Wave Motion and Waves on a String

1. Wave velocity (propagation speed): $v = \frac{\omega}{k}$ where $k = \frac{2\pi}{\lambda}$ and $\omega = 2\pi\nu$

2. Particle velocity (oscillation speed): $v_p = \frac{\partial y}{\partial t} = -A\omega\cos(kx-\omega t)$

Conceptual distinction:

• Wave velocity ($v$): Constant speed at which wave pattern moves along string; independent of time and position
• Particle velocity ($v_p$): Speed of individual particles moving perpendicular to string; varies with:
  • Time (oscillates sinusoidally)
  • Position (different for particles at different x positions)
  • Amplitude (larger A means faster particle motion)
  • Frequency (higher ω means faster particle motion)

Example: In a stadium wave, people (particles) move up and down while the wave pattern moves horizontally around the stadium.

Reynolds number (Re): A dimensionless quantity that predicts fluid flow patterns by comparing inertial forces to viscous forces.

Expression: $\text{Re} = \frac{\rho vD}{\eta} = \frac{vD}{\nu}$

Where:

• ρ = fluid density (kg/m³)
• v = fluid velocity (m/s)
• D = tube diameter or characteristic length (m)
• η = dynamic viscosity (kg/m·s or Pa·s)
• ν = kinematic viscosity (m²/s)

Physical insight:

• Re < 2000: Laminar flow (smooth, orderly)
• 2000 < Re < 4000: Transitional flow
• Re > 4000: Turbulent flow (chaotic, with eddies)

Reynolds number indicates whether inertial effects (which promote turbulence) or viscous effects (which dampen turbulence) dominate the flow behavior.

Example: Blood flow in smaller vessels has low Re (laminar), while in the aorta it can have higher Re (potentially turbulent).

When an observer moves at the same velocity as a wave:

• The wave pattern appears stationary to the observer
• The string appears to flow through the wave pattern in the opposite direction
• The crests and troughs remain fixed relative to the observer

This moving reference frame provides key analytical advantages:

• It transforms a dynamic problem into a static geometric problem
• Allows for direct analysis of the forces acting on string elements at different points
• Simplifies the derivation of the wave velocity equation: $v = \sqrt{F/\mu}$, where $F$ is tension and $\mu$ is linear mass density

Example: A guitar string vibrating with traveling waves would appear as a stationary pattern if you could move alongside it at the wave speed.

When two wave pulses meet on a string:

1. The pulses follow the principle of superposition - the resulting displacement at any point equals the algebraic sum of the individual displacements
2. During overlap, the combined amplitude can be either:
   • Larger than either pulse alone (constructive interference)
   • Smaller than either pulse (destructive interference)
   • Zero at some points if one pulse is inverted relative to the other
3. After passing through each other, both pulses emerge unchanged in shape and continue their original motion
4. No energy is lost or exchanged between the pulses during this interaction

Example: If an upward pulse meets a downward pulse of equal amplitude, there will be a moment when the string appears straight, but both pulses will continue in their original directions afterward.

Resonance mechanism in a string fixed at both ends:

1. Initial excitation: Wave of amplitude $A$ is produced at one end (e.g., by a tuning fork)
2. First reflection: Wave travels distance $L$, reflects with inversion at fixed end
3. Second reflection: Wave travels back distance $L$, reflects again with inversion
4. Phase analysis: Original wave has traveled $2L$ and undergone two inversions, returning to original phase
5. Constructive interference: If $2L = n\lambda$ (or $L = n\lambda/2$), the reflected wave constructively interferes with new waves
6. Amplitude growth: Each successive reflection adds in phase, building amplitude to $2A$, $3A$, etc.
7. Limiting factors: Energy losses (air resistance, internal friction) eventually balance energy input
8. Steady state: Standing wave pattern with fixed nodes and antinodes forms with stable amplitude

Only specific frequencies that satisfy $f_n = \frac{n}{2L}\sqrt{\frac{F}{\mu}}$ will resonate.

