Wave Motion and Waves on a String

For a string fixed at one end (x=0) and free at the other end (x=L):

Allowed frequencies: $\nu_n = \frac{(2n+1)v}{4L} = \frac{2n+1}{4L}\sqrt{\frac{F}{\mu}}$ where $n = 0,1,2,3...$

• Fundamental frequency: $\nu_0 = \frac{v}{4L}$ (when $n=0$)
• First overtone: $\nu_1 = \frac{3v}{4L} = 3\nu_0$ (when $n=1$)
• Second overtone: $\nu_2 = \frac{5v}{4L} = 5\nu_0$ (when $n=2$)

Key differences from string fixed at both ends:

• Only odd harmonics of the fundamental are allowed
• Fundamental is 1/2 the frequency of a same-length string fixed at both ends
• Free end must be an antinode, fixed end must be a node

• A wave pulse travels along a stretched string with constant speed while maintaining its shape (assuming small amplitudes) • Key properties:

• The string material does not move along the direction of wave propagation
• Only the disturbance (energy) travels forward
• Each string particle only moves up and down temporarily
• After the pulse passes, particles return to their original positions

• This demonstrates the fundamental difference between wave motion and particle motion:

• In wave motion: energy transfers without material transport
• In particle motion: matter physically moves from one location to another

Example: When you speak, sound waves travel through air without the air particles traveling from your mouth to the listener's ear.

Wave reflection rules:

1. Fixed end (clamped boundary):

   • Wave is reflected with inversion of shape
   • Wave is reflected with 180° phase shift
   • Reflected amplitude equals incident amplitude
   • A node forms at the fixed end

2. Free end (unconstrained boundary):

   • Wave is reflected without inversion
   • No phase shift occurs
   • Reflected amplitude equals incident amplitude
   • An antinode forms at the free end

3. Boundary between media with different wave velocities:

   • If wave enters region with lower velocity (higher density): reflection with inversion
   • If wave enters region with higher velocity (lower density): reflection without inversion
   • Transmitted wave never undergoes inversion

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.

Nodes:

• Points where the string remains completely stationary (zero amplitude)
• Occur at regular intervals of λ/2 (half wavelength) apart
• Particles at nodes experience zero displacement but maximum tension variation
• In a string fixed at both ends, nodes always occur at the fixed points

Antinodes:

• Points where the string vibrates with maximum amplitude
• Located midway between consecutive nodes, also spaced λ/2 apart
• Particles at antinodes experience maximum displacement but minimum tension variation

Vibration pattern:

• All particles between adjacent nodes vibrate in phase with each other
• Adjacent segments (between consecutive nodes) vibrate in opposite phases
• All particles cross their equilibrium positions simultaneously
• Energy does not propagate past nodes, remaining confined to segments

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

To identify the vibrational mode of a standing wave on a string:

Visual identification:

• Fundamental mode: One antinode (bulge) in the middle, no internal nodes
• First overtone: Two antinodes, one internal node at exact center
• Second overtone: Three antinodes, two internal nodes dividing string into thirds

Mathematical relationships:

• If frequency = f₁, then it's the fundamental mode (1st harmonic)
• If frequency = 2f₁, then it's the first overtone (2nd harmonic)
• If frequency = 3f₁, then it's the second overtone (3rd harmonic)

General formula: fn = n·f₁ where n is the harmonic number

Pattern of nodes: For the nth harmonic, expect (n-1) internal nodes spaced at intervals of L/n along the string

Wavelength relationship: λn = 2L/n, where L is string length and n is the harmonic number

Note: When measuring, the standing wave pattern remains stationary with fixed nodes and oscillating antinodes.

For a string fixed at one end and free at the other:

• The fixed end is always a node (N)
• The free end is always an antinode (A)
• Only odd harmonics are possible: fundamental, 3rd harmonic, 5th harmonic, etc.
• The fundamental frequency is v/4L (where v is wave velocity and L is string length)
• The harmonic series follows the pattern: f, 3f, 5f, 7f, etc.
• Node placement: One node in fundamental mode, two nodes in first overtone, three nodes in second overtone
• The wavelength of the fundamental mode is 4L

In contrast, a string fixed at both ends:

• Both ends are nodes
• All harmonics are possible (both odd and even)
• The fundamental frequency is v/2L
• The harmonic series follows: f, 2f, 3f, 4f, etc.

A slit can be used to differentiate between transverse and longitudinal waves:

For transverse waves:

• The wave transmission depends on the orientation of the slit relative to the oscillation direction
• Rotating the slit will produce varying degrees of transmission
• When the slit is perpendicular to the oscillation direction, the wave is completely blocked

For longitudinal waves:

• The wave passes through the slit regardless of the slit's orientation
• Rotating the slit has no effect on transmission
• The particles oscillate in the same direction as wave propagation

Practical test: If rotating a slit affects the transmitted intensity of a wave, the wave must be transverse. If rotation has no effect, the wave is likely longitudinal.

Example: Light can be blocked by crossed polarizers, confirming it's a transverse wave. Sound passes through slits regardless of orientation, confirming it's longitudinal.

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.