4.2 Wave equation and vibrating strings

The wave equation is a powerful tool for understanding vibrations in strings and other physical systems. It connects displacement, position, and time, allowing us to model how waves travel and interact.

By exploring solutions like D'Alembert's and examining standing waves, we can describe complex vibrations. This helps us understand musical instruments, sound propagation, and other wave phenomena in the real world.

Wave Equation and Solutions

Derivation and Form of the Wave Equation

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• Derived from Newton's second law and Hooke's law for an elastic string
• Relates the displacement $u(x,t)$ of a string to its position $x$ and time $t$
• Takes the form $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$, where $c$ is the wave speed
• Wave speed $c$ depends on the string's tension $T$ and linear mass density $\mu$ as $c = \sqrt{\frac{T}{\mu}}$

D'Alembert's Solution and Initial/Boundary Conditions

• D'Alembert's solution expresses the wave as a sum of two traveling waves: $u(x,t) = f(x-ct) + g(x+ct)$
  • $f(x-ct)$ represents a wave traveling to the right with speed $c$
  • $g(x+ct)$ represents a wave traveling to the left with speed $c$
• Initial conditions specify the string's displacement $u(x,0)$ and velocity $\frac{\partial u}{\partial t}(x,0)$ at time $t=0$
• Boundary conditions describe the string's behavior at its endpoints $(x=0$ and $x=L)$
  • Fixed ends: $u(0,t) = u(L,t) = 0$ for all $t$
  • Free ends: $\frac{\partial u}{\partial x}(0,t) = \frac{\partial u}{\partial x}(L,t) = 0$ for all $t$

Types of Waves

Standing Waves

• Occur when two identical waves travel in opposite directions and interfere
• Resulting wave appears to be standing still, with nodes (points of no displacement) and antinodes (points of maximum displacement)
• Displacement of a standing wave can be described by $u(x,t) = A \sin(kx) \cos(\omega t)$, where $A$ is the amplitude, $k$ is the wavenumber, and $\omega$ is the angular frequency
• Examples include vibrating strings (guitar, violin) and air columns (organ pipes, flutes)

Traveling Waves

• Waves that propagate through a medium without changing shape
• Can be described by the general form $u(x,t) = f(x \pm ct)$, where the sign depends on the direction of travel
• Energy is transported along with the wave
• Examples include sound waves, light waves, and waves on a string (before reflections)

Harmonic Oscillations

• Periodic motion where the restoring force is directly proportional to the displacement
• Displacement follows a sinusoidal pattern: $x(t) = A \cos(\omega t + \phi)$, where $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase
• Can be modeled using the simple harmonic oscillator equation: $\frac{d^2x}{dt^2} + \omega^2 x = 0$
• Examples include mass-spring systems, pendulums, and LC circuits

Eigenfrequencies and Mode Shapes

Eigenfrequencies

• Natural frequencies at which a system tends to oscillate when disturbed
• For a string of length $L$ with fixed ends, eigenfrequencies are given by $f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}}$, where $n = 1, 2, 3, ...$
• Each eigenfrequency corresponds to a specific mode of vibration (mode shape)
• Determined by the system's physical properties (length, tension, mass density) and boundary conditions

Mode Shapes

• Patterns of displacement associated with each eigenfrequency
• For a string with fixed ends, mode shapes are described by $u_n(x) = A_n \sin(\frac{n \pi x}{L})$, where $n = 1, 2, 3, ...$
• The mode number $n$ determines the number of nodes and antinodes
  • $n = 1$: fundamental mode (one antinode)
  • $n = 2$: first overtone (two antinodes)
  • $n = 3$: second overtone (three antinodes)
• Superposition of mode shapes can describe any arbitrary vibration of the string