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

Average power transmitted: $P_{av} = \frac{1}{2}\omega^2A^2\frac{F}{v} = 2\pi^2\mu\nu^2A^2v$

Derivation steps:

1. Power = Force × Velocity
2. Force component = $F\frac{\partial y}{\partial x}$
3. Particle velocity = $\frac{\partial y}{\partial t}$
4. Instantaneous power = $-F\frac{\partial y}{\partial x}\frac{\partial y}{\partial t}$
5. Average over one cycle gives $P_{av}$

Dependency relationships:

• Amplitude (A): $P_{av} \propto A^2$ (quadratic relationship)
• Frequency (ν): $P_{av} \propto \nu^2$ (quadratic relationship)
• Tension (F): $P_{av} \propto \sqrt{F}$ (through its effect on v)
• Linear density (μ): $P_{av} \propto \sqrt{\mu}$ (through its effect on v)

Example: Doubling amplitude increases power by factor of 4, while doubling frequency increases power by factor of 4.

Mathematical representation of standing waves: $y = 2A\sin(kx)\cos(\omega t)$

For a string fixed at both ends (length $L$) in the $n$th normal mode:

• Wave equation: $y = 2A\sin\left(\frac{n\pi x}{L}\right)\cos(\omega t)$
• Nodes occur at positions where $\sin\left(\frac{n\pi x}{L}\right) = 0$
  • $x = 0, \frac{L}{n}, \frac{2L}{n},...,L$
  • Total of $n+1$ nodes (including endpoints)
• Antinodes occur at positions where $\sin\left(\frac{n\pi x}{L}\right) = \pm 1$
  • $x = \frac{L}{2n}, \frac{3L}{2n},...,\frac{(2n-1)L}{2n}$
  • Total of $n$ antinodes
• Distance between successive nodes or successive antinodes: $\frac{L}{n} = \frac{\lambda}{2}$

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.

General rule for wave reflection at boundaries:

• If a wave enters a region where the wave velocity is smaller, the reflected wave is inverted
• If a wave enters a region where the wave velocity is larger, the reflected wave is not inverted
• The transmitted wave is never inverted regardless of the direction of travel
• For strings: wave velocity depends on tension (F) and linear mass density (μ) as v = √(F/μ)

This principle applies across wave phenomena, including mechanical waves on strings, sound waves at interfaces between different media, and electromagnetic waves at boundaries between different materials.

Allowed frequencies:

• A string fixed at both ends can only support standing waves where both ends are nodes
• This boundary condition requires: L = n(λ/2), where n = 1, 2, 3, ...
• The allowed frequencies are therefore: fn = (n·v)/(2L) = n·f₁

Fundamental frequency (n=1):

• f₁ = v/(2L) = (1/2L)√(T/μ)
• Where T is tension and μ is linear mass density
• Represents the lowest frequency at which the string can form a standing wave
• Has only one antinode located at the center of the string

Harmonics:

• For a string fixed at both ends, all harmonics are present (no missing frequencies)
• Second harmonic (n=2): f₂ = 2f₁ (first overtone) - has 2 antinodes, 3 nodes
• Third harmonic (n=3): f₃ = 3f₁ (second overtone) - has 3 antinodes, 4 nodes
• Each harmonic adds one additional node and antinode to the pattern

Physical significance:

• The harmonic series produces the characteristic musical quality of stringed instruments
• Higher tensions increase all frequencies proportionally (√T)
• Shorter strings produce higher frequencies (1/L)

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

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 sonometer is an apparatus consisting of:

• A wooden resonance box
• Fixed bridges at the ends (A and B)
• An auxiliary wire (C) fixed between bridges
• An experimental wire (D) with one end fixed and the other supporting weights
• Movable bridges that can adjust the vibrating length

Used to verify three laws of transverse vibration:

1. Law of length: Frequency ∝ 1/L (tested by moving bridges to change length)
2. Law of tension: Frequency ∝ √F (tested by changing weights on the hanger)
3. Law of mass: Frequency ∝ 1/√μ (tested by changing the experimental wire)

Resonance is detected using paper riders that jump off the wire when standing waves form.

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.