The amplitude response follows the principle of mechanical resonance:

1. Physics mechanism:

   • Tuning fork produces waves with frequency f
   • Waves travel at velocity v = √(T/μ) where T is tension and μ is linear mass density
   • Waves reflect at both ends with phase inversions
   • The system builds energy when reflections are in phase with new waves

2. Mathematical relationship:

   • Maximum amplitude occurs when L = nλ/2 = n(v/2f)
   • Fundamental frequency: f₀ = (1/2L)√(T/μ)
   • Energy from fork accumulates in the string until losses equal energy input

3. Real-world application:

   • In musical instruments, this principle allows specific string lengths to produce desired pitches
   • Engineers use this principle to avoid resonance in structures that could cause catastrophic failure
   • The same principle explains why pushing a swing at its natural frequency builds amplitude effectively

Node and antinode distribution in a string fixed at one end:

Fundamental mode:

• 1 node (at fixed end)
• 1 antinode (at free end)
• The string forms 1/4 of a wavelength

First overtone:

• 2 nodes (at fixed end and at 2L/3 from fixed end)
• 2 antinodes (at L/3 from fixed end and at free end)
• The string forms 3/4 of a wavelength

Second overtone:

• 3 nodes (at fixed end, L/5, and 3L/5 from fixed end)
• 3 antinodes (at 2L/5, 4L/5, and at free end)
• The string forms 5/4 of a wavelength

Physical principle: This pattern occurs because the boundary conditions must be satisfied - the displacement must be zero at the fixed end (node) and maximum at the free end (antinode). The wave equation solutions that satisfy these boundary conditions only allow standing waves where the string length equals (2n+1)λ/4, where n=0,1,2,... The displacement is given by y = A sin(kx)cos(ωt) where k=(2n+1)π/2L, producing the characteristic node-antinode patterns.

Wave polarization refers to the orientation of the oscillation direction in a transverse wave:

• In a transverse wave, particles oscillate perpendicular to the direction of wave propagation
• When a transverse wave encounters a slit:
  • If the slit is parallel to the oscillation direction, the wave passes through
  • If the slit is perpendicular to the oscillation direction, the wave is blocked
  • If the slit is at an angle, only the component of oscillation parallel to the slit passes through

This principle applies to various transverse waves:

• For a string: only vibrations aligned with the slit orientation can pass through
• For electromagnetic waves (light): only electric field components aligned with a polarizing filter pass through

Real-world application: Polarized sunglasses filter light oscillating in horizontal planes to reduce glare from reflective surfaces.

Wave behavior at string boundaries depends on the relative mass (linear density μ) of the strings:

From Lighter to Heavier String:

• Reflection: Partial reflection with pulse inversion (upside-down)
• Transmission: Partial transmission with reduced amplitude
• Wave Speed: Decreases (v = √(F/μ) is smaller in heavier string)

From Heavier to Lighter String:

• Reflection: Partial reflection with no inversion (same orientation)
• Transmission: Partial transmission with increased amplitude
• Wave Speed: Increases (v = √(F/μ) is larger in lighter string)

Key Principle:

• Reflection characteristics are determined by the change in wave impedance at the boundary
• Inversion occurs when wave velocity decreases at the boundary

Application:

Musical instruments utilize these principles - varied string thickness creates controlled reflection patterns that contribute to the instrument's distinctive sound.

Wave Pulse Reflection Comparison

| Parameter | Fixed End Reflection | Free End Reflection | |-----------|--------------------------|-------------------------| | Pulse Orientation | Inverted (flipped upside-down) | Non-inverted (same orientation) | | Phase Change | 180° phase shift | No phase shift (0°) | | Amplitude | Same magnitude, opposite sign | Same magnitude, same sign | | Energy | Conserved | Conserved |

Physical Mechanisms

Fixed End:

• Fixed boundary cannot move (displacement = 0)
• Incident wave creates tension forces that the fixed end counters with equal and opposite forces
• Resulting destructive interference at the boundary produces an inverted return pulse
• Example: Wall anchor or immovable support

Free End:

• Boundary can move freely (tension = 0)
• The element at the free end overshoots normal maximum displacement due to lack of restraining force
• No opposing force exists to invert the pulse
• Example: String attached to a light frictionless ring on a rod

Visual Representation:

Fixed End:    ∧     →     →     ∨
              |                 |
              ▼                 ▲

Free End:     ∧     →     →     ∧
              |                 |
              ▼                 ▼

Note: These reflection principles are fundamental to understanding standing waves, resonance, and boundary conditions in wave mechanics.