19. Waves

Wave Fundamentals

The relationship between frequency (f), wave speed (v), and wavelength (λ) is given by the equation:
- f = v / λ
For a string of length L clamped at both ends, the wavelength is given by:
- λ = 4L; thus, f = v / (4L)
- This means that the fundamental frequency (first mode of vibration) is inversely proportional to the length of the string, indicating shorter strings vibrate at higher frequencies.

For a string clamped at both ends:
- The allowed frequencies are integer multiples of the fundamental frequency (f₁):
  - fₙ = n(f₁), where n = 1, 2, 3, ...
For a tube closed at one end and open at the other:
- The allowed frequencies are given by odd integer multiples of the fundamental frequency:
  - fₙ = (2n - 1)(f₁), where n = 1, 2, 3, ...
- This restriction results from the boundary conditions imposed by the closed end (node) and the open end (antinode).

The wave function describes the displacement of the medium at point x and time t:
- Displacement of the string at point x:
  - ⍵ = ψ(x, t)
The wave equation is represented as:
- d²ψ/dx² = (1/v²)(d²ψ/dt²)
The wave speed (v) can be determined using the formula:
- v² = T / μ, where:
  - T = tension in the string;
  - μ = mass per unit length of the string.
- This equation highlights that wave speed depends on the medium's properties.

The general form of the solution to the wave equation can be expressed as:
- ψ(x, t) = A cos(kx - ωt)
Where:
- ω = angular frequency; k = wave number.
The relationship between angular frequency and wave speed is:
- ω = kv
Amplitude A can theoretically take any value, but practical limitations exist in real waves, preventing infinite large displacements.

The velocity of a wave is related to wavelength (λ) and period (T) by:
- v = λ / T, indicating the wave speed can also be determined by the time it takes to complete one cycle.
In three-dimensional waves (like sound waves), intensity is defined as:
- Intensity = Power / Area;
- I = P / (4πr²) for a spherical sound wave source, where P is the power and r is the distance from the source.

The loudness of sound can be expressed in decibels (β) using the formula:
- β = 10 log₁₀(I / I₀)
Where I₀ is the reference intensity, typically set at 10⁻¹² W/m². This logarithmic scale means a small increase in the decibel level is associated with a large increase in intensity, making it a useful scale for representing sound pressure levels.

The Doppler Effect describes the change in frequency (f') of a wave in relation to an observer moving relative to the wave source:
- Source moving towards observer:
  - f' = f / (1 - (u/v))
- Observer moving towards source:
  - f' = f(1 + (u₀/v))
Here, u is the speed of the source, and u₀ is the speed of the observer, demonstrating how relative motion affects perceived frequencies.

Interference occurs when two or more waves overlap, leading to:
- Constructive interference:
  - Occurs when the path difference = mλ, resulting in an increase in amplitude.
- Destructive interference:
  - Occurs when the path difference = (m + 1/2)λ, resulting in a decrease in amplitude.
According to the superposition principle, the total displacement at any point is the sum of the individual displacements of the combined waves.

When a wave reflects off a fixed end:
- It inverts upon reflection while preserving the same amplitude. This principle is crucial for understanding standing waves formed in fixed mediums.

For tubes, the relationship between length and wavelength varies with their ends:
- Tube closed at one end:
  - Length (L) is related to the wavelength by L = λ/4 for the fundamental frequency (first harmonic).
- Tube open at both ends:
  - There is no node at the ends, resulting in L = λ/2, allowing all frequencies as integer multiples of the fundamental.