves rarely travel as pure, infinite single frequencies. Instead, they travel as a collection or "packet" of waves with slightly different frequencies. The group velocity is the speed at which the overall envelope or wave packet travels. It is the velocity at which energy and information are transmitted.

v_g = dω / dk

Derivation of the Relation Between Phase and Group Velocities

We know that ω = v_p * k. Differentiating this expression with respect to k:

v_g = dω / dk = d(v_p * k) / dk

v_g = v_p + k * (dv_p / dk)

We can convert this relation to express it in terms of wavelength λ. Since k = 2 * π / λ, differentiating both sides yields:

dk = -(2 * π / λ²) * dλ

Therefore, we can rewrite the derivative term as:

k * (dv_p / dk) = (2 * π / λ) * (dv_p / dλ) * (dλ / dk)

k * (dv_p / dk) = (2 * π / λ) * (dv_p / dλ) * (-λ² / (2 * π))

k * (dv_p / dk) = -λ * (dv_p / dλ)

Substituting this back into our original expression gives the fundamental relation:

v_g = v_p - λ * (dv_p / dλ)

Dispersive vs. Non-Dispersive Media

6. Energy of a Vibrating String

A vibrating string executing standing waves holds physical energy distributed along its length. This total energy is the sum of its Kinetic Energy (associated with the motion of the string particles) and its Potential Energy (associated with the work done to stretch the string against its tension).

Let the equation of the standing wave on a string of length L fixed at both ends be:

y = A * sin(k * x) * cos(ω * t)

Kinetic Energy (K.E.)

Consider a small element of length dx and mass dm = μ * dx. The transverse velocity of this element is:

u = dy / dt = -A * ω * sin(k * x) * sin(ω * t)

The kinetic energy of this small element is:

dK = (1 / 2) * dm * u² = (1 / 2) * (μ * dx) * [-A * ω * sin(k * x) * sin(ω * t)]²

dK = (1 / 2) * μ * A² * ω² * sin²(k * x) * sin²(ω * t) * dx

To find the total Kinetic Energy K of the entire string, we integrate this expression from x = 0 to x = L:

K = (1 / 2) * μ * A² * ω² * sin²(ω * t) * ∫ [from 0 to L] sin²(k * x) dx

Since the string is fixed at both ends, k * L = n * π. The integral of sin²(k * x) over this interval is L / 2. Therefore:

K = (1 / 4) * μ * L * A² * ω² * sin²(ω * t)

Potential Energy (P.E.)

Potential energy is stored when the string is stretched from its straight equilibrium state. The increase in length of a small segment dx when curved is ds - dx, where:

ds = √[dx² + dy²] = dx * √[1 + (dy/dx)²] ≈ dx * [1 + (1/2) * (dy/dx)²]

The work done against tension T to cause this stretch represents the potential energy stored in the element:

dU = T * (ds - dx) = (1 / 2) * T * (dy/dx)² * dx

From our standing wave equation, the spatial derivative is:

dy / dx = A * k * cos(k * x) * cos(ω * t)

Substituting this derivative into the potential energy expression:

dU = (1 / 2) * T * A² * k² * cos²(k * x) * cos²(ω * t) * dx

Integrating from 0 to L to find the total Potential Energy U:

U = (1 / 2) * T * A² * k² * cos²(ω * t) * ∫ [from 0 to L] cos²(k * x) dx

The value of this integral is also L / 2. Recognizing that v = √(T / μ) and ω = v * k, we can substitute T * k² = μ * ω²:

U = (1 / 4) * μ * L * A² * ω² * cos²(ω * t)

Total Energy (E)

The total energy is the sum of the kinetic and potential energies:

E = K + U = (1 / 4) * μ * L * A² * ω² * [sin²(ω * t) + cos²(ω * t)]

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

E = (1 / 4) * μ * L * A² * ω²

Conclusion: Although kinetic energy and potential energy fluctuate individually over time, their sum remains constant. Energy oscillates continuously between purely kinetic (when the string is flat and moving fastest) and purely potential (when the string is momentarily at rest at its maximum displacement).

7. Normal Modes of Stretched Strings

When a string of length L is fixed at both ends, standing waves can only form at specific, discrete frequencies. These discrete vibrational patterns are known as the normal modes of the string.

Boundary Conditions

Since both ends are fixed, the displacement must be zero at both boundary points:

• At x = 0, y = 0
• At x = L, y = 0

This restricts the allowed wavelengths to those that can fit an integer number of half-wavelengths within the length L:

L = n * (λ / 2)

λ_n = 2 * L / n

Where n = 1, 2, 3... is the mode number.

Resonant Frequencies

Since wave velocity is v = √(T / μ), the allowed frequencies (f_n = v / λ_n) are:

f_n = (n / 2 * L) * √(T / μ)

• Fundamental Mode (First Harmonic, n = 1): This is the lowest frequency at which the string can vibrate, containing a single loop with nodes at both ends and one antinode in the middle.

  f_1 = (1 / 2 * L) * √(T / μ)

• Second Harmonic (First Overtone, n = 2): The string vibrates in two loops, with a node in the exact center.

  f_2 = 2 * f_1 = (1 / L) * √(T / μ)

• Third Harmonic (Second Overtone, n = 3): The string vibrates in three loops, with two internal nodes.

  f_3 = 3 * f_1

The normal modes contain all integer multiples of the fundamental frequency (f_1, 2*f_1, 3*f_1...).

8. Plucked and Struck Strings

The specific mix of normal modes (harmonics) produced when exciting a string depends heavily on how and where the string is disturbed. We analyze two primary cases: plucking and striking.

Plucked Strings

A string is plucked when it is pulled aside at a specific point and then released from rest.

• Initial Conditions: At t = 0, the initial velocity is zero everywhere, and the string's shape is a triangle with its peak at the pluck point.
• Harmonic Content: Because the initial shape has a sharp bend, it contains a large number of high-frequency harmonics, giving it a bright, clear sound.
• Young-Helmholtz Law: If a string is plucked at a distance d from one of its ends, any harmonic that would require a node at that exact point d cannot be excited. For example, if plucked in the exact center (d = L/2), all even harmonics (2nd, 4th, 6th...) will be entirely absent from the resulting sound.

Struck Strings

A string is struck when it is hit with a hard object (like a piano hammer) over a very short duration.

• Initial Conditions: At t = 0, the displacement of the string is zero everywhere, but it is given an instantaneous initial velocity over a localized striking zone.
• Harmonic Content: The deformation of the string is smoother compared to a plucked string because the hammer has finite width. As a result, very high harmonics are weaker, giving the struck string a warmer and more rounded tone.
• Node Elimination: Similar to plucking, if the string is struck at a point that corresponds to a node for a particular harmonic, that harmonic will not be present in the resulting vibration. Piano manufacturers utilize this rule by striking strings at roughly 1/7th of their length to eliminate the harsh-sounding 7th harmonic.

9. Melde’s Experiment

Melde's experiment is an elegant laboratory setup designed to study standing waves on a stretched string and verify the laws of transverse vibrations. It uses an electrically driven tuning fork to maintain constant vibrations in a string under tension.

Experimental Setup

One end of a light string is tied to one of the prongs of a tuning fork. The other end passes over a frictionless pulley and is attached to a pan carrying weights. The tension T in the string is determined by the total suspended mass (T = m * g).

The Two Configurations

Melde's experiment can be conducted in two distinct physical orientations:

Laws Verified

Using Melde's experiment, students can verify three fundamental laws of transverse waves:

1. Law of Length: The resonant length is proportional to the wavelength (L * √T = Constant when frequency is fixed).
2. Law of Tension: The square of the frequency of vibration is directly proportional to the tension (T / L² = Constant).

10. Longitudinal Standing Waves and Normal Modes

Just like transverse waves, longitudinal waves can also superimpose to form standing waves. This occurs commonly in air columns, such as those inside pipes and musical wind instruments.

Boundary Conditions in Air Columns

• Closed End: The air molecules are physically blocked from moving. This creates a displacement node (zero movement) which corresponds to a pressure antinode (maximum pressure variation).
• Open End: The air molecules are completely free to move in and out of the tube. This creates a displacement antinode (maximum movement) which corresponds to a pressure node (constant atmospheric pressure).

1. Closed Organ Pipe (Closed at one end, Open at the other)

A closed organ pipe has a displacement node at the closed end and a displacement antinode at the open end.

• Fundamental Mode (First Harmonic): The simplest wave pattern that can fit this profile is a quarter-wavelength.

  L = λ_1 / 4

  λ_1 = 4 * L

  f_1 = v / (4 * L)

• Third Harmonic (First Overtone):

  L = 3 * λ_3 / 4

  f_3 = 3 * v / (4 * L) = 3 * f_1

• Fifth Harmonic (Second Overtone):

  f_5 = 5 * f_1

Crucial Rule: Closed organ pipes can produce only odd harmonics (1st, 3rd, 5th, 7th...). Even harmonics cannot be formed under these boundary conditions.

2. Open Organ Pipe (Open at both ends)

An open organ pipe has displacement antinodes at both ends, with at least one node in the center.

• Fundamental Mode (First Harmonic): The simplest pattern is a half-wavelength.

  L = λ_1 / 2

  λ_1 = 2 * L

  f_1 = v / (2 * L)

• Second Harmonic (First Overtone):

  L = λ_2

  f_2 = v / L = 2 * f_1

• Third Harmonic (Second Overtone):

  f_3 = 3 * f_1

Crucial Rule: Open organ pipes produce all integer harmonics (1st, 2nd, 3rd, 4th...). Consequently, open pipes have a richer, more complex sound profile than closed pipes of comparable size